# implied volatility

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Python for Finance by Yuxing Yan

Finance Inputting data from a text file Inputting data from an Excel file Inputting data from a CSV file Retrieving data from a web page Inputting data from a MATLAB dataset Several important functionalities Using pd.Series() to generate one-dimensional time series Using date variables Using the DataFrame Return estimation Converting daily returns to monthly returns Converting daily returns to annual returns Merging datasets by date Forming an n-stock portfolio T-test and F-test Tests of equal means and equal variances Testing the January effect [v] 156 157 158 159 160 161 163 163 163 164 167 168 169 169 170 171 171 173 174 176 176 177 178 179 180 180 181 182 182 183 183 185 187 190 191 192 193 194 195 Table of Contents Many useful applications 52-week high and low trading strategy Roll's model to estimate spread (1984) Amihud's model for illiquidity (2002) Pastor and Stambaugh (2003) liquidity measure Fama-French three-factor model Fama-MacBeth regression Estimating rolling beta Understanding VaR Constructing an efficient frontier Estimating a variance-covariance matrix Optimization – minimization Constructing an optimal portfolio Constructing an efficient frontier with n stocks Understanding the interpolation technique Outputting data to external files Outputting data to a text file Saving our data to a binary file Reading data from a binary file Python for high-frequency data Spread estimated based on high-frequency data More on using Spyder A useful dataset Summary Exercise Chapter 9: The Black-Scholes-Merton Option Model Payoff and profit/loss functions for the call and put options European versus American options Cash flows, types of options, a right, and an obligation Normal distribution, standard normal distribution, and cumulative standard normal distribution The Black-Scholes-Merton option model on non-dividend paying stocks The p4f module for options European options with known dividends Various trading strategies Covered call – long a stock and short a call Straddle – buy a call and a put with the same exercise prices A calendar spread [ vi ] 196 196 197 198 199 204 206 207 210 211 212 214 215 217 220 221 221 222 222 222 227 228 230 232 232 237 238 242 243 243 247 248 250 251 252 253 254 Table of Contents Butterfly with calls Relationship between input values and option values Greek letters for options The put-call parity and its graphical representation Binomial tree (the CRR method) and its graphical representation The binomial tree method for European options The binomial tree method for American options Hedging strategies Summary Exercises Chapter 10: Python Loops and Implied Volatility Definition of an implied volatility Understanding a for loop Estimating the implied volatility by using a for loop Implied volatility function based on a European call Implied volatility based on a put option model The enumerate() function Estimation of IRR via a for loop Estimation of multiple IRRs Understanding a while loop Using keyboard commands to stop an infinitive loop Estimating implied volatility by using a while loop Nested (multiple) for loops Estimating implied volatility by using an American call Measuring efficiency by time spent in finishing a program The mechanism of a binary search Sequential versus random access Looping through an array/DataFrame Assignment through a for loop Looping through a dictionary Retrieving option data from CBOE Retrieving option data from Yahoo!

Fortunately, this is true since the value of an option price is an increasing function of the volatility. In particular, we will cover the following topics: • What is an implied volatility? • Logic behind the estimation of an implied volatility • Understanding the for loop, while loop, and their applications • Nested (multiple) loops • The estimation of multiple IRRs • The mechanism of a binary search • The estimation of an implied volatility based on an American call • The enumerate() function • Retrieving option data from Yahoo! Finance and from Chicago Board Options Exchange (CBOE) • A graphical presentation of put-call ratios Python Loops and Implied Volatility Definition of an implied volatility From the previous chapter, we know that for a set of input variables—S (the present stock price), X (the exercise price), T (the maturity date in years), r (the continuously compounded risk-free rate), and sigma (the volatility of the stock, that is, the annualized standard deviation of its returns)—we could estimate the price of a call option based on the Black-Scholes-Merton option model.

Thus, our expected implied volatility is 0.25. The logic of this program is that we use the trial-and-error method to feed our Black-Scholes-Merton option model with many different sigmas (volatilities). For a given sigma (volatility), when the difference between our estimated call price and the given call price is less than 0.01, we stop. That sigma (volatility) will be our implied volatility. The output from the earlier program is shown as follows: (49, 0.25, -0.0040060797372882817) >>> The first number, 49, is the value of the variable i, and 0.25 is the implied volatility. The last value is the difference between our estimated call value and the given call price of \$3.30. Implied volatility function based on a European call Ultimately, we could write a function to estimate the implied volatility based on a European call.

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Frequently Asked Questions in Quantitative Finance by Paul Wilmott

Although we have the Black-Scholes formula for option values as a function of volatility, there is no formula for the implied volatility as a function of option value, it must be calculated using some bisection, Newton- Raphson, or other numerical technique for finding zeros of a function. Now plot these implied volatilities against strike, one curve per expiration. That is the implied volatility smile. If you plot implied volatility against both strike and expiration, as a three-dimensional plot, that is the implied volatility surface. Often you will find that the smile is quite flat for long-dated options, but getting steeper for short-dated options. Since the Black-Scholes formulæ assume constant volatility (or with a minor change, time-dependent volatility) you might expect a flat implied volatility plot. This appears not to be the case from real option-price data.

This is a simple and popular model, but it does not capture the dynamics of implied volatility very well. Stochastic volatility: Since volatility is difficult to measure, and seems to be forever changing, it is natural to model it as stochastic. The most popular model of this type is due to Heston. Such models often have several parameters which can either be chosen to fit historical data or, more commonly, chosen so that theoretical prices calibrate to the market. Stochastic volatility models are better at capturing the dynamics of traded option prices better than deterministic models. However, different markets behave differently. Part of this is because of the way traders look at option prices. Equity traders look at implied volatility versus strike, FX traders look at implied volatility versus delta. It is therefore natural for implied volatility curves to behave differently in these two markets.

Journal of Finance 69 771-818 Wilmott, P 2006 Paul Wilmott On Quantitative Finance, second edition. John Wiley & Sons What is the Volatility Smile? Short Answer Volatility smile is the phrase used to describe how the implied volatilities of options vary with their strikes. A smile means that out-of-the-money puts and out-of-the-money calls both have higher implied volatilities than at-the-money options. Other shapes are possible as well. A slope in the curve is called a skew. So a negative skew would be a download sloping graph of implied volatility versus strike. Example Figure 2-9: The volatility ‘smile’ for one-month SP500 options, February 2004. Long Answer Let us begin with how to calculate the implied volatilities. Start with the prices of traded vanilla options, usually the mid price between bid and offer, and all other parameters needed in the Black-Scholes formulæ, such as strikes, expirations, interest rates, dividends, except for volatilities.

A Primer for the Mathematics of Financial Engineering by Dan Stefanica

It is interesting to note that computing the implied volatility is a straightforward way of showing that the lognormal assumption, and the BlackScholes formulas derived based on this assumption, are not correct. At any point in time, several options with different strikes and maturities may be traded. If the lognormal assumption were true, then the implied volatilities corresponding to all these options should be equal. However, this does not happen. Usually, the implied volatility of either deep out of the money or deep in the money options is higher than the implied volatility of at the money options. This phenomenon is called the volatility smile. Another possible pattern for implied volatility is the volatility skew, when, e.g., the implied volatility of deep in the money options is smaller than the implied volatility of at the money options, which in turn is smaller than the implied volatility of deep out of the money options.

Assume that the dividend yield q and the constant interest rate r can be estimated from market data. The implied volatility is the unique value of the volatility parameter 0' in the lognormal model that makes the Black-Scholes value of the option equal to the price the option traded at. If we look at (8.64) and (8.65) as functions of only one variable, 0', finding the implied volatility requires solving the nonlinear problem f(x) = 0, (8.66) 268 CHAPTER 8. LAGRANGE MULTIPLIERS. NEWTON'S METHOD. 8.6. IMPLIED VOLATILITY where x = a and 269 Table 8.7: Pseudocode for computing implied volatility (8.67) for the call option, and f(x) = Ke- rT N(-d2 (x)) - Se-qTN(-d1(x)) - P, (8.68) for the put option. Here, In d1 = (~) + (r - q + ~) T xVT ~ d ; 2 In (~) + (r ~ q - ~) T ~ xVT The value of a thus computed is the implied volatility corresponding to a given option price.

d2 = o--VT - t, we find that ad1 ar ad2 ar' and therefore, p(C) = K(T - t)e-r(T-t) N(d2), which is the same as formula (3.74). Implied volatility (3.89) The implied volatility can also be derived from the given price P of a put option, by solving PBS(S, K, T, o-imp, r, q) = P, (3.90) where PBS(S, K, T, 0-, r, q) is the Black-Scholes value of a put option. Note that, as functions of volatility, the Black-Scholes values of both call and put options are strictly increasing since 1 dt Se- qT VT - - e - T V2if > O' ' (3.91) cf. (3.70) and (3.71). (Throughout this section, to keep notation simple, we assume that the present time is t = 0.) Therefore, if a solution o-imp for (3.89) exists, it will be unique, and the implied volatility will be well defined. Similarly, equation (3.90) has at most one solution. For the implied volatility to exist and be nonnegative, the given value C of the call option must be arbitrage-free, i.e., (3.92) The bounds for the call option price from (3.92) can be obtained by using the Law of One Price; cf.

The Rise of Carry: The Dangerous Consequences of Volatility Suppression and the New Financial Order of Decaying Growth and Recurring Crisis by Tim Lee, Jamie Lee, Kevin Coldiron

The liquidity price for an asset can be expressed through a chart of volatility curves. For realized volatility the curve is of realized volatilities as measured over different time horizons; for liquidity to have a positive price, shorter-term realized volatilities need to exceed longer-term realized volatilities. For implied volatility the curve is of implied volatilities at different forward points; for liquidity to have a positive price, further forward implied volatilities need to exceed nearer forward implied volatilities—and all implied volatilities must exceed all realized volatilities. The slope of realized volatility means that the asset displays mean-reverting behavior. While this mean reversion implies meaningful predictability of price movements, the perspective taken here is that this predictability is not an inefficiency.

Last month’s volatility, most of the time, is a pretty good guess at the distribution of tomorrow’s price change, even though it might not be useful for thinking about daily price changes in a year’s time. Therefore implied volatility, the market’s expectation for future realized volatility, tends to move in line with realized volatility—albeit that implied volatility is normally higher than realized volatility by some margin. In modern financial markets implied volatility can be sold directly. The simplest and most popular way to do this is through VIX futures or through exchange-traded notes (ETNs) that correspond to simple VIX futures strategies. The VIX is an index representing the implied volatility for the S&P 500 over the next 30 days, derived from the prices of options on the S&P 500 index.2 VIX futures are monthly-expiring contracts that settle at the VIX.3 2.

This chart depicts three critical features of volatility. First, the top line shows that implied volatilities are greater further forward. This seems intuitive—it expresses the requirement that shorting VIX futures be profitable at all points along the VIX term structure. (The horizontal axis for implied volatility is at the top of the chart and goes from spot VIX out to VIX futures five months forward.) Realized 28 VIX 1 Month 2 Months Implied 3 Months 4 Months 5 Months 26 24 VIX Roll Down 22 Direction of Volatility Premiums 20 18 16 Momentum “Anomaly” Implied-Realized/ Bid-Ask Spread Short-Term Mean Reversion 14 Long-Term Value 12 10 Instantaneous 1 Day 1 Week 1 Month 1 Year 7 Years FIGURE 9.4 Equilibrium volatility premium term structure Chart shows implied volatility spot and forward on the upper gray line, read off the top horizontal axis, and realized volatility across measuring horizons on the lower black line, read off the bottom horizontal axis.

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Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues by Alain Ruttiens

Contrary to yield curves, for which the curve is said to be “normal” when going upwards (cf. Chapter 2, Section 2.1), a “normal” implied volatility curve is going down with higher maturities. This can be explained by a long-term mean reversion (cf. Section 12.2) effect: the longer the maturity, the lower the volatility due to the mean reversion feature. Furthermore, shorter maturity volatilities are more volatile than longer maturity ones. So we may speak of an “implied volatility cone”, involving various observed implied volatility curves for a given underlying, showing that the range of possible implied volatilities is usually broader for shorter maturities, as in Figure 12.5. Figure 12.4 Example of an implied volatility curve Figure 12.5 Typical shapes of an implied volatility curve Similarly as a yield curve allows computing forward rates (cf.

Figure 12.3 Example of autocorrelations calculation (The results in column “lag 1” are obtained by computing the correlation between a data series and the same series, lagged by 1 week; similarly for the “lag 2” column, lagging the data by 2 weeks, etc.). 12.1.2 Volatility curve With respect to the Black–Scholes formula for option pricing, and related pricing models, the implied volatility to be used is a constant, whatever the option maturity is. Practically speaking, the market is using different implied volatilities for different maturities: in other words, the volatility estimate (by the option market maker) is not necessarily the same for the next 3 months as for the next 3 years, for example. Hence, the use of implied volatility curves (or “volatility structure”), just as yield curves (or term structure). For example, in Figure 12.4 is the implied volatility curve of options (of ATM and near to ATM strikes) on the S&P 500, as of 05/03/2011. These implied volatility curves are changing over time, just as with yield curves. Contrary to yield curves, for which the curve is said to be “normal” when going upwards (cf.

Figure 12.7 Kurtosis feature Figure 12.8 Implied volatility in function of the delta of the option Figure 12.9 A volatility smirk, or sneer Finally, in practice, both features may well coincide, leading to patterns such as in Figure 12.10, showing the implied volatilities of calls on the S&P 500 maturing in 1 week, 1 month + 1 week, 2 months + 1 week and 3 months + 1 week (data for 04/22/11, source: Bloomberg). Figure 12.10 Implied volatilities of calls on the S&P 500 We could wonder about the problem – for an options market maker – of determining an adequate implied volatility level, if such phenomena have to be taken into account. But let us not forget that, after all, to anticipate a future volatility level is in any case some delicate job: with or without smiles and the like, it will always involve some arbitrary dimension, almost impossible to reasonably model. 12.1.4 Implied volatility surface In the previous section, we have considered different implied volatilities in function of the option delta.

The Concepts and Practice of Mathematical Finance by Mark S. Joshi

We have So = 100, and r = 5%, and QC(s)= 20% fort < 1/2, 15% 10% forl/2<t < 1, forl <t. Find the implied volatility of a call option struck at 110 with the following maturities: 0.5, 1, 1.5, 2. Exercise 6.11 A stock, St, follows geometric Brownian with time-dependent volatility. We have So = 100, and r = 5%, and 10% 0-j(S) = 15% 20% fort < 1/2, for 1/2 < t < 1, for 1 < t. Find the implied volatility of a call option struck at 110 with the following maturities: 0.5, 1, 1.5, 2. Exercise 6.12 A stock, St, follows geometric Brownian with time-dependent volatility. We have So =100, and r = 0%. Call options struck at 100 with maturities 0.5, 1 and 2 have implied volatilities of 10%, 15% and 20%. Find a piecewise constant volatility function that is consistent with these implied volatilities. Exercise 6.13 A stock follows geometric Brownian motion with drift a and volatility o-, and there is a riskless bond with growth rate r.

As the Gamma is non-negative the value is convex as a function of spot. 15.11 Key points 385 15.10.3 Floating smiles One nice consequence of the homogeneity of call prices is that the implied volatility smile floats. Thus if strike is K and spot is S then the implied volatility function, a (S, K), that is, the implied volatility of a call option struck at K given that spot is S satisfies 6(S,K)=g( ), (15.25) for some function g. To see this observe that if C(S, K, T) = BS(S, K, or, T), then it also true for any A > 0 that C(AS, AK, T) = BS(AS, a,K, a, T), as the ), passes through everything. This means that 6(),S, AK, T) is independent of A, which is equivalent to saying that it is a function of K/S. We call K/S the moneyness as it expresses how much the option is in or out of the money as a ratio. The implied volatility is only a function of moneyness i.e. the smile will always look the same qualitatively.

Price options on 1-, 5- and 10-year swaps starting in 1, 5, and 10 years with a variety of strikes. Plot the implied volatility smile of the swaptions in each case. (See Project 12 for some discussion of how to implement an implied volatility function.) What can we conclude about log-normality? B.14 Project 12: Jump-diffusion models In this project, we investigate how to implement a pricer for a jump-diffusion model, see what sort of smiles are implied and look at pricing variations for exotic options. Vanilla options Implement a pricer for vanilla options for a jump-diffusion model with log-normal jumps. Implement a Monte Carlo pricer also and check they give the same answers. Implied volatility Implement an implied volatility function - this is a function which inverts the Black-Scholes price function to get the unique volatility which gives the correct price for the option.

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My Life as a Quant: Reflections on Physics and Finance by Emanuel Derman

Before 1987, in contrast, more light-heartedly naive options markets were happy to charge about the same implied volatility for all strikes, as illustrated by the dashed line in Figure 14.1. Figure 14.1 A typical implied volatility smile for three-month options on the Nikkei index in late 1994. The dashed line shows the lack of skew that was common prior to the 1987 crash. It was not only three-month implied volatilities that were skewed. A similar effect was visible for options of any expiration, so that implied volatility varied not only with strike but also with expiration. We began to plot this double variation of implied volatility in both the time and strike dimension as a two-dimensional implied volatility surface. A picture of the surface for options on the Standard & Poor's (S&P) 500 index is illustrated in Figure 14.2.

Even today, when no one believes that the Black-Scholes model is absolutely the best way to estimate option value, and even though more sophisticated traders sometimes use more complex models, the Black-Scholes model's implied volatilities are still the market convention for quoting prices. Options are generally less liquid than stocks, and implied volatility market data is consequently coarse and approximate. Nevertheless, Dave pointed out to me what I was already dimly aware of. There was a severe skew in the implied volatilities, so that three-month options of low strike had much greater implied volatilities than three-month options of higher strikes.You can see a sketch of this asymmetry in Figure 14.1. This lopsided shape, though it's commonly called "the smile," is more of a smirk. With implied volatility as your measure of value, low-strike puts are the most expensive Nikkei options. Anyone who was around on October 19, 1987 could easily guess why.

In the options world as well, price alone is an insufficient measure of value; it's impossible to tell whether Y300 for an at-the-money put is more attractive than Y40 for a deep out-of-the-money put. A better measure of value is the option's implied volatility. The Black-Scholes model views a stock option as a kind of bet on the future volatility of a stock's returns. The more volatile the stock, the more likely the bet will pay off, and therefore the more you should pay for it.You can use the model to convert an option price into the future volatility the stock must have in order for the option price to be fair. This measure is called the option's implied volatility. It is, so to speak, an option's view of the stock's future volatility. The Black-Scholes model was the market standard. When I sat next to Dave in Tokyo that day, his computer screen showed the prices quoted in Black-Scholes implied volatilities. Even today, when no one believes that the Black-Scholes model is absolutely the best way to estimate option value, and even though more sophisticated traders sometimes use more complex models, the Black-Scholes model's implied volatilities are still the market convention for quoting prices.

Mathematical Finance: Theory, Modeling, Implementation by Christian Fries

.: Linear interpolation for decreasing implied volatility. Conclusion: An arbitrarily fast decrease of implied volatility is not possible. 6.2.2.2. Lineare Interpolation for increasing Implied Volatilities For the example of increasing implied volatility we consider the prices i 2 3 Strike Ki 0.75 1.25 Price V(Ki ) 0.2897 0.2532 Implied Volatility σ(Ki ) 0.2 0.8 Figure 6.5 shows the linear increasing interpolation of the implied volatilities. Interpolated prices Interpolated volatlities Probability density 0,50 1,5E-2 volatility price 0,30 0,20 0,10 density 0,75 0,40 0,50 0,25 1,0E-2 5,0E-3 0,0E0 0,00 0,75 1,00 1,25 0,75 strike 1,00 strike 1,25 0,75 1,00 1,25 underlying value Figure 6.6.: Linear interpolation for increasing implied volatility. At first sight the density implied by the linear interpolation of the implied volatilities exhibits no flaw (positive, no point measure).

.: Spline interpolation of option prices (upper row) and implied volatilities (lower row). 6.2.2. Example (2): Interpolation of two Prices The example from Section 6.2.1 of a linear interpolation of implied volatilities may suggest that the problem arises at the joins of the linear interpolation, i.e. the behavior at K2 = 0.75 and K3 = 1.25. One could hope that a local smoothing would solve this problem. Instead we will give an example that a linear grow of implied volatility may be inadmissible alone, although the interpolated prices are admissible. We consider two prices V(K2 ) > V(K3 ). The monotony ensures that these two prices alone do not allow for an arbitrage. 6.2.2.1. Lineare Interpolation for decreasing Implied Volatilities Given are the prices i 2 3 Strike Ki 0.75 1.25 Price V(Ki ) 0.4599 0.0018 Implied Volatility σ(Ki ) 0.9 0.1 The implied volatility decreases with the strike K.

The linear interpolation of implied (Black-Scholes) volatilities may lead Interpolated prices Interpolated volatlities Probability density 2,5E-2 0,50 0,30 0,20 0,60 density volatility price 0,40 0,50 0,40 0,0E0 -2,5E-2 0,10 0,50 1,00 1,50 2,00 0,50 1,00 1,50 2,00 0,50 1,00 1,50 2,00 strike strike underlying value Figure 6.3.: Linear interpolation of implied volatilities. to prices allowing for arbitrage: At the edges K = 0.75 and K = 1.25 we have a point measure, the measure for S (T ) = 1.25 is negative. Thus the implied measure is not a probability measure, see Figure 6.2, right. From Lemma 72 we have: There is no arbitrage free pricing model that generates this interpolated price curve. On the other hand, linear interpolation of implied volatilities also has a pleasant property: If the given prices correspond to prices from a Black-Scholes model, i.e. if the implied volatilities are constant, then, trivially, the interpolated prices correspond to the same Black-Scholes model. 6.2.1.3. Spline Interpolation of Prices respective Implied Volatilities To remove the disadvantage of degenerated densities (i.e. the formation of point measures) as it appears for a linear interpolation, we move to a smooth (differentiable) interpolation method, e.g. spline interpolation.

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Market Risk Analysis, Quantitative Methods in Finance by Carol Alexander

If n1 is the number of days to expiry of the prompt futures contract and n2 is the number of days to expiry of the next futures contract, then the linearly interpolated 30-day futures price on that day is n2 − 30 P1 + 30 − n1 P2 n2 − n1 For instance, if the prompt futures price is 10 and it has 5 days until expiry, and the next futures price is 12 and it has 36 days to expiry, then the constant maturity futures price is 6 × 10 + 25 × 12 = 11 613 31 Similarly, linear interpolation can be applied to construct constant maturity implied volatility series from the implied volatilities of options of different maturities. The next example illustrates this application. Example I.5.3: Interpolating implied volatility Suppose we have two options with the same strike but different maturities: option 1 has maturity 10 days and option 2 has maturity 40 days. If the implied volatility of option 1 is 15% and the implied volatility of option 2 is 10%, what is the linearly interpolated implied volatility of an option with the same strike as options 1 and 2 but with maturity 30 days? Solution Under the assumption that log returns are i.i.d. it is variances that are additive, not volatilities.

But the regression R2 is only 0.1995 and there is a high degree of multicollinearity between the equity index return and the change in implied volatility. In fact in our sample their correlation is −0829, and the square of this is 0.687, which far exceeds the regression R2 . To remedy the multicollinearity problem we can simply drop one of the collinear variables from the regression, using either the equity return or the change in implied volatility in the model, but not both. If we drop the implied volatility from the regression the estimated model becomes ŝ = 002618 − 21181 r − 05312 R 00610 −10047 −139069 The equity return alone is actually a more significant determinant of changes in credit spread than the model (I.4.57) would indicate. And if we drop the equity return from the regression, the estimated model becomes ŝ = −00265 − 21643 r + 05814 −06077 −10086 126154 Hence, the equity implied volatility alone is a very significant determinant of the credit spread.

To give just a few common examples:2 • • • • • The Black–Scholes–Merton (Black and Scholes, 1973; Merton, 1973) model gives an analytic solution for the price of a standard European option under certain (rather unrealistic) assumptions about the behaviour of asset prices. However, it is not possible to invert the Black–Scholes–Merton formula so that we obtain an analytic solution for the implied volatility of the option. In other words, the implied volatility is an implicit function, not an explicit function of the option price (and the other variables that go into the Black–Scholes–Merton formula such as the strike and the maturity of the option). So we use a numerical method to find the implied volatility of an option. The allocations to risky assets that give portfolios with the minimum possible risk (as measured by the portfolio volatility) can only be determined analytically when there are no specific constraints on the allocations such as ‘no more than 5% of the capital should be allocated to US bonds’.

Mathematical Finance: Core Theory, Problems and Statistical Algorithms by Nikolai Dokuchaev

© 2007 Nikolai Dokuchaev Implied and Historical Volatility 135 If a market is exactly the Black-Scholes market with constant σ, then σimp does not depend on (T, K), and it is equal to this σ (which is also the historical volatility). However, in the real market, σimp depends usually on (T, K) (and on the type of option), and the implied volatility differs from the historical volatility. In this case, we can conclude that the Black-Scholes model does not describe the real market perfectly, and its imperfections can be characterized by the gap between the historical and implied volatilities. Varying K and T gives different patterns for implied volatility. Similarly, the evolving price S(t) gives different patterns for implied volatility for different t for a given K. The most famous pattern is the so-called volatility smile (or volatility skew) that describes dependence of σimp on K. Very often these patterns have the shape of a smile (or sometimes skew).

This investor may try to calculate volatility by solving the equation with respect to σ, where is the market price of the option. (It follows from Lemma 7.3 that VBS(σ) is a strictly increasing function in σ, so this equation is solvable.) The solution σ=σimp of the equation is called implied volatility. Definition 7.5 A value σimp is said to be implied volatility at time t=0 for the call option given K, r, T, if the current market price of the option at time t=0 can be represented as HBS,c(S(0), K, T, σimp, r), where HBS,c(S(0), K, T, σ, r) is the Black-Scholes price for call, where K is the strike price, σ is the volatility, r is the risk-free rate, and T is the terminal time. The definition for the implied volatility for a put option is similar. © 2007 Nikolai Dokuchaev Implied and Historical Volatility 135 If a market is exactly the Black-Scholes market with constant σ, then σimp does not depend on (T, K), and it is equal to this σ (which is also the historical volatility).

Assume also that European call options on these two stocks, with the same strike price K and same expiration time be the T>0, have market prices C(i) at time t=0, i=1, 2. Let C(1)>C(2). Let corresponding implied volatilities, i=1, 2. Indicate which statement is most correct and explain your answer: Problem 7.8 Solve Problem 7.7 assuming put options instead of call options. Problem 7.9 Assume that r= 0.05, S(0)=1, and T=1. Using a code that represents the Black-Scholes price of a call option as a function of volatility, draw a graph that represents the option price with this parameter as a function of the volatility. Further, assume that the price of the call option with strike price K=1 is 0.25. Estimate the implied volatility using the figure. Problem 7.10 Calculate the implied volatility using a code. Assume that the price of a call option with strike price K=100 is 25, and r=0.05, S(0)=100, T=1.

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Expected Returns: An Investor's Guide to Harvesting Market Rewards by Antti Ilmanen

Thus, real-world deviations from BSM assumptions should explain observed smile effects and other apparent violations of no-arbitrage pricing. (The smile refers to the appearance of a plot of implied volatility on the y-axis against strike prices or option moneyness on the x-axis, where moneyness is the degree to which an option is in the money.) The BSM analysis taught investors to infer implied parameters such as implied volatility from market prices, always given a particular model or set of assumptions. In the risk-neutral world, implied volatility reflects the market’s volatility expectations. In the real world, this implied parameter may also reflect some risk premia in addition to volatility expectations. Thus, analogous to the way a term spread reflects the market’s rate expectations plus required bond risk premia, implied volatilities also reflect the market’s volatility expectations plus some volatility-related risk premia.

Figure 19.2 shows that volatility spikes tend to coincide with the worst market meltdowns. Figure 19.3 is a stylized graph that makes three points about option skew (implied volatility levels across strike prices):• Index options have much lower implied volatilities than typical single-stock options, thanks to diversification, but their skew is much more pronounced. • At least this was the case after the 1987 crash. Before the crash, the implied volatilities of index options exhibited a symmetric smile rather than a one-side smirk as both OTM calls and puts had higher volatilities than ATM options. After the crash, realized fluctuations of the index have been reasonably symmetric but implied volatilities disagree: they implicitly forecast a much higher probability of a large downside move than of a large upside move.Figure 19.1.

Any decomposition requires a specified model and empirical estimation, and any resulting estimates are noisy. As a convenient simplification, it is common to interpret excess implied volatility over realized volatility as a volatility risk premium. Empirically, this approximation works well, at least for assessing historical average premia. If we recognize the time-varying nature of the volatility premium, careful estimation of the two components becomes important, preferably using more robust “model-free” approaches. By market convention, implied volatilities are often quoted based on the B-S formula and reflect its underlying assumptions. Unequal B-S implied volatilities across strike prices (smile or smirk) may be interpreted as market recognition of prevalent non-normalities and other deviations from B-S assumptions.

Trading Risk: Enhanced Profitability Through Risk Control by Kenneth L. Grant

How does all of this pertain to the applicability of options implied volatility to our efforts to estimate portfolio risk? As it turns out, because implied volatility is quoted in the exact manner that historical volatility is expressed (i.e., in annualized terms), for the purposes of 88 TRADING RISK estimating price dispersion, the terms can be used interchangeably. In fact, implied volatility can be viewed very explicitly as the options market’s attempt to predict what historical volatility will be in the future. We can therefore substitute options volatilities for historicals in order to size our associated exposure. Perhaps surprisingly, we can do so without ever trading a single option. For example, just as is the case with historical volatility, if we take a \$50,000 bet in a given security with a 10% implied volatility, we would expect the position in question to fluctuate at an annualized, one-standard-deviation rate of \$5,000.

Again, we don’t have to trade options to arrive at this estimate but can simply derive data from the options markets and apply it against our position in cash securities. Options implied volatility thus has the advantage over historical volatility of encompassing not only historical time series information but also “qualitative” data and prospective economic inputs into its estimates of price dispersion. The measure gains further credibility through the fact that options traders are actually risking financial capital on the basis of implied volatility valuations. Take my word for it—this type of reality dose does wonders for the accuracy of financial models. From these perspectives, implied volatility arguably is the superior measure. However, like everything else in our statistical tool kit, it has shortcomings. First, because the implied volatility statistic is derived entirely from the manner in which options are priced, it is subject to the same idiosyncrasies that characterize the options markets themselves.

For example, during periods of extreme market duress, prices and volatilities for all types of options are often bid up to levels beyond what would be justified by the underlying economic data; and the use of implied volatility as an exposure input might lead to inaccuracies in these instances. In addition, there is a well-known concept within the universe of option trading, referred to as the volatility skew, or smile, that describes the tendency for out-of-the-money options to trade at higher implied volatilities than those that are at, near, or in the money. It is therefore necessary to understand the relationship between the strike price and the underlying market price (i.e., the “moneyness”) for the option on whose implied volatility you are relying. As a practical matter, I recommend the volatility for the at-the-money strike as the best approximation of future price dispersion, as this is the part of the options market that is least impacted by market inefficiencies.

Risk Management in Trading by Davis Edwards

If a coin is flipped twice, 162 RISK MANAGEMENT IN TRADING S&P 500 Implied Volatility 7/ 1/ 20 13 7/ 1/ 20 12 7/ 1/ 20 11 7/ 1/ 20 10 7/ 1/ 20 09 7/ 1/ 20 08 7/ 1/ 20 07 7/ 1/ 20 06 7/ 1/ 20 05 7/ 1/ 20 04 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Simple Historical Volatility Crude Oil FIGURE 6.7 01 3 28 6/ /2 28 6/ /2 01 2 01 1 28 6/ /2 28 /2 01 0 9 6/ 28 /2 00 8 Implied Volatility 6/ 28 /2 00 7 6/ /2 28 6/ /2 28 6/ 00 6 00 5 00 /2 28 6/ 6/ 28 /2 00 4 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Simple Historical Volatility Implied Volatility KEY CONCEPT: HOW SENSITIVE DOES VAR NEED TO BE? One of the highly touted advantages of exponential historical volatility, GARCH, and implied volatility compared to simple historical VAR models is that the more complicated models react much more quickly to changes.

The easiest method of calculating historical volatility. Exponentially Weighted Historical Volatility. More recent estimates are weighted more than older estimates. 154 RISK MANAGEMENT IN TRADING Volatility Equally Weighted Historical Volatility Implied Volatility Historical Volatility Exponentially Weighted Historical Volatility GARCH Historical Volatility FIGURE 6.4 ■ ■ Types of Volatility Estimates GARCH Historical Volatility. Short for Generalized Auto‐Regression Conditional Heteroskedacity, GARCH introduces a mean‐reverting term into an exponentially weighted calculation. Implied Volatility. Implied volatility from options markets is used to estimate returns. Equally Weighted Historical Volatility For a single asset, parametric VAR requires estimating a single parameter— the volatility of asset prices. The easiest way to do this is to calculate the historical volatility of the returns, weighting all observations equally.

For example, someone looking to reduce risk might be willing to pay an above market price to reduce some exposure. However, there is only one reason to sell options and take on additional risk—because the price of the option is sufficiently high. An empirical comparison of equal weighted historical volatility to forward implied volatility shows that both methods give similar estimates. Market‐implied volatility generally has more day‐to‐day variation than historically calculated volatility. It also responds (both up and down) more quickly than historical volatility. (See Figure 6.7, Implied Volatility.) More sensitive numbers can be helpful in some cases to give an early warning about possible market volatility. However, it can also cause operational problems that exacerbate risky situations since a major use of VAR is to set position limits and calculate regulatory capital requirements.

Solutions Manual - a Primer for the Mathematics of Financial Engineering, Second Edition by Dan Stefanica

Assume that the risk free interest rate is constant at 6%. (i) Compute the implied volatility with six decimal digits accuracy, using the bisection method on the i口terval [0.0001 , 1]' the secant method with initial guess 0.5 , and Newton's method with initial guess 0.5. 190 CHAPTER 8. LAGRANGE (ii) Let MULTIPLIER丘 NEWTON'S METHOD. σ问 be the implied volatility previously computed method. Use the formula σ usi吨 Newtor内 r-.JV2在 C 一与庄s r-.J百万 imp ,approx to compute an approximate value compute the relative error σimp， apprωfor the implied volatility, and |σzmp， αpprox 一 σ vzmp σimp Solution: (i) Both the secant method with ♂ -1 = 0.6 and Xo = 0.5 and Newton's method with initial guess Xo = 0.5 converge in three iterations to an implied volatility of 39.7048%. The approximate values obtained at each iteration are given below: k Secant :Nlethod Newton's Method 。

(7.21) Solutions to Supplemental Exercises Problem 1: Compute Consider the following change of variables: V(S , I , t) Show that F(y , t) I = F(y , t) , where y = satis且es the + (T - t) InS , 2T2 8 y2 D followi吨 PDE: 主主十 σ2(T - t)2 ~2: + (r 一到丘 t 8F δt where ,- T \2) f儿z 伽 = {何?们}R2 I x ~ 0 , 1 三时 2 ， 1 才三 2} Sol仰on: The change of variables 8 = xy and t rF = 0 7. For the same maturity, options with different strikes are traded simultaneously. The goal of this problem is to compute the rate of change of the implied volatility as a function of the strike of the options. In other words , assume that S , T , q and r are given , and let C (K) be the (known) value of a call option with maturity T and strike K. Assume that options with all strikes K exist. Define the implied volatility σimp(K) as the unique solution to ~ is equivalent to when x 主 o and y ~三 O. This change of variables maps the domain D into the recta吨Ie n = [1 , 2] x [1 , 2]. It is easy to see that 6. One way to see that American calls on non-dividend-paying assets are never optimal to exercise is to note that the Black-Scholes value of the European call is always greater than the intrinsic premium S - K , for S>K.

SOLUTIONS TO SUPPLEMENTAL EXERCISES which is what we wanted to show. 177 己 θVθFInSθF δtθt T θν7 δV 1θF θI T δV 1 T-t θF θu? δS S T θ2V θν? 1 ( (T - t)2 θ2F 一一_. θS2 Problem 7: For the same maturity, options with different strikes are traded simultaneously. The goal of this problem is to compute the rate of change of the implied volatility ωa function of the strike of the options. In other words , assume that S , T , q and r are given , and let C(K) be the (know叫 value of a call option with maturity T and strike K. Assume that options with all strikes K exist. Defi 缸 fine 也 t he implied volatility σ 叽inη1以 K) 创 as the unique solution to C(K) = CBs(K， σimp(K)) ， S2 飞 T-t θF\ 一一一一. T2 θy2 T θy) Then , the PDE (7.27) for V(S , I , t) becomes the followi鸣 PDE for F(y , t): o= θVθ V 一一 + 1 ')~')护 VθV 1口 S 一一 +一 σL. SL. 一一τ +γS一一 ' 2- .- 8S 2 θS rV θ t θI θ FInS θF 一一一一一一一十 θt T θy __1δF lnS~~ -T θσimp(K) θu 十 1σ2 i( (T - t)Z 一一一一一一一一一…一一一 θ2F T-t θFV+T-tθF I + r 一一一一一一一一 2\ T2 δy2 T θF σ2(T - t)2 θ2F , 一一+ n ~ ~十 θt' 2T2 where CBs(K， σ 叽仰nη以 1 阮cl挝 S hoωoles 臼s value of a call 叩 0 pt挝io ∞ n with strike K on an unde 臼rl勾i坊抖 y 甘切 切 rin 吨 1鸣 ga 部ss附 创t ，f'01 e 扣如 low 叭in 吨 ga log 伊 nω10ωr m 口 口 r丑na 1恼al 叫 model 呐 w it由 h volatility σ叫 (K).

pages: 321

Finding Alphas: A Quantitative Approach to Building Trading Strategies by Igor Tulchinsky

Bollen and Whaley and Gârleanu et al. attribute the “shape of observed volatility skew and its predictive ability to the buying pressure due to the information possessed by option traders.” Bollen and Whaley find that “contemporaneous changes in daily implied volatilities are driven by changes in net buying pressure.” Options traders with expectations of positive news create an excess of buy-call trades and/or sell-put trades, which causes 172 Finding Alphas Volatility skew 30 Implied volatility 28 26 24 22 20 18 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250 Strike price Figure 23.4 Sample volatility skew in equity options markets prices and implied volatilities of call options, relative to put options, to rise. Similarly, options traders with expectations of negative news create an excess of sell-call trades and/or buy-put trades, which causes the prices and implied volatilities of put options, relative to call options, to rise. Thus, when options traders expect information about the probability of a negative event, the demand for out-of-the-money put options increases relative to the demand for at-the-money call options, thereby increasing the volatility skew.

OCC does not guarantee the accuracy, adequacy, completeness or availability of information and is not responsible for errors or omissions or for the results obtained from the use of such information. 1 Stock Returns Information from the Stock Options Market171 Average month-end open interest in millions Contracts exercised in millions 750 531.8 556.1 562.2 458.9 449.1 375 279.5 305.1 263.5 275.5 295.1 300 273.1 92.54% 600 225 450 150 300 150 75 0 423.3 94.24% 22.0 10 Equity 11 12 13 14 7.46% 0 10 11 12 13 14 25.8 5.76% Index Futures not depicted in graphs Figure 23.3 Options open interest and contracts exercised 2010–2014 Source: © 2018, The Options Clearing Corporation. Used with permission. All rights reserved. VOLATILITY SKEW A useful source of information on the direction of the options market is the implied volatility of stock options. This is the value for the volatility of the underlying instrument such that, when the value is input in an option pricing model (such as Black–Scholes), the model will return a theoretical value equal to the current market price of the option. In the case of equity options, a plot of the implied volatility against the strike price gives a skewed surface. The volatility skew is the difference in implied volatility between out-of-the-money, at-the-money, and in-themoney options. The volatility skew is affected by sentiment and supply– demand relationships, and provides information on whether fund managers prefer to write calls or puts.

According to the authors, although the information advantage as reflected in the predictive ability of volatility skews is greater for negative news than for positive news, the predictive ability of the options market applies to news surprises of a range of magnitudes. This phenomenon can be used to find stocks to short long–short equity alphas on longer and shorter time scales. Figure 23.5 shows the performance of an alpha on the Russell 1000 universe of stocks. The alpha uses the slope of the implied volatility curve to measure the skew. The idea is to buy stocks that have shown a decrease in the slope of the implied volatility curve (or decrease in volatility skew), and vice versa. 2 Alpha = −(change in slope of the implied volatility curve). 174 Finding Alphas VOLATILITY SPREAD The put-call parity relation states that in perfect markets, the following equality holds for European options on non-dividend-paying stocks: C P S D.K where C and P are the current call and put prices, respectively; D is the discount factor; K is the strike price; and S is the spot price.

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Efficiently Inefficient: How Smart Money Invests and Market Prices Are Determined by Lasse Heje Pedersen

This insight also means that, for any option price Ct, there exists a corresponding volatility σ that justifies the price in the sense that, if we plug this σ into the Black–Scholes–Merton formula, then the formula spits out the right price. This level of stock volatility is called the implied volatility. According to the Black–Scholes–Merton model, the implied volatility of all options on the same underlying stock should be the same, namely the stock’s true volatility. Therefore, option prices can be more easily compared by looking at their implied volatilities. If one option has a higher implied volatility, it is more expensive relative to its fundamental Black–Scholes–Merton value—and a candidate for short-selling. Option arbitrageurs look to short-sell options with implied volatility above their assessed true volatility and buy options with implied volatility below the true volatility. Of course, it must be recognized that the option’s market price can differ from the model-implied fundamental value because of a possible arbitrage opportunity, or because the model is wrong, or because the estimate of the true volatility is wrong, or some combination of these things.

INTEREST-RATE VOLATILITY TRADING AND OTHER FIXED-INCOME ARBITRAGES Some fixed-income traders also trade interest-rate related option instruments. For instance, they trade swaptions, caps, floors, and options on bond futures. This involves both directional volatility trades and relative value trades. A directional volatility trade means comparing a derivative’s implied volatility with the arbitrageur’s own prediction of actual volatility and buying the derivative if the implied volatility is low, while hedging the interest rate risk with bonds, bond futures, or swaps. Conversely, if the implied volatility is high, the arbitrageur will reverse the trade and short the derivative. Fixed-income arbitrage traders also perform relative-value volatility trades, in which they compare the pricing of different derivatives and go long–short based on the relative attractiveness. Finally, fixed-income arbitrageurs pursue a variety of other trades such as municipal bond spreads, emerging market bonds, the bond futures basis relative to the cash market (based on cheapest-to-deliver considerations), structured credit, and break-even inflation trading.8 14.10.

Clearly, the Black–Scholes–Merton model rests on strong assumptions that are not satisfied in the real world. In particular, real stock prices can suddenly jump and the volatility varies over time, features that are not captured by the standard Black–Scholes–Merton model (but can be captured in extensions of the basic model). Such potential jumps in the stock price can explain why implied volatilities tend to be higher for out-of-the-money put options, especially for index options, a tendency called the implied volatility “smirk.” Hence, this smirk is not just an arbitrage opportunity but also a reflection of a real crash risk. As in the binomial model, we can derive the option replicating portfolio in the Black–Scholes–Merton model. If a hedge fund short-sells an option, it will hedge its position by buying Δt shares, where Since Δt is changing over time, the hedge fund must keep adjusting the number of shares held, which is called dynamic hedging.

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Handbook of Modeling High-Frequency Data in Finance by Frederi G. Viens, Maria C. Mariani, Ionut Florescu

APPENDIX B.2: STEP-BY-STEP EXPLANATION OF THE CONSTRUCTION OF VIX USING STOCHASTIC VOLATILITY QUADRINOMIAL TREE METHOD Here we use quadrinomial tree model to compute the price of a synthetic options with exact 30 days maturity using distribution of implied volatility obtained from S&P500 as input. Then by Black and Scholes (1973) formula, we obtain the implied volatility of this synthetic option. We want to study whether or not this implied volatility multiplied with 100 can better reﬂect the market volatility. References 115 There are four steps in the construction of this VIX as follows: • Compute the implied volatilities of entire option chain on SP500 and construct an estimate for the distribution of current market volatility. The implied volatility is calculated by applying Black–Scholes formula. • Use this estimated distribution as input to the quadrinomial tree method. Obtain the price of an at-the-money synthetic option with exactly 30-day maturity. • Compute the implied volatility of the synthetic option based on Black–Scholes formula once the 30-day synthetic option is priced. • Obtain the estimated VIX by multiplying the implied volatility of the synthetic option by 100.

The model intrinsically has an entire distribution describing the volatility and therefore providing a number is nonsensical. However, since a number we need to provide the idea of this approach is to produce the price of a synthetic one-month option with strike exactly the spot price and calculate the implied volatility value corresponding to the price we produce. The real challenge is to come up with a stochastic volatility distribution characteristic of the current market conditions. In the current work, we take the simplest approach possible. We use a proxy for this ϕt calculated directly from the implied volatility values characterizing the option chains. This is used in conjunction with a highly recombining quadrinomial tree method to compute the price of options. The quadrinomial tree method is described in details in Florescu and Viens (2008). In Appendix B, we describe a one-step quadrinomial tree construction. 5.2.2 DIFFERENCE BETWEEN CBOE PROCEDURE AND QUADRINOMIAL TREE METHOD 1.

. • Compute the implied volatility of the synthetic option based on Black–Scholes formula once the 30-day synthetic option is priced. • Obtain the estimated VIX by multiplying the implied volatility of the synthetic option by 100. Please note that the most important step in the estimation is the choice of proxy for the current stochastic volatility distribution. REFERENCES Black F, Scholes M. The valuation of options and corporate liability. J Polit Econ 1973;81:637–654. Bollen N, Whaley R. Does net buying pressure affect the shape of implied volatility functions? J Finance 2004;59(2):711–753. CBOE. The new CBOE volatility index-vix. White papers, CBOE; 2003, http://www. cboe.com/micro/vix/vixwhite.pdf. Demeterﬁ K, Derman E, Kamal M, Zou J. More than you ever wanted to know about volatility swaps. Technical report, Goldman Sachs Quantitative Strategies Research Notes; 1999, http://www.ederman.com/new/docs/gs-volatility_swaps.pdf.

pages: 353 words: 88,376

The Investopedia Guide to Wall Speak: The Terms You Need to Know to Talk Like Cramer, Think Like Soros, and Buy Like Buffett by Jack (edited By) Guinan

During these times, holders of illiquid securities may find themselves unable to unload them at all or unable to do so without losing a lot of money. 133 134 The Investopedia Guide to Wall Speak Related Terms: • Intrinsic Value • Law of Supply • Marketable Securities • Law of Demand • Liquidity Implied Volatility (IV) What Does Implied Volatility Mean? The estimated volatility of the price of a security. Investopedia explains Implied Volatility In general, implied volatility increases when the market is bearish and decreases when the market is bullish. This is due to the common belief that bearish markets are more risky than bullish markets. In addition to known factors such as market price, interest rate, expiration date, and strike price, implied volatility is used in calculating an option’s premium. IV can be derived from a model such as the Black Scholes Model. Implied volatility sometimes is referred to as vols. Related Terms: • Beta • Options • Volatility • Black Scholes Model • Stock Option In the Money What Does In the Money Mean?

VIX is the ticker symbol for the Chicago Board Options Exchange (CBOE) Volatility Index, which numerically expresses the market’s expectation of 30-day volatility; it is constructed by using the implied volatilities of a wide range of S&P 500 Index options. The results are meant to be forward-looking and are calculated by using both call and put options The VIX is a widely used measure of market risk and often is referred to as the investor fear gauge. There are three variations of the volatility indexes: (1) the VIX, which tracks the S&P 500, (2) the VXN, which tracks the Nasdaq 100, and (3) the VXD, which tracks the Dow Jones Industrial Average. Investopedia explains VIX (CBOE Volatility Index) The first VIX Index was introduced by the CBOE in 1993 and was a weighted measure of the implied volatility of eight S&P 100 at-themoney put and call options. In 2003, it was expanded to use options that were based on a broader index, the S&P 500 Index, which provides a more accurate picture of investors’ expectations of future market volatility.

See Bank guarantee Haircut, 127 Head and shoulders pattern, 127-128 Hedge, 72, 128-129 Hedge fund, 3, 129-130 Hedge ratio. See Delta High-yield bond, 130. See also Junk bond Historical cost, 130-131 Historical volatility. See Standard deviation Holder of letter of credit, 159 Holder of record, 248 Hostile takeover, 131, 225, 295 House call. See Margin call HR(10) plan. See Keogh plan Humped yield curve, 324-325 Hyperinflation, 131-132 IB. See Investment bank (IB) Illiquid (asset), 133-134 Implied volatility (IV), 134 In the money, 117, 134, 144 Income, 210, 252, 301. See also Revenue Income statement, 17, 40, 54, 135, 196, 208, 235-236 Index, 135-136. See also specific indexes such as Standard & Poor’s 500 Index (S&P 500) Index fund, 136-137, 221 Index futures, 137 Indicators. See Trend analysis Individual retirement account (IRA), 137-138. See also Retirement plans; specific types of IRAs Inelastic, 138 Inflation, 131-132, 139, 278, 304-305 Inflation GDP.

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Derivatives Markets by David Goldenberg

You can also do it in Microsoft Excel, which is a useful exercise in the End of Chapter Exercise 3. There is another method which follows directly from Black–Scholes. It is called the implied volatility method and it generates the implied volatility estimator. The idea behind implied volatility is that the Black–Scholes formula embodies an implicit volatility estimator. If we compare market option prices to Black–Scholes model option prices, we can extract the Black–Scholes implicit volatility estimator. Since option prices incorporate a wide variety of forward views of volatility, implied volatility could be a better estimator of unknown volatility than the historical estimator, which is a backward looking estimator. B. The Implied Volatility Estimator Method Volatility is one of the key parameters in the Black–Scholes formula, but it is unobservable. Why not let the model generate estimates of σ that are consistent with the assumption that the market prices options using the Black–Scholes formula?

In order to implement it, all we have to do is plug all the parameters, except σ, into the Black–Scholes formula. Then, if we take the market’s (not the model’s) option price we can equate Ct,Black–Scholes to Ct,Market, and obtain a non-linear equation in σ that can be iteratively solved for the implied volatility estimator, which we will denote by σIV. The brief version of this procedure is, Ct,Black–Scholes=Ct,Market implies σIV. End of Chapter Exercises 4 and 5 implement this procedure. The IV estimator turns out empirically to be a better estimator than the historical σ, which is probably not too surprising. Unfortunately, whether we use the historical volatility estimator or the implied volatility estimator, we are still stuck with the constant σ assumption. If σ is constant, then it is also constant across options with different exercise prices and σIV should not depend upon which exercise price K is used to estimate it.

There is no immediate and completely adequate empirical fix for the constant σ assumption, except to throw out Black–Scholes’ assumption of a stationary log-normal diffusion, and search for a viable (smile-consistent) underlying stochastic process among the vast set of alternatives, many of which will lead to incomplete markets. Black–Scholes and its modifications, however, still have tremendous appeal, especially among traders, who use Black–Scholes calibrated to an implied volatility surface. Traders use ATM options to imply volatility, since these are the most liquid, and therefore most informative about future volatility. Furthermore, there are exotic and American options for which the log-normal GBM remains the workhorse. This is for the simple reason that it is difficult (or so far impossible) to price these complex options for any processes other than a standard GBM. Black–Scholes usually appears as a component of the option prices for these option types; for example, for American options.

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The Crisis of Crowding: Quant Copycats, Ugly Models, and the New Crash Normal by Ludwig B. Chincarini

To execute this trade, one might buy 5-year call options on 10-year swaps in Deutschemarks and sell 1-year call options on 10-year swaps in the right proportion to hedge away unwanted risks, such as risks from overall interest rate movements. Figure 4.7 graphs the difference between the implied volatility of 5-year options and 1-year options. When this went up, LTCM made money. When this went down, LTCM lost money. FIGURE 4.7 The Difference between Implied Volatility on 5-Year Options and 1-Year Options on 10-Year Euro Swap Rates Source: Goldman Sachs. Around June 1998, the implied volatilities of short-term and long-term options were about the same. Then, as LTCM predicted, the implied volatility on the 5-year increased, making LTCM profits. Then came the Russian crisis, when Russia defaulted on its debt. The volatility spread crashed and took LTCM’s position along with it. The same volatility spread collapse took place in the 2008 crash.

That implies that LTCM received an option premium of about \$1.01 billion for selling the straddle.29 According to LTCM partners, they had a vega of \$25 million on the S&P 500 and another \$25 million vega shared in the European equity markets. LTCM’s idea was good, but in August and September 1998 it was not practical. Figure 4.10 plots the implied volatility (according to market prices) of 12-month options on the S&P 500 and the Nikkei, as well as the rolling 20-day and rolling 5-year historical volatility of the S&P 500. FIGURE 4.10 The Implied Volatility of 12-Month Options on the S&P 500 and the Nikkei 225 Source: Goldman Sachs. It’s clear that the Russian default and LTCM’s crisis pushed both short-term actual volatility and one-year options’ implied volatility sharply up. On August 3, 1998, the implied volatility on short-term options was 24%, 20-day historical volatility was 16%, and 5-year historical volatility was 12%. By August 31, these three numbers rose to 32%, 32%, and 13% respectively.

With a formula that relates an option’s price to the underlying security’s volatility, a trader could convert the option’s price into a volatility consistent with that price. This is called implied volatility. The Black-Scholes formula, discovered in 1973, is most commonly used for this purpose. It is named after one of LTCM’s principals, Myron Scholes, and the late Goldman Sachs partner Fischer Black. LTCM made volatility trades in both fixed income and equities. In the fixed-income arena, they noticed in 1998 that the implied volatility of 5-year options (i.e., options with five years to maturity) on German-denominated swaps was trading much lower than actual realized volatility. Option prices were trading with an implied volatility of 3 basis points per day, while the realized volatility in the marketplace was closer to 5 basis points. These were essentially options on German interest rates, and the market’s volatility assessment was out of step with actual movements in German interest rates.21 LTCM wanted to go long on volatility at the 5-year mark.

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New Market Wizards: Conversations With America's Top Traders by Jack D. Schwager

An alternative interpretation is that the market is simply assuming that volatility in the period remaining until the option’s expiration will be different from the recent past volatility, called the historical volatility. The volatility assumption embedded in the market price is called the implied volatility. If option prices are a better predictor of future volatility than is the recent past volatility, then the question of whether an option is overpriced or underpriced is not only irrelevant but actually misleading. In essence, the question Joe Ritchie / 357 posed above is equivalent to asking whether there is any reason to assume that the strategy of buying options priced below their fair value and selling those that are above their fair value has any merit.] Implied volatility seems better to me. Conceptually or empirically? To me it seems pretty obvious conceptually. The implied volatility is a statement of what all the players in the market, having cast their votes, believe is a fair price for future volatility.

So when you first started in option trading, you were looking for options that were out of line with their theoretical value. That’s right. That raises an interesting question. Since theoretical values are based on historical volatility, doesn’t that approach imply that historical volatility is a better predictor of future volatility than implied volatility? [For a detailed discussion of the concepts underlying this question, see the Joe Ritchie interview, pages 356-574.] No. Actually, empirical studies have shown that implied volatility is better than historical volatility in predicting the actual future volatility. 376 / The New Market Wizard Then how could you make money by trading based on mispricings relative to your model? The real key is relative value. It doesn’t matter what model you use, as long as you apply it consistently across all option prices.

I always tried to be relatively hedged. in a takeover situation, however, you might think that you are hedged, but the price move occurs so quickly that you really aren’t. You mentioned that speculators are usually on the buy side of options. In general, do you believe there is a mispricing that occurs because people like to buy options? If you compare historical graphs of implied volatility versus historical volatility across a spectrum of markets, you will see a distinct tendency Blair Hull / 379 for implied volatility being higher—a pattern that suggests that such a bias exists. Does that imply that being a consistent seller of options is a viable strategy? I believe there’s an edge to always being a seller, but I wouldn’t trade that way because the implied risk in that approach is too great. But to answer your question, generally speaking, I believe the buyer of options has the disadvantage.

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Tools for Computational Finance by Rüdiger Seydel

For the ith update in the algorithm carry out a rounding error analysis. What is your judgement on the algorithm? Exercise 1.5 Implied Volatility For European options we take the valuation formula of Black and Scholes of the type V = v(S, τ, K, r, σ), where τ denotes the time to maturity, τ := T −t. For the deﬁnition of the function v see Appendix A4, equation (A4.10). If actual market data of the price V are known, then one of the parameters considered known so far can be viewed as unknown and ﬁxed via the implicit equation V − v(S, τ, K, r, σ) = 0 . (∗) In this calibration approach the unknown parameter is calculated iteratively as solution of equation (∗). Consider σ to be in the role of the unknown parameter. The volatility σ determined in this way is called implied volatility and is zero of f (σ) := V − v(S, τ, K, r, σ). Assignment: a) Implement the evaluation of VC and VP according to (A4.10). b) Design, implement and test an algorithm to calculate the implied volatility of a call.

Assignment: a) Implement the evaluation of VC and VP according to (A4.10). b) Design, implement and test an algorithm to calculate the implied volatility of a call. Use Newton’s method to construct a sequence xk → σ. The derivative f (xk ) can be approximated by the diﬀerence quotient f (xk ) − f (xk−1 ) . xk − xk−1 c) For the resulting secant iteration invent a stopping criterion that requires smallness of both |f (xk )| and |xk − xk−1 |. Calculate the implied volatilities for the data T − t = 0.211 , S0 = 5290.36 , r = 0.0328 Exercises 55 and the pairs K, V from Table 1.3 (for more data see www.compfin.de). Enter for each calculated value of σ the point (K, σ) into a ﬁgure, joining the points with straight lines. (You will notice a convex shape of the curve. This shape has lead to call this phenomenon volatility smile.)

(1.38) dSt = rSt dt + σSt dWtγ . (1.39) Then (1.37) becomes Comparing this SDE to (1.33), notice that the growth rate µ is replaced by the risk-free rate r. Together the transition consists of µ P W → r → Q → Wγ which is named risk-neutral valuation principle. The advantage of the “risk-free measure” Q that corresponds to (1.38) is that the discounted process e−rt St is drift-free, 6 For the implied volatility see Exercise 1.5. 1.7 Stochastic Diﬀerential Equations 37 d(e−rt St ) = e−rt σSt dWtγ . This property of having no drift is an essential ingredient of a no-arbitrage market and a prerequisite to modeling options. For a thorough discussion of the continuous model, martingale theory is used. (More background and explanation is provided by Appendix B3.) Let us summarize the situation in a remark: Remark 1.14 (risk-neutral valuation principle) For modeling options the return rate µ is replaced by the risk-free interest rate r, µ = r.

Analysis of Financial Time Series by Ruey S. Tsay

In options markets, if one accepts the idea that the prices are governed by an econometric model such as the Black–Scholes formula, then one can use the price to obtain the “implied” volatility. Yet this approach is often criticized for using a specific model, which is based on some assumptions that might not hold in practice. For instance, from the observed prices of a European call option, one can use the Black–Scholes formula in Eq. (3.1) to deduce the conditional standard deviation σt . The resulting value of σt2 is called the implied volatility of the underlying stock. However, this implied volatility is derived under the log normal assumption for the return series. It might be very different from the actual volatility. Experience shows that implied volatility of an asset return tends to be larger than that obtained by using a GARCH type of volatility model. Although volatility is not directly observable, it has some characteristics that are commonly seen in asset returns.

The price is Pt = P0 exp[(µ − σ 2 /2)t + σ wt ]. (6.25) 6.9 JUMP DIFFUSION MODELS Empirical studies have found that the stochastic diffusion model based on Brownian motion fails to explain some characteristics of asset returns and the prices of their derivatives (e.g., the “volatility smile” of implied volatilities; see Bakshi, Cao, and Chen, 1997, and the references therein). Volatility smile is referred to as the convex function between the implied volatility and strike price of an option. Both out-ofthe-money and in-the-money options tend to have higher implied volatilities than at-the-money options especially in the foreign exchange markets. Volatility smile is less pronounced for equity options. The inadequacy of the standard stochastic diffusion model has led to the developments of alternative continuous-time models. For example, jump diffusion and stochastic volatility models have been proposed in the literature to overcome the inadequacy; see Merton (1976) and Duffie (1995).

(a) Horizon GARCH SVM Log return 1 0.66 0.53 2 0.66 0.78 (b) Horizon GARCH SVM 3 0.66 0.92 4 0.66 0.88 5 0.66 0.84 4 18.34 19.65 5 18.42 20.13 Volatility 1 17.98 19.31 2 18.12 19.36 3 18.24 19.35 441 EXERCISES eter uncertainty in producing forecasts. In contrast, the GARCH model assumes that the parameters are fixed and given in Eq. (10.26). This is an important difference and is one of the reasons that GARCH models tend to underestimate the volatility in comparison with the implied volatility obtained from derivative pricing. Remark: Besides the advantage of taking into consideration parameter uncertainty in forecast, the MCMC method produces in effect a predictive distribution of the volatility of interest. The predictive distribution is more informative than a simple point forecast. It can be used, for instance, to obtain the quantiles needed in Value at Risk calculation. 10.10 OTHER APPLICATIONS The MCMC method is applicable to many other financial problems.

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Investing Demystified: How to Invest Without Speculation and Sleepless Nights by Lars Kroijer

When looking at the price of an option on a stock market index the only variable that is not readily observable is the expected volatility (the other inputs are: the strike price of the option, the current price of the index, time to maturity and the interest rate). Using the Black-Scholes option pricing formula we can obtain the implied volatility. Looking at the implied volatility for options with various maturities we can see how volatile traders expect the market to be in future. In the past, the implied volatility of index options have been better predictors of future market volatility than using the historical volatility of the stock market. For the S&P 500 index you can look at the VIX index, which gives the implied volatility for that market for the coming month, but expect the implied volatility to be very different depending on the market, maturity and strike price you are looking at. 2 You can look up the probabilities associated with various standard deviations and get a fuller explanation of standard deviation in general, on Wikipedia.

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Red-Blooded Risk: The Secret History of Wall Street by Aaron Brown, Eric Kim

And we can take the derivative of bond price with respect to yield to get a first order idea of the volatility of a bond. The Black-Scholes-Merton model works the same way. Two options with similar terms will have similar implied volatilities, but not necessarily similar prices. So we can use implied volatilities of options we know the prices of to estimate the implied volatilities, and hence the prices, of options whose prices we do not know. We can graph option implied volatility versus time or moneyness (the ratio of the strike price of an option to the underlying price) and get the same kind of insights we get from yield curves and credit curves. We can take the derivative of option price with respect to implied volatility, known as vega, which some forgotten trader thought was a Greek letter. All of this is pure mathematics; it does not require any economic assumptions.

You can regard an option as a derivative and treat its price as something you can derive mathematically from the underlying stock price, or you can regard the option price as something that trades up and down on its own, correlated to the stock price but not determined by it. Both views are valid for different purposes. In this context, it is important to understand that the Black-Scholes-Merton option pricing model is not really a pricing model. It tells us one thing we don’t know, the price of an option, in terms of another thing we don’t know, the volatility of the underlying stock. Solving for the implied volatility (the volatility that gives the option its market price) from the price is exactly analogous to solving for the yield to maturity of a bond from its price. Both are conversions, like from Fahrenheit to Centigrade temperatures. They are mathematical transformations with no economic content. The reason people solve for the yield to maturity of bonds is that two bonds with similar terms and credit qualities will have similar yields, but not necessarily similar prices.

Hedge funds Herbert, Zbigniew Heteroskedasticity Hirsh, Michael Historical simulation VaR History of Statistics, The (Stigler) Hoffer, Richard Hong Kong on Air (Cohen, Muhammad) House of Cards (Cohan, William) How Big Banks Fail and What to Do about It (Duffie) Hugh-Jones, Stephen Humphrey, Caroline Iceberg Risk (Osband) IGT. See Investment growth theory (IGT) Ilmanen, Antti Implied volatility Index funds Inflation Inside the Black Box (Narang) Inside the House of Money (Drobny) Internet. See Bubble investors Investment growth theory (IGT): EMH and equations, MPT and IGT CAPM fairness and IGT CAPM and MPT CAPM virtues of Inviting Disaster (Chiles) Is God a Mathematician? (Livio) Jackknife, the Bootstrap, and Other Resampling Plans, The (Efron) Jackpot Nation (Hoffer) Jessup, Richard John Bogle on Investing (Bogle) Johnson, Barry Johnson, Simon JPMorgan Junk bonds Kahneman, Daniel Kamensky, Jane Kaplan, Michael Kassouf, Sheen Kelly, John Kelly bets/levels of risk Kelly principles/investors Keynes, John Maynard Key performance indicators (KPIs) Key risk indicators (KRIs) King of a Small World (Bennet) Korajczyk, Robert Knetsch, Jack Knight, Frank Kraitchik, Maurice Krüger, Lorenz Laplace, Pierre-Simon Lehman Brothers Leitzes, Adam Lepercq de Neuflize Leverage Levine, David Levinson, Horace Lewis, Michael Limits of Safety, The (Sagan) Liquidity Livio, Mario Logic of Failure, The (Dorner) Long-Run Collaboration on Games with Long-Run Patient Players, A (Fudenberg and Levine) Loss aversion Lowenstein, Roger Mackay, Charles Madoff, Bernie Mallaby, Sebastian Man with the Golden Arm, The (Bennet) Managed futures Managing risk Mandelbrot, Benoit Market: “beating” the efficiency (see also Efficient markets theory) equilibrium (see Equilibrium) portfolio prices return sympathy Mark-to-market accounting Markowitz, Harry.

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Hedge Fund Market Wizards by Jack D. Schwager

When volatility gets very low in a market, we consider that a very interesting time to start looking for ways to get long volatility, both because volatility is very cheap in an absolute sense and because the market certainty and complacency reflected by low volatility often implies an above-average probability of increased future volatility. Do you favor long-dated options? Often, the longer the duration of the option, the lower the implied volatility, which makes absolutely no sense. We recently bought far out-of-themoney 10-year call options on the Dow as an inflation hedge. Implied volatility on the index is very low. The Dow companies would be in the best position to pass along higher prices. There is also an interest rate bet implicit in buying long-term options that can be quite interesting when interest rates are very low, as they are now. By being long 10-year call options, we are taking exposure on the risk-free rate implicit in the option pricing models.

I have listened to every podcast since the program’s inception, and I highly recommend it. 4If this comment is unintelligible to you, don’t worry. A primer on mortgage-backed securities and their role in the financial crises is provided later in this chapter before our conversation related to Cornwall’s short trade in collaterized debt obligations (CDOs). 5Mai explained that the typical quoting convention for implied volatility in interest rate markets, known as “normalized volatility,” is the number of absolute basis points reflecting a one-standard-deviation event, as opposed to the standard convention of quoting implied volatility in other asset classes in terms of percentage changes in the underlying security. Normalized volatility of 100 basis points equals a much smaller volatility, as measured in “traditional” percentage terms, when rates are high than when they are low—a characteristic that may have been an additional factor amplifying the anomaly. 6The expected value is the sum of the probability of each outcome multiplied by its value.

A simple example of this anomaly would be a rates trade we did in 2010. At the time, the current Brazilian interest rate was around 8 percent and the 6-month forward rate was over 12 percent. The 6-month forward option prices were distributed around the forward rate of over 12 percent. In other words, the option prices implicitly assumed the 6-month forward rate as the expected level. The implied volatility at the time was around 100 basis points normalized, which meant the market was assigning the odds of nothing happening for the next six months or so— that is, rates staying near 8 percent—as over a four-standard-deviation event.5 We did not have conviction about the future direction of Brazilian interest rates, much less the actual levels. But we thought the assumption that a spot rate in six months near the current spot rate was a greater than four-standard-deviation event—an assumption that was embedded in the option price—represented a mispricing.

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Inside the House of Money: Top Hedge Fund Traders on Profiting in a Global Market by Steven Drobny

I actually had a big argument with another hedge fund manager who later told me that he was on the other side of that trade selling me the zero strike options via the bank because I was overpaying for volatility. I just had to laugh. I said to him,“Volatility has nothing to do with it.Volatility is a stupid concept. We all know that it probably will not happen and that probability is very, very high, but Long Term Capital also thought there was a 1 in 6 billion chance that their portfolio could blow up.” Implied volatility is based on historical volatility, but who cares about historicals? They’re irrelevant.The point is, things can happen for the first time that aren’t in your distribution so they can’t be priced. If it’s never happened before, how can you hedge yourself? The only way to hedge the unknown is to cut off tail risk completely. What was your favorite trade of all time? The trade I remember the most only made a small amount of money but it was the first time I took on a large position based on my belief that I understood the market better than the market.

When things move up by whatever definition you use, you should sell and when they move back down, you should buy. On average, over time you’re going to make money or earn risk premia. In sum, you overpay for options but don’t mind overpaying? Correct, especially when you move out past one-month options.There is a tendency to believe that people overpay for options because the research shows that implied volatility is higher than realized volatility.That has to be the case for the seller to be willing to write you an option—he’s got to make some money.The difference is, he’s going to delta hedge and you’re not, so 1.30 Dollars per Euro 1.20 1.10 1.00 0.90 M ay Ju 01 nJu 01 lAu 01 gSe 01 pOc 01 tNo 01 vDe 01 cJa 01 nFe 02 bM 02 ar Ap 02 rM 02 ay Ju 02 nJu 02 lAu 02 gSe 02 pOc 02 tNo 02 vDE 02 cJa 02 nFe 03 bM 03 ar Ap 03 rM 03 ay Ju 03 nJu 03 lAu 03 gSe 03 pOc 03 tNo 03 vDe 03 c03 0.80 FIGURE 4.2 Euro, 2001–2003 Source: Bloomberg.

THE FAMILY OFFICE MANAGER 57 you are going to have to pay a little bit extra so that work gets compensated.You have to realize ex ante that yes, you are overpaying.The interesting thing is that you are not delta hedging nor are you paying the seller to do all the work.The market is.The seller is making his money off the delta hedge, and you are paying him a little bit by paying him more than what realized volatility is, but ex ante no one really knows what realized is. We had a study done on the foreign exchange options market going back to 1992, where one-year straddle options were bought every day across a wide variety of currency pairs.We found that even though implied volatility was always higher than realized volatility over annual periods, buying the straddles made money. It’s possible because the buyer of the one-year straddles is not delta hedging but betting on trend to take the price far enough away from the strike that it will cover the premium for the call and the put. Over time, there’s been enough trend in the market to carry price far enough away from the strike of the one-year outright straddle to more than cover the premium paid.

The Trade Lifecycle: Behind the Scenes of the Trading Process (The Wiley Finance Series) by Robert P. Baker

In order to look at the shape of market data over time, we need to use a threedimensional object, usually referred to as a surface. A typical example is implied volatility. Although volatility is generally regarded as a reflection of the way prices move, it can also be a source of market data. This is because premium prices of an option infer the volatility (known as the implied volatility). The typical behaviour of options is that they are have greater implied volatility when deeply in or out-of-themoney. In or out-of-the-money is a function of price. So in order to model the market data we need three dimensions – value, price and time. By observing market data for implied volatility across different prices and times, we can construct an implied volatility surface. It is important to recognise that future curves and surfaces generated from current market data are only a market-implied snapshot of how asset prices will move.

This involves them asking, say, 20 market makers for their prices, disregarding the top two and bottom two and taking an average of the middle 16 to arrive at a representative set of market data. For end of day, the middle office will have used data supplied by the traders. But at end of month, this is corrected to the independent value. If the traders have (deliberately or otherwise) used skewed values, these will be corrected by means of a provision. For example: A trader’s P&L comes out as EUR 1 million using his implied volatility data. At end of month, using the independent market data it is revised to EUR 700,000. So a figure of negative EUR 300,000 is added to his accounts as a provision to correct the P&L. He now carries that provision for the coming month until the next time the independent market data is used. It generally takes somewhere in the region of three to four working days each month to derive and verify the end of month valuations and to report them to the relevant audience.

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High-Frequency Trading: A Practical Guide to Algorithmic Strategies and Trading Systems by Irene Aldridge

Market crashes and other negative shocks have been shown to result in higher subsequent volatility levels than rallies and other news favorable to the market. This asymmetric property of volatility generates skews in volatility surfaces constructed of option-implied volatilities for different option strike prices. Engle and Patton (2001) cite the following example of volatility skews: the implied volatilities of in-the-money put options are lower than those of at-themoney put options. Furthermore, the implied volatilities of at-the-money put options are lower than the implied volatilities of out-of-the-money options. Finally, volatility forecasts may be influenced by external events, such as news announcements. In foreign exchange, for example, price and return volatility of a particular currency pair increase markedly during macroeconomic announcements pertaining to one or both sides of the currency pair.

T 115 116 HIGH-FREQUENCY TRADING This chapter discusses the following topics: r r r r Various properties of tick data Econometric techniques specific to tick data estimation How trading systems can make better trading decisions using tick data How trading systems can apply traditional econometric principles PROPERTIES OF TICK DATA The highest-frequency data is a collection of sequential “ticks,” arrivals of the latest quote, trade, price, and volume information. Tick data usually has the following properties: r A timestamp r A financial security identification code r An indicator of what information it carries: r Bid price r Ask price r Available bid volume r Available ask volume r Last trade price r Last trade size r Option-specific data, such as implied volatility r The market value information, such as the actual numerical value of the price, available volume, or size A timestamp records the date and time at which the quote originated. It may be the time at which the exchange or the broker-dealer released the quote, or the time when the trading system has received the quote. The quote travel time from the exchange or the broker-dealer to the trading system can be as small as 20 milliseconds.

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Market Sense and Nonsense by Jack D. Schwager

Instead of using only correlation for this task, investors should base their decisions on the following more comprehensive and descriptive combination of four statistics: 1. Correlation. 2. Beta. 3. Percentage of up months in down markets. 4. Average return in down markets. 1 Figure 9.1 is a hypothetical, simplified illustration. In actual markets, the pattern would not be symmetrical, since declining prices would likely increase implied volatility, further exacerbating losses, while rising prices would likely reduce implied volatility, mitigating losses. 2 The hypothetical fund returns examples (Funds A, B, and C) used in this chapter are artificial and not meant to be representative of any actual funds. The return statistics have been created specifically to highlight some key concepts related to the properties of correlation. 3 Although this is an artificial and unrealistic return series, it is useful in helping to illustrate the concept that high correlation does not necessarily imply a large price impact. 4 Mathematically, beta is equal to correlation times the ratio of the investment standard deviation to the benchmark standard deviation.

(In contrast, the time remaining until expiration and the relationship between the current market price and the strike price can be exactly specified at any juncture.) Thus, volatility must always be estimated on the basis of historical volatility data. The future volatility estimate implied by market prices (i.e., option premiums), which may be higher or lower than the historical volatility, is called the implied volatility. On average, there is a tendency for the implied volatility of options to be higher than the subsequent realized volatility of the market till the options’ expiration. In other words, options tend to be priced a little high. The extra premium is necessary to induce option sellers to take the open-ended risk of providing price insurance to option buyers. This situation is entirely analogous to home insurance premiums being priced at levels that provide a profit margin to insurance companies—otherwise, they would have no incentive to assume the open-ended risk.

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Why Stock Markets Crash: Critical Events in Complex Financial Systems by Didier Sornette

The standard procedure is then to see what the market forces decide for the option price and then determine the implied volatility by inversion of the Black and Scholes formula for option pricing [294]. Basically, the implied volatility is a measure of the market risks perceived by investors. Figure 7.4 presents the time evolution of the implied volatility of the S&P 500, taken from [84]. The perceived market risk is small prior to the crash, jumps up abruptly at the time of the crash, and then decays slowly over several months. This decay to “normal times” of perceived risks is compatible with a slow power law decay decorated by log-periodic 237 autopsy of major c r a s h e s 90 80 70 σ 2 (S&P 500) 60 50 40 30 20 10 0 87.6 87.8 88.0 88.2 88.4 Time (year) 88.6 88.8 Fig. 7.4. Time evolution of the implied volatility of the S&P 500 index (in logarithmic scale) after the October 1987 crash, taken from [84].

In other words, we should be able to document the existence of a critical exponent as well as log-periodic oscillations on relevant quantities after the crash. Such a signature in the volatility of the S&P 500 index, implied from the price of S&P 500 options (which are derivative assets with price varying as a function of the price of the S&P 500), can indeed be seen in Figure 7.4. The term “implied volatility” has the following meaning. First, one must recall what an option is: this ﬁnancial instrument is nothing but an insurance that can be bought or sold on the market to insure oneself against unpleasant price variations. The price of an option on the S&P 500 index is therefore a function of the volatility of the S&P 500. The more volatile and the more risky is the S&P 500, the more expensive is the option.

The question we ask is whether the cooperative herding behavior of traders might also produce market evolutions that are symmetric to the accelerating speculative bubbles that often end in crashes. This symmetry is performed with respect to a time inversion around a critical time tc such that tc − t for t < tc is changed into t − tc for t > tc . This symmetry suggests looking at decelerating devaluations instead of accelerating valuations. A related observation has been reported in Figure 7.4 in relation to the October 1987 crash showing that the implied volatility of traded options relaxed after the October 1987 crash to its long-term value, from a maximum at the time of the crash, according to a decaying power law with decelerating log-periodic oscillations. It is this type of behavior that we document now, but for real prices. The critical time tc then corresponds to the culmination of the market, with either a power law increase with accelerating log-periodic oscillations preceding it or a power law decrease with decelerating log-periodic 276 chapter 7 oscillations after it.

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Asset and Risk Management: Risk Oriented Finance by Louis Esch, Robert Kieffer, Thierry Lopez

In practice, it is generally taken for a reference period of one year and expressed as a percentage. This concept of volatility can be seen from two points of view: historical volatility and implied volatility. Historical volatility is simply the annualised standard deviation on the underlying equity return, obtained from daily observations of the return in the past: n 1 σR = J · (Rt − R)2 n t=1 Here, the factor J represents the number of working days in the year; n is the number of observations and Rt is the return on the underlying equity. It is easy to calculate, but the major problem is that it is always ‘turned towards the past’ when it really needs to help analyse future developments in the option price. For this reason, the concept of implied volatility has been introduced. This involves using a valuation model to estimate the dispersion of the return of the underlying equity for the period remaining until the contract matures.

This involves using a valuation model to estimate the dispersion of the return of the underlying equity for the period remaining until the contract matures. The value of the option premium is determined in practice by the law of supply and demand. In addition, this law is linked to various factors through a binomial model of valuation: pt = f (St , K, T − t, σR , RF ) or through Black and Scholes (see Section 5.3). The resolution of this relation with respect to σR deﬁnes the implied volatility. Although the access is more complicated, this concept is preferable and it is this one that will often be used in practice. 5.2.3 Sensitivity parameters 5.2.3.1 ‘Greeks’ The premium is likely to vary when each of the parameters that determine the price of the option (spot price, exercise price, maturity etc.) change. The aim of this paragraph is to study the indices,7 known as ‘Greeks’, which measure the sensitivity of the premium to ﬂuctuations in some of these characteristics through the relation pt = f (St , K, τ, σR , RF ). 7 In the same way as duration and convexity, which measure the sensitivity of the value of a bond following changes in interest rates (see Chapter 4). 156 Asset and Risk Management Here, we will restrict ourselves to examining the most commonly used sensitivity coefﬁcients: those that bring the option price and namely the underlying equity price time, volatility and risk-free rate into relation.

If the reduced normal density is termed φ 1 2 φ(x) = (x) = √ e−x /2 2π we arrive, by derivation, at: (C) = CS = (d1 ) + 1 √ St φ(d1 ) − Ke−rτ φ(d2 ) St σR τ It is easy to see that the quantity between the square brackets is zero and that therefore (C) = (d1 ), and that by following a very similar logic, we will arrive at a put of: (P ) = (d1 ) − 1. The above formula provides a very simple means of determining the number of equities that should be held by a call issuer to hedge his risk (the delta hedging). This is a common use of the Black and Scholes relation: the price of an option is determined by the law of supply and demand and its ‘inversion’ provides the implied volatility. The latter is therefore used in the relation (C) = (d1 ), which is then known as the hedging formula. Options 173 The other sensitivity parameters (gamma, theta, vega and rho) are obtained in a similar way: φ(d1 ) (C) = (P ) = √ St σR τ  St σR φ(d1 )  − rKe−rτ (d2 )   (C) = − 2√τ  S σ φ(d )   (P ) = − t R√ 1 + rKe−rτ (−d2 ) 2 τ V (C) = V (P ) = τ St φ(d1 ) ρ(C) = τ Ke−rτ (d2 ) ρ(C) = −τ Ke−rτ (−d2 ) In ﬁnishing, let us mention a relationship that links the delta, gamma and theta parameters.

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The New Science of Asset Allocation: Risk Management in a Multi-Asset World by Thomas Schneeweis, Garry B. Crowder, Hossein Kazemi

Expected return models are often based on multifactor models. In this case, a 4-factor model is used to predict risk and return on each predetermined allocation. The four factors are: Strategic, Tactical, and Dynamic Asset Allocation ■ 103 1. Current level of credit risk premium (CR) compared to its historically normal level 2. Current level of term premium (TP) compared to its historically normal level 3. Current level of S&P 500 implied volatility as measured by VIX compared to its historically normal level 4. Recent return to each allocation Estimation Strategy. A quantitative approach is adopted to estimate the lead-lag relationship between the performance of each allocation and the factors mentioned above. E [Rt +1 ] = f (CRt , TPt , Rt , VIXt ) ■ In this case, five years of monthly returns are used to estimate the model. The estimated relationship is back tested to ensure its robustness and stability.

Exhibit 9.1 provides one sample portfolio allocation across multiple asset classes. Consider the case of a family business, which currently has an investment of \$200 million in a well diversified portfolio of traditional global equity and fixed income assets as well as alternative investments. The five-year historical volatility on the portfolio’s pro-forma return has been 10%, while during the same period the average implied volatility of U.S. equity market has been around 18%. This means that the portfolio’s volatility has been about 55% of VIX. Once the portfolio is constructed, the portfolio manager will need to monitor the VIX. If there is a significant increase in VIX, the portfolio manager will use index futures to hedge out some of the portfolio’s volatility such that its expected volatility remains close to the target.

In light of the growing investment interest, the CBOE has recently introduced a number of buy-write indices based on a variety of equity indices such as the S&P 500, the Dow Jones 207 Risk Budgeting and Asset Allocation Industrial Average, the NASDAQ 100 and the Russell 2000. In addition, a number of funds based on a buy-write strategy have been introduced over the last five years.2 As illustrated in Kapadia and Szado (2007), the excess risk-adjusted performance of the passive buy-write strategy is primarily derived from selling calls at an implied volatility that exceeds the subsequently realized volatility. In fact, they find that if the calls were sold at the Black-Scholes price corresponding with the realized volatility, the buy-write strategy would underperform the underlying index. In this sense, the buywrite is providing something more than a simple return distribution truncation; it is also providing an additional source of returns—the option volatility risk premium.

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The Rise of the Quants: Marschak, Sharpe, Black, Scholes and Merton by Colin Read

We can observe the movement of options prices based on current market conditions to derive an implied volatility, and analysts can at least compare current volatility trends against historical patterns. While the Black-Scholes equation demonstrated that measures of volatility affect options prices, the expected rate of return µ of the underlying security does not. This means that analysts can differ in their valuation of a stock, but not in their valuation of its associated option. In fact, Monte Carlo simulations of options prices as the underlying stock price is allowed to evolve show that the option price remains remarkably stable for different rates of drift of the underlying stock. This tendency of options to measure market volatility rather than market strength has been enshrined through the Volatility Index (VIX) tallied by the CBOE. This implied volatility is calculated by solving for the volatility that justified the prevailing options price, based on the underlying security price.

Alternatively, an investor can book some profits immediately in the security by selling a call on a bond. If the bond price rises above a certain level, the seller of the call must sacrifice the underlying bond and in turn sacrifice the gain above the exercise price, but is able to book some profit with certainty. Assuming that these fixed-income options are properly priced in an efficient market, we can even calculate the implied volatility by solving the Black-Scholes equation for the volatility necessary to generate the prevailing price. This technique allows the Federal Reserve to measure point volatility in bond markets, or analysts to obtain a measure for the perceived level of volatility in a securities market. Playing with financial fire While options can be used as a legitimate way to share and hedge risk, their highly leveraged nature can also be dangerous.

Principles of Corporate Finance by Richard A. Brealey, Stewart C. Myers, Franklin Allen

BEYOND THE PAGE ● ● ● ● ● VIX—USA, Europe, and Japan brealey.mhhe.com/c21 Calculating Implied Volatilities So far we have used our option pricing model to calculate the value of an option given the standard deviation of the asset’s returns. Sometimes it is useful to turn the problem around and ask what the option price is telling us about the asset’s volatility. For example, the Chicago Board Options Exchange trades options on several market indexes. As we write this, the Standard and Poor’s 500 Index is about 1250, while a one-year at-the-money call on the index is priced at 126. If the Black–Scholes formula is correct, then an option value of 126 makes sense only if investors believe that the standard deviation of index returns is about 25% a year.16 The Chicago Board Options Exchange regularly publishes the implied volatility on the Standard and Poor’s index, which it terms the VIX (see the box on page 551).

If the Black–Scholes formula is correct, then an option value of 126 makes sense only if investors believe that the standard deviation of index returns is about 25% a year.16 The Chicago Board Options Exchange regularly publishes the implied volatility on the Standard and Poor’s index, which it terms the VIX (see the box on page 551). There is an active market in the VIX. For example, suppose you feel that the implied volatility is implausibly low. Then you can “buy” the VIX at the current low price and hope to “sell” it at a profit when implied volatility has increased. FINANCE IN PRACTICE ● ● ● ● ● The Perfect Payday* On an October day in 1999 the shares of the giant insurer United Health Group sank to their lowest level of the year. That may have been bad news for investors but it was good news for William McGuire, the chief executive, for the company granted him options to buy the stock in the future at that low price. If the options had been dated a month later when the stock price was 40% higher, those options would have been far less valuable. Lucky coincidence?

Yermack, “Good Timing: CEO Stock Option Awards and Company News Announcements,” Journal of Finance 52 (1997), pp. 449–476, and in E. Lie, “On the Timing of CEO Stock Option Awards,” Management Science 51 (2005), pp. 802–812. BEYOND THE PAGE ● ● ● ● ● An uncertainty index brealey.mhhe.com/c21 You may be interested to compare the current implied volatility that we calculated earlier with Figure 21.6, which shows past measures of implied volatility for the Standard and Poor’s index and for the Nasdaq index (VXN). Notice the sharp increase in investor uncertainty at the height of the credit crunch in 2008. This uncertainty showed up in the price that investors were prepared to pay for options. FIGURE 21.6 Standard deviations of market returns implied by prices of options on stock indexes.

How I Became a Quant: Insights From 25 of Wall Street's Elite by Richard R. Lindsey, Barry Schachter

American retail investors wanted to bet that the Nikkei would fall from its then-stratospheric heights, but there was nothing they could trade (the U.S. government, which upholds the right of its citizens to bear arms, draws the line at derivatives). The loophole was that American investors could trade Canadian warrants after the warrants had traded for 90 days in Canada, so Gordon Capital created Nikkei put warrants, quantoed into U.S. or Canadian dollars. I remember smugly marveling at the huge implied volatilities those dumb investors paid for the warrants. However, the dummies got the last laugh when the Nikkei finally crashed. (Fortunately, it was not at our expense. We were hedged.) This taught me that there a lot of different ways to make (and lose) money in finance. Things ended badly at Gordon Capital due to deals it was my privilege to unravel. The essence of these deals (as is the case with so many supposed arbitrages) was the sale of insurance.

Utilizing a database of closing daily FX rates over several years, I set up a simulation study of how FX options written for many different strikes, starting dates, and tenors would have fared using BlackScholes delta hedging, rehedging only at the end of each day. The results showed that there was almost no dependence between profit and loss and where FX rates ended the day. There was a great deal of variance in P&L, but this could virtually all be attributed to whether realized volatility was higher or lower than the implied volatility at which we assumed the option would be priced. I could, therefore, use these simulation results to demonstrate that hedging based upon the Black-Scholes theory could produce reasonable results without assuming continuous trading and with a frequency of hedging that would not involve ruinous transactions costs. It showed that success of a market making operation in options would be a relatively simple matter of whether volatility could be priced at reasonable levels and should not be difficult to analyze.

At least it was fascinating in the early 1990s. Option modeling in those days was Black-Scholes based, though the market knew a lot about fat-tails, and so winger options would trade above Black-Scholes values. Everyone also knew that option volatility, an important parameter in the Black-Scholes model, moved around like crazy. In those days, the market concentrated on measuring, interpreting and forecasting at-the-money option implied volatilities. Although matter-of-fact today, we recognized something was amiss here. Volatility moving around was a given. At-the-money options, however, have zero volatility curvature. But out-of-the-money options do not. [Out-ofthe-money options, hedged with at-the-money options, were therefore sources of free gamma, unless the out-of-the-money options were priced to reflect this gamma.] Indeed, there are two sources of gamma in an option: price gamma and volatility gamma.

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The Invisible Hands: Top Hedge Fund Traders on Bubbles, Crashes, and Real Money by Steven Drobny

You could have constructed a scenario from that and have had a good idea what correlations would be. But in some cases, the logical correlations were overridden by market technical factors. At the time of the Lehman collapse we owned some long-dated fixed income options where volatility was trading at all-time lows. But volatility actually fell further, a rather strange occurrence, since all other implied volatilities in the marketplace increased sharply for obvious reasons. Yet the reason for this was very technical: These cheap long-dated, long vega positions were held by leveraged players who were forced out of their positions under stress, and there were few natural buyers on the other side. That is a typical example of why you want to have your cash cushion to be able to put on or add to positions in times of stress as we did in this example.

To use an actual example from my past, buying a stock with an annual volatility of 30 percent with a plan to risk 1 percent and a goal of making 10 percent in a week is madness. Statistically, it is almost guaranteed to get stopped out. Yet, this kind of logic is very prevalent in macro, and admittedly I was guilty of it as well. You leave pennies on the table by overusing stops without understanding the implied volatility required to keep you in the trade. My approach has evolved such that I am now more concerned about how my overall portfolio does in what I call “the Titanic scenario,” where everything goes down, fundamental logic escapes the market, and risk aversion rules. I am concerned with how much I lose at the organization level if there is a repeat of the Asian crisis, if the bond market sells off like it did in 1994, or if the dot-com boom/bust or other cathartic experiences reproduce themselves.

In that scenario, I do not want to hold my store of value in a mechanism where value is defined by a politician’s judgment or conscience. If your pension fund’s base currency is gold and gold falls out of favor reverting to its supply and demand price of \$42 an ounce, will your pension fund be down 95 percent? I would be an active user of options if I were running a real money portfolio. We have had a significant retrenchment in implied volatility, creating opportunities to manage exposures using option premium. The direction depends on the skews in the volatility market at any given time. Having a hard asset base exposure provides a potential natural hedge because people will still need the basic elements that drive economic existence. So you envision being a very active and tactical pension fund manager? In this day and age, you have to be tactical.

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The Misbehavior of Markets: A Fractal View of Financial Turbulence by Benoit Mandelbrot, Richard L. Hudson

In the end, the paper was rewritten and published in the Journal of Political Economy—but only after two friends from the University of Chicago, Fama and Merton Miller, lobbied the journal’s editors to give it a second look. Their article appeared in print just after the opening bell on the Chicago Board Options Exchange in 1973. It met an eager audience. Within a few years, options dealers had incorporated its esoteric terminology, of “deltas” and “implied volatilities,” into their daily language. Texas Instruments began advertising its latest calculator as just the thing for a quick Black-Scholes calculation on the fly. An entire industry grew. With the help of the Black-Scholes formula and its many subsequent amendments, corporate financiers now routinely buy insurance, or hedge, against unwanted market problems, and not just in stocks. For instance, when General Electric signs a contract to deliver turbines to a British electricity company, it will buy pound “put” options whose value will rise if the pound falls.

Normally, to calculate an options price, you plug in a few numbers, including your estimate of how much the underlying stock price or currency rate fluctuated in the past; the suggested price falls out the back end of the formula. But if you run the equation in reverse, plugging real market prices into its back and pulling from its front the volatility that those prices would imply, you get a nonsense: a range of different volatility forecasts for the same options. A graphic example is below. It shows the implied volatility for several different flavors—different maturities and different strike prices—of the same kind of option. If Black-Scholes were right, this would be a very boring picture, one flat line for all the varieties. Instead, you see a whole range of errors, wandering across the chart. Indeed, the mistakes have a Rococo structure of their own, worthy years of study. In the options industry, where mistakes can cost millions, that is exactly what they have received.

Heresies of finance deceptive markets as flexible time as future volatility odds estimate as inevitable market bubbles as market time/place equality as market uncertainty as prices leap as risky markets as ten turbulent markets as value limited as Hermite, Charles Herodotus Heterogeneous markets Hilbert, David Hippocratic Oath Hölder, Ludwig Otto Hollywood Houthakker, Hendrik S. House fires Human Behavior and the Principle of Least Effort (Zipf) Hurst, Harold Edwin Abu Nil as birth of brief biography of Cairo arrival of formula of New York rainfall measured by water storage project of Hydrology. See Nile river flooding Implied volatility Income cotton price curve compared to curve of Pareto’s study of Index of Market Shocks Inflation Information theory probability in Initiator fractal geometry Insurer profits Intellectual property Internet bubble Bachelier influencing Cisco in risk seen in valuing options with Introduction to Mathematical Probability (Upensky) Investment bubbles inevitability of market behavior with research need for IQ bell curve of James, Jessica January effect Jegadeesh, Narashimhan Johns Hopkins Joseph Joseph effect introduction to multifractal model with Journal Business Journal of Political Economy Journal of the Royal Statistical Society Joyce, James Kendall, Maurice G.

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Ideally your starting point is a collection of unstructured, raw data that you are going to process in a way that will lead to informative features. 2.2 Essential Types of Financial Data Financial data comes in many shapes and forms. Table 2.1 shows the four essential types of financial data, ordered from left to right in terms of increasing diversity. Next, we will discuss their different natures and applications. Table 2.1 The Four Essential Types of Financial Data Fundamental Data Market Data Analytics Alternative Data Assets Liabilities Sales Costs/earnings Macro variables . . . Price/yield/implied volatility Volume Dividend/coupons Open interest Quotes/cancellations Aggressor side . . . Analyst recommendations Credit ratings Earnings expectations News sentiment . . . Satellite/CCTV images Google searches Twitter/chats Metadata . . . 2.2.1 Fundamental Data Fundamental data encompasses information that can be found in regulatory filings and business analytics. It is mostly accounting data, reported quarterly.

When alphabets are too small, information is discarded and entropy is underestimated. Figure 18.1 Distribution of entropy estimates under 10 (top), 7 (bottom), letter encodings, on messages of length 100 Distribution of entropy estimates under 5 (top), and 2 (bottom) letter encodings, on messages of length 100 Second, we can use the above equation to connect entropy with volatility, by noting that . This gives us an entropy-implied volatility estimate, provided that returns are indeed drawn from a Normal distribution. 18.7 Entropy and the Generalized Mean Here is a practical way of thinking about entropy. Consider a set of real numbers x = {xi}i = 1, …, n and weights p = {pi}i = 1, …, n, such that 0 ≤ pi ≤ 1, ∀i and . The generalized weighted mean of x with weights p on a power q ≠ 0 is defined as For q < 0, we must require that xi > 0, ∀i.

Compute the high-low volatility estimator (Section19.3.3.) on E-mini S&P 500 futures: Using weekly values, how does this differ from the standard deviation of close-to-close returns? Using daily values, how does this differ from the standard deviation of close-to-close returns? Using dollar bars, for an average of 50 bars per day, how does this differ from the standard deviation of close-to-close returns? Apply the Corwin-Schultz estimator to a daily series of E-mini S&P 500 futures. What is the expected bid-ask spread? What is the implied volatility? Are these estimates consistent with the earlier results, from exercises 2 and 3? Compute Kyle's lambda from: tick data. a time series of dollar bars on E-mini S&P 500 futures, where bt is the volume-weighted average of the trade signs. Vt is the sum of the volumes in that bar. Δpt is the change in price between two consecutive bars. Repeat exercise 5, this time applying Hasbrouck's lambda.

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The Devil's Derivatives: The Untold Story of the Slick Traders and Hapless Regulators Who Almost Blew Up Wall Street . . . And Are Ready to Do It Again by Nicholas Dunbar

The banks that lent money to hedge funds solemnly promised the Fed and other regulators that they would not make this mistake again.10 Yes, Wall Street proclaimed, derivatives markets had seized up temporarily in September 1998, and Alan Greenspan had been forced to cut the federal funds rate by an emergency three-quarters of a percentage point to get them started again. But the positive, information-transmitting qualities of new derivatives markets had not gone away. In the same way that implied volatility from option prices became a useful fear gauge for the market, transparency and consensus about more complex derivatives pricing would bring in new radar signals, as well as an assurance that the markets were safe. To allay fears that their secret webs of over-the-counter derivatives had increased the uncertainty in the market, the dealers took several measures. One idea was called netting.

Morgan’s secret weapon was that it already had such a CDO recipe in place—a trading formula it had invented to price its groundbreaking BISTRO deal in 1998. That was meant to shift credit risk off J.P. Morgan’s balance sheet, but Bill Winters and his derivatives marketers wanted to adapt it so that they could create new deals for clients, and Anshu Jain and Rajeev Misra were following close behind. The recipe was called the Gaussian copula, and just as the “implied volatility” of the Black-Scholes formula provided a shorthand for the market’s perception of risk, this model became common currency among dealers who began calling themselves correlation traders. Moody’s and the other ratings agencies had already come up with a crude way of estimating how bundles of bonds or loans might default together, by modeling them as dice or coins according to the binomial expansion technique (BET).

But what was going on in the middle? Like Moody’s BET, the Gaussian copula appeared to be just another actuarial rule of thumb, only in this case, Li was using it to connect one set of market prices to another. The difference was that while the Moody’s BET was an inflexible rule book administered by a ratings agency, the mysterious correlation parameter in the Gaussian copula put the trader in control. Like the implied volatility that emerged from the Black-Scholes formula as the market’s fear gauge, here was the tantalizing prospect of a new risk shorthand in CDO prices that supposedly broke open the complexity of the CDO. The Gaussian copula quickly became the lingua franca of credit derivatives traders, who dubbed it the Black-Scholes of credit. A few quants were troubled by that comparison. They pointed out that somewhere along the circuitous route between Merton and Li, the replication-and-arbitrage recipe that enforced the market mechanism was dropped.

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Stigum's Money Market, 4E by Marcia Stigum, Anthony Crescenzi

Because of this fact and the fact that all the variables from the Black-Scholes formula are readily available except the volatility, traders often quote options in terms of implied volatilities. When they buy an option, they are buying volatility. In this sense, options can be thought of as a bet on the future volatility of the underlying stock. This is why professional options traders are often known as “vol traders.” Can we calculate implied volatility for different options with the same underlying asset? The answer is yes, but the results actually contradict one of the major assumptions of the Black-Scholes formula. The Black-Scholes formula includes the volatility of the underlying asset. This volatility should be constant regardless of the terms of any options that are written on it. However, it is an empirical fact that the implied volatilities of in-the-money and out-of-the-money options are typically higher than the implied volatilities of at-the-money options.

Traders will typically choose some historical time horizon and use a weighted average of historical volatilities to get an unbiased estimator of the volatility. However, this is highly dependent on the horizon and weighting chosen. Implied Volatility The above discussion suggests plugging our estimate of volatility into the Black-Scholes formula and comparing the resulting price with the actual price in the market to decide whether we think the option is cheap or expensive. An alternative method would be to take the actual market price of the option (from financial news sources) and the four easily obtainable variables to “back out” the implied volatility: the volatility that the market is incorporating into the current price. Specifically, given the readily available information on an option, the price of the underlying stock, and the risk-free rate, what is the volatility of the underlying stock that would make the Black-Scholes formula true?

However, it is an empirical fact that the implied volatilities of in-the-money and out-of-the-money options are typically higher than the implied volatilities of at-the-money options. This is known as the volatility smile. Typically, the implied volatility also depends on other characteristics of the option such as its maturity. SHORTCOMINGS OF BLACK-SCHOLES One of the failures of the Black-Scholes formula—an empirical one—is discussed above. Empirically, we cannot match all the prices of options on an underlying stock by using a single volatility for the underlying stock. A volatility that correctly prices at-the-money options will drastically underprice deep in-the-money and deep out-of-the-money options. The Black-Scholes formula makes the restrictive assumption that the underlying asset has a lognormal distribution. This also implies that instantaneous returns to the underlying asset are normally distributed. However, it has been empirically shown that stock returns follow a distribution that has fatter tails than does a normal distribution.

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Capital Ideas: The Improbable Origins of Modern Wall Street by Peter L. Bernstein

A volatility estimate this far above past experience was unusual: most of the time, people expect the near future to look pretty much the way the recent past has looked. The same experiment revealed that the implied volatility of individual stocks reflected the differences in their fundamental characteristics. American Telephone and Telegraph’s implied monthly volatility was only 10.8 percent, compared to 19.3 percent for Chrysler, a company that was in deep trouble at that moment. The implied volatility for UAL was way up at 22.8 percent, reflecting the uncertainty about the airline’s takeover prospects. Note that each of these stocks had an implied volatility above the 7.4 percent of the S&P 500. That should come as no surprise. The index is a widely diversified portfolio; an individual stock, no matter how stable, is still a totally undiversified portfolio.

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Smart Money: How High-Stakes Financial Innovation Is Reshaping Our WorldÑFor the Better by Andrew Palmer

In his entertaining book on the physiology of trading, John Coates, a trader turned neuroscientist, examines the effects of testosterone and cortisol on risk appetite and aversion. One of his experiments, on the employees of a London trading floor, showed that cortisol levels in traders’ saliva jump by as much as 500 percent in a day. Cortisol is a hormone produced in response to stress: Coates found that increases in its levels were directly correlated to a financial-market measure called “implied volatility,” which functions as a gauge of uncertainty. And these, remember, are the professionals.4 All of which suggests that the logical, efficient part of our brain is not always in charge. It is extremely hard to stick to an optimal portfolio allocation when the world is going to hell. When volatility spikes, fear rises. People panic, sell assets they regard as risky, and rush for safer ones, like government bonds and cash.

See Credit-default swap Cecchetti, Stephen, 79 Church-tower principle, 207 Cigarettes, as means of payment, 5 Clark, Geoffrey Wilson, 144 Clearinghouse, 39 ClearStreet, 210 Clinical drug trials, indemnification of, xii–xiii Coates, John, 116 Code, simplification of, 63 Cohen, Ronald, 91–95, 97, 106, 108, 112 Coins, history of, 4 Collateral, xiv, 7, 38, 65, 76, 150, 177, 185, 204–206, 215 Collateralized-debt obligations (CDOs), 43, 234–235 Collective Health, 104 College graduates, earning power of, 170–171 Commenda, 7–8, 19 Commercial paper, 185 Commodity Futures Trading Commission, 54 CommonBond, 182, 184, 197 Confusion de Confusiónes (de la Vega), 24 Congressional Budget Office, 99, 169 Consumer Financial Protection Bureau overdraft fees and prepaid cards, concern about, 203–204 report on reverse mortgages, 141 survey on payday borrowing, 200 CoRI, 132 Corporate debt, in United States, 120 Corporate finance, 237–238 Correlation risk, 165 Cortisol and testosterone, effect of on risk appetite and aversion, 116 Counterparty risk, 22 Credit, industrialization of, 206 Credit Card Accountability, Responsibility, and Disclosure (Credit CARD) Act of 2009, 203 Credit cards, 203 Credit-default swap (CDS), 37, 64–65, 75, 124, 169, 238 Credit ratings, 24, 120–121, 233–236 Credit-reporting firms, 24 Credit risk, 200, 201, 237, 238 Credit scores, 47–49, 201, 216–217 Creditworthiness, xiv, 10, 12, 47, 121, 197, 202, 204, 216 Crowdcube, 152–155, 158–159, 162 Damelin, Errol, 208 Dark Ages, banking in, 11 Dark pools, 60 DCs (defined-contribution schemes), 129, 131 DE Shaw, 163 Debit cards, 204 Debt, 6, 7, 70, 149, 164 Decumulation, 138–139 Defined-benefit schemes, 129, 131 Defined-contribution (DC) schemes, 129, 131 Dependent variable, 201 Deposit insurance, 13, 43–44 Derivatives, 3, 9–10, 29–32, 38, 40 Desai, Samir, 189 Development-impact bonds, 103 Diabetes, cost of in United States, 102 Dimensional Fund Advisors, 129 Direct lending, 184 Discounting, 19 Disposition effect, 25 Diversification, 8, 12, 20, 117–119, 196, 236 Doorways to Dreams (D2D), 213–214 Dot-com boom, 148 Dow Jones Industrial Average, 40 Dow Jones Transportation Average, 40 Drug development, investment in, vii-viii, 114–115 Drug-development megafund adaptive market hypothesis and, 115–117 Alzheimer’s disease, 122 credit rating, importance of, 120–121 diversification and, 117, 119–120, 122 drug research, improvement of economics of, 114–115 financial engineering, need for, 119 guarantors for, 121 orphan diseases and, 118–119, 122 reactions to, 118 securitization and, 117–119, 122 Dumb money, comparison of to smart money, 155–158 Dun and Bradstreet, 24 Durbin Amendment (2010), 204 Dutch East India Company (VOC), 14–15, 38 E-Mini contracts, 54–55 Eaglewood Capital, 183–184 Ebola outbreak (2014), mortality rate of, 230 Ebrahimi, Rod, 210–211 Ecology, finance and, 113 Economist 2013 conference, xv on railways, 25 on worth of residential property, 70 Educational equity adverse selection in, 174, 175, 182 CareerConcept, 166 differences in funding rates, 176 enforceability, 177 in Germany, 166 Gu, Paul, 172, 175–176 income-share legislation, US Senate and, 172 information asymmetry, 174 Lumni, 165, 168, 175 Oregon, interest in income-share agreements, 172, 176 Pave, 166–168, 173, 175, 182 peer-to-peer insurance, 182 problems with, 167–168, 173–174 providers and recipients, contact between, 160, 175 risk-based pricing model, 176 student loans, 169–171 Upstart, 166–168, 173, 175, 182 Yale University and, 165 Efficient-market hypothesis, 115 Endogeneity, 239 Epidemiology, finance and, 113 Eqecat, 222 Equity, 7–8, 149–150, 186–187 Equity-crowdfunding in Britain, 154 Crowdcube, 152–155, 158–159, 162 Friendsurance, 182–183 Equity-crowdfunding in Britain (continued) herding, 159–160 social insurance, 182–183 Equity-derivatives contracts, 29 Equity-sharing, 7–8 Equity-to-assets ratio, 186 Eren, Selcuk, 73 Eroom’s law, 114 Essex County Council, 95 Eurobond market, 32 European Bank for Reconstruction and Development, 169 Exceedance-probability curve, 231–232, 232 figure 3 Exxon, 169 Facebook, 174 Fair, Bill, 47 False substitutes, 44 Fama, Eugene, 115 Fannie Mae, 48, 78, 85, 168 Farmer, Doyne, 60, 63 Farynor, Thomas, 16 FCIC (Financial Crisis Inquiry Commission), 50 Federal Deposit Insurance Corporation (FDIC), 186, 200 Federal Reserve Bank of New York, 170, 204, 205 Feynman, Richard, 115 Fibonacci (Leonardo of Pisa), 19 FICO score, 47–49 Films to rent, study of hyperbolic discounting, 133–134 Finance bailouts, 35–36 banks, purpose of, 11–14 collective-action problem in, 62 computerization of, 31–32 democratization of, 26–28 economic growth and, 33–34 fresh ideas, need for, xviii, 38–39, 80, 85–86 globalization and, 30, 225 heuristics, use of in, 45–50 illiteracy, financial, 134–135 importance of, 10 information, importance of, 10–11 inherent failings in, 241 misconceptions about, xiii–xvi panic, consequences of, 44 regulatory activity, results of, 33 risk assessment, 24, 45, 77–78 risk management, 55, 117–118, 123 as solution to real-world problems, 114 standardization, 39–41, 45, 47, 51 unconfirmed trades, backlog of, 64–65 use of catastrophe risk modeling in, 233–239 See also High-frequency trading (HFT); Internet Finance, history of bank, derivation of word, 12 Book of Calculation (Fibonacci), 19 call options, 10 Code of Hammurabi, 8 coins, 4 commodity forms of exchange, 4–5 credit and debt, 5–7 in Dark Ages, 11 democratization, 26–28 deposits, 6 derivatives, 29–32, 38 Dutch East India Company (VOC), 14–15, 38 early financial contracts, 5 early forms of finance, 3 equity contracts, 7–8 fire insurance, 16–17 first futures market, 29, 39–40 forward contracts, 38 in Greece, 11 industrialization and, 3, 27–28 inflation-protected bonds, 26 insurance, 8–10, 16–17, 20–22 interest, origin of, 5 in Italy, 9, 14 life annuities, 20–22 maritime trade and, 7–8, 14, 17, 23 payment, forms of, 4–5 put options, 9–10 railways, effect of on, 23–25 in Roman Empire, 7, 8, 11, 36 securities markets, 14 stock exchanges, 14, 24–25 Finance, innovation in absence of, xvi–xvii credit and debt, 5–7 derivatives, 9–10, 29–32 diffusion, pattern of, 45 drivers of, 22–26 equity, 7–8 importance of, 66, 242–243 insurance, 8–9, 16–17, 20–22 lessons from, 32–34 mathematical insights, 18–20 payment, forms of, 4–5 risks of, 145 stock exchanges, 14–16 Finance and the Good Society (Shiller), 242 Financial Crisis Inquiry Commission (FCIC), 50 Financial crisis of 2007–2008 causes of, xv, 34, 69 effects of, xx–xi future of finance, effect on, 243 mortgage debt, role of in, 69–70 new regulations since, 185, 187 Financial Times, quote from Chuck Prince in, 62 Fire insurance, early, 16–17 Fitch Ratings, 24 Flash Boys (Lewis), 57 Flash crash, 54–56, 63 Florida, hurricane damage in, 223, 225 Florida, new residents per day in, 225 Foenus nauticum, 8 Forward contracts, 38 Forward transactions, 15 France collapse of Mississippi scheme in, 36 eighteenth century life annuities in, 20–21 government spending in, 99 Freddie Mac, 48, 85 Fresno, California, social-impact bond pilot program in, 103–104 Friedman, Milton, 165 Friendsurance, 182–183 Fundamental sellers, 54–55 Funding Circle, 181–182, 189, 197 Futures, 29, 39–40 Galton Board, 17, 18 figure 1 Gaussian copula, 235 Geithner, Timothy, 64–65 Genentech, xii General Motors, bailout of, xi Geneva, Switzerland, annuity pools in, 21–22 Gennaioli, Nicola, 42, 44 Ginnie Mae, 168 Girouard, Dave, 166 Glaeser, Edward, 74 Globalization, finance and, 30, 225 Goldman Sachs, 61, 98, 156, 235 Google Trends, 218 Gorlin, Marc, 218 Government spending, rise in, 99–100 Governments, support for new financial products by, 168–169 Grameen Bank, 203 Greece, forerunners of banks in, 11 Greenspan, Alan, 236 Greenspan consensus, 236 Grillo, Baliano, 9 Gu, Paul, 162–164, 166, 172, 175–176 Guardian Maritime, 151 Haldane, Andy, 188 Halley, Edmund, 19–20 Hamilton, Alexander, 35–36 Hammurabi, Code of, 5, 8 Health conditions, SIB early detection programs for, 102–104 Health-impact bonds, 103–104 Hedge funds, 123, 158, 183 Hedging, 30–31, 54, 124, 129, 131, 156, 206, 227 Heiland, Frank, 73 Herding, 24, 159–160 Herengracht Canal properties, Amsterdam, real price level for, 74 Heuristics, 45–50 HFRX, 157–158 High-frequency trading (HFT) benefits of, 58 code, simplification of, 63 flash crash, 54–56 latency, attempts to lower, 53 pre-HFT era, 59–61 problems with, 56–58, 62–63 Hinrikus, Taavet, 190–191 HIV infection rates, SIB program for reduction of, 103 Holland, tulipmania in, 33, 36 Home equity, 139–140 Home-ownership rates, in United States, 85, 170 Homeless people, SIB program for, 96–97 Housing boom of mid-2000s, 148–149 Human capital contracts, 165, 167, 173–174, 176, 177 defined, 6 as illiquid asset, 177 Hurricane Andrew, effect of on insurers, 223–224, 225 Hurricane Hugo, 223 Hyperbolic discounting, 133–134, 211 IBM, 169 If You Don’t Let Us Dream, We Won’t Let You Sleep (drama), 111 IMF (International Monetary Fund), 125–126 Impact investing, 92 Implied volatility, 116 Impure altruism, 109–110 Income-share agreements, 167, 172–178 Independent variables, 201 Index funds, 40 India, CDS deals in, 37 India, social-impact bonds (SIBs) in, 103 Industrialization, effect of on finance, 3, 27–28 Inflation-protected Treasury bills, 131 Information asymmetry, 174 Innovator’s dilemma, 189 Instiglio, 103 Insurance, 8–10, 16–17, 142, 223–225 Insurance-linked securities, 222 Interbank markets, x Interest, origin of, 5 Interest-rate swaps, 29 International Maritime Bureau Piracy Reporting Centre, 151 International Monetary Fund (IMF), 125–126 International Swaps and Derivatives Association (ISDA), 40 Internet, role of in finance creditworthiness, determination of, 172–173, 202, 218 direct connection of suppliers and consumers, xviii, 32 equity crowdfunding, 152–155 income-share agreements, 172–173 ROSCAs, 210 small business loans, 216 speed and ease of borrowing, 189 student loans, 166–167 Intertemporal exchange, 6 Intuit, 218 Investment grade securities, 121 Ireland, banking crisis in, xiv–xv, 69 Isaac, Earl, 47 ISDA (International Swaps and Derivatives Association), 40 ISDA master agreement, 40 Israel, SIBs in, 97 Italy discrimination against female borrowers in, 208 financial liberalization and, 34 first securities markets in, 14 maritime trade partnerships in, 7–8 J.

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Money Mavericks: Confessions of a Hedge Fund Manager by Lars Kroijer

A simple calculation shows that buying rolling three-month put options on the S&P500 at 8 per cent out-of-the-money would only have been profitable over the past 20 years if you had ceased buying options when the implied standard deviation of the option was above 23 per cent – not a great result considering the markets were frequently down a lot during this period. Above that implied volatility threshold the options were simply too expensive for it to be a consistently profitable strategy. There is probably an argument to be made that investors who are comfortable trading options could benefit from buying deep out-of-the-money put options on the market when implied volatilities are low and thus be protected against shock events in their diversified portfolio at a manageable cost, but it is clearly not a strategy for everyone. In summary, for those without edge (and that would be most people) we probably have to accept that most our financial investments will correlate in a downturn and we should adjust our risk appetite accordingly.

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The Fear Index by Robert Harris

This has been running, in one form or another, for seventeen years. It’s a ticker, for want of a better word, tracking the price of options – calls and puts – on stocks traded in the S and P 500. If you want the math, it’s calculated as the square root of the par variance swap rate for a thirty-day term, quoted as an annualised variance. If you don’t want the math, let’s just say that what it does is show the implied volatility of the market for the coming month. It goes up and down minute by minute. The higher the index, the greater the uncertainty in the market, so traders call it “the fear index”. And it’s liquid itself, of course – there are VIX options and futures available to trade, and we trade them. ‘So the VIX was our starting point. It’s given us a whole bunch of useful data going back to 1993, which we can pair with the new behavioural indices we’ve compiled, as well as bringing in our existing methodology.

Quarry shot a look at Hoffmann. Ju-Long said, ‘We started accumulating VIX futures back in April, when the index stood at eighteen. If we had sold earlier in the week we would have done very well, and I assumed that’s what would happen. But rather than following the logical course and selling, we are still buying. Another four thousand contracts last night at twenty-five. That is one hell of a level of implied volatility.’ Rajamani said, ‘I’m seriously worried, frankly. Our book has gone all out of shape. We’re long gold. We’re long the dollar. We’re short every equity futures index.’ Hoffmann looked from one to another – from Rajamani to Ju-Long to van der Zyl – and suddenly it was clear to him that they had caucused beforehand. It was an ambush – an ambush by financial bureaucrats. Not one of them was qualified to be a quant.

Systematic Trading: A Unique New Method for Designing Trading and Investing Systems by Robert Carver

Table 34 shows that with the cheapest futures you can use a relatively tight stop, with a holding period of three days. However this would be far too expensive if you’re using spread bets. I mentioned a particular spread betting system in the introductory chapter which held positions for around a week. This would have a turnover of 52 round trips per year. The standardised cost I’m using for spread bets is 0.01 SR, so this gives costs of 52 × 0.01 = 0.52 annually. Earlier (page 150) I calculated the implied volatility target of that system at 160%. This implies you’d need to make 0.52 × 160% = 83% a year on your trading capital before costs – just to break even! Table 34 shows that with spread bets you can’t use stops with holding periods of less than about six weeks. Furthermore it’s unlikely that any trader will be happy with a holding period of more than six months. So the slowest turnover you can get is 2 round trips a year, which is the figure I used earlier in the chapter when looking at instrument selection. 133.

Euro\$ T-note Estxx V2TX MXP Corn Combined forecast (K) 15.8 20 -2.1 -11.7 2 -11.6 Subsystem position, contracts (L) 29.0 21.9 -1.2 -34.3 3.4 -12.3 11.7% 11.7% 20% 10% 23.3% 23.3% K × J ÷ 10 Instrument weight (M) 255 Systematic Trading Instrument Diversification Multiplier (N) 1.89 1.89 1.89 1.89 1.89 1.89 Portfolio instrument position, contracts (O) 6.40 4.83 -0.45 -6.45 1.49 -5.42 6 5 0 -6 1 -5 L×M×N Rounded target position contracts (P = round O) Notable positions are a large long in US treasury notes, which had been rallying strongly for a month giving positive momentum, and also had a carry forecast of over 19. The V2X also had a substantial short carry forecast of -20, and although volatility spiked substantially over the last two weeks, the very slowest EWMAC rules were still short after the steady fall in implied volatility for most of 2014. 1 December 2014 Let’s move forward to 1 December. The system has made cumulative profits of around \$10,000 so your trading capital is \$260,000 and annualised cash volatility target is \$52,000. There has now been a pronounced rally in the equity markets so we’re now modestly long Euro Stoxx, with volatility easing and V2X falling in price. Interestingly the V2X forecast hasn’t strengthened as much as you might expect, primarily because the carry forecast has weakened.

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A Demon of Our Own Design: Markets, Hedge Funds, and the Perils of Financial Innovation by Richard Bookstaber

In retrospect, it was clear that we were getting dangerously close to critical mass in terms of the effect of portfolio insurance on the market, the point where one drop in the market would trigger another, each tied to portfolio insurance hedging and each magnifying the previous one. IGNORING THE CASSANDRAS There were warning signs before any of this happened, of course. By the beginning of October there had been rising concerns that things had gone too far, even though the market had retrenched from its August highs. One indication on the technical side was that the premiums for put options were increasing because of a rise in the implied volatility of these options. On the intellectual side was John Kenneth Galbraith, who wrote an article for the January 1987 Atlantic Monthly entitled “The 1929 Parallel” that stated bluntly: “The market at this stage is inherently unstable.” Galbraith, who had lived through the first Crash, cited the market’s spectacular and constant rise, in part because of “the present commitment to seemingly imaginative, and eventually disastrous, innovation in financial structure.”

Thus the dynamic stop-loss strategy at the core of portfolio insurance was designed to transform a portfolio into one giant call option; it replicated the payoff from a call option that had an exercise price equal to the floor. 262 ccc_demon_261-270_notes.qxd 2/13/07 1:47 PM Page 263 NOTES 3. Another person who detected the emerging problem was Sandy Grossman, a brilliant researcher who was one of the early academics to move into the hedge fund world. In the months before the 1987 crash, Sandy noticed that option implied volatility was expanding relative to the volatility of the underlying stocks and that this differential was trying to tell the markets something: There was a lot of liquidity demand chasing these options, and the actual cost of hedging was probably higher than it appeared. The markets had something to say, but no one was listening. Instead, the higher option volatility meant more portfolio insurance moved toward dynamic futures hedges rather than doing the replication with the better-tracking, but apparently more expensive, exchange-traded options.

Stock Market Wizards: Interviews With America's Top Stock Traders by Jack D. Schwager

(In contrast, the time remaining until expiration and the relationship between the current market price and the strike price can be exactly specified at any juncture.) Thus, volatility must always be estimated on the basis of historical volatility data. The future volatility estimate implied by market prices (i.e., option premiums), which may be higher or lower than the historical volatility, is called the implied volatility. INDEX accounting, 84, 89, 91, 94, 324 Blake, Gil, 189, 197-98 Bloomberg financial services, 57, 59 acquisition finance, 250-52, 253 Advanced Research Projects Agency (ARPA), 262 AIDS drugs, 1 74 Amazon.com, 150-52, 272-73, 323 America Online (AOL), 43-44, 235, 259 Amerigon, 62 arbitrage, 132-33, 255-56, 267, 284 ARPAnet, 262 Asian financial crisis, 18 assets: growth of, 23-24, 207, 312-13 liquidation of, ]65 return of, ], 31 transfer of, 189 value of, 41, 63, 248-49, 253 audits, 84,91 Balance Bars, 66 balance sheets, 42, 51, 85, 268 Bankers Trust, 57, 58 banking, 141, 243-44, 249 Bombay (clothing store), 67-68 bonds: convertible, 257 government, 8, 247-48 illiquid, 7, 8, 25 interest rates and, 9, 24, 67, 105, 133-34, 135, 269, 277 junk, 82 market for, 9-10, 1 10-1 1, 144-45, 285, 309 price of, 7-8, 110-11, 144-45, 285 book value, 44, 149, 150, 165, 167 Brandywine Fund, 58-59 brokerage firms, 55-56, 61-62 Buffett, Warren, 40, 42, 157 business plans, 68-69, 91, 94, 1 18, 122, 316, 324 Business Week, \27n Canada, 1-6, 9-10, 36 capital: bankruptcy, 12-14, 24, 105-6, 122, 139, 145-46, appreciation of, 23 loss of, 68 163 Bannister, Roger, 291 Bear Stearns, 127, 131, 135-36, 138, 142 Beat the Dealer (Thorpe), 266 Beat the Market (Thorpe), 266 Bender, John, 221-38 background of, 221-25 fund managed by, 221, 222 losses of, 225 as novice trader, 225-27 profits of, 221-22, 234 strategy of, 221, 226-38, 303, 304, 306, 312 Bezos, Jeff, 272-73 preservation of, 44, 141, 217 venture, 10, 205, 207, 222, 303 capitalization: large, 34, 43-44, 150, 198-99, 320 medium, 58 revenue vs., 36-37, 45, 52 small, 24, 47, 57, 58, 59, 68, 69, 78, 198-99, 281 for trading, 10, 114-19, 120, 142, 146, 147,205, 207, 222, 303 cash flow, 43, 44, 45, 51, 149, 248 catalysts, 44-46, 52, 60, 61, 62-63, 89, 94, 114, bid/ask differentials, 134-35 215-17,220, 279-81, 307, 325 Black & Decker, 62-63 central processing unit (CPU), 261 blackjack, 266-67 Black-Scholes model, 221, 227-34 certificates, stock, 69-70 chart patterns, 181-84 331 INDEX chief executive officers (CEOs), 49, 91, 241, 244-45, 250 Cook, Marvin, 95, 96, 123-24, 126 Cook, Terri, 97, 99, 106 funds managed by, 128-29, 138, 142-47 Ingram's, 272 as novice trader, 127-38 initial public offerings (IPOs), 24, 25, 250-52, chief financial officers (CFOs), 57, 58, 59, 60, 62, 64, Cramer's commentary, 218 67, 71, 72,94, 142, 324 Church, George J., 320 Cisco, 22 Cray, Seymour, 261-62 profits of, 128-29, 132, 141-42, 147 strategy of, 138-47, 252, 300, 301, 304, 305, 306 279-81 innovation, 147 currency trading, 5, 9, 202-3 Cyrus J.

., 91-92 Gulf War, 81, 113 Einstein, Albert, 236 Elsie the Cow, 98-100 head-and-shoulders pattern, 265 Enamalon, 68 enhanced index funds, 34-36 equity trading, 6, 144-45, 257 hearing aids, 242 hedge funds, 16, 22, 59-60, 61, 80, 81-84, 146-47, 200-202, 207, 295, 306 strategy of, 241-53, 301, 306, 315 historical volatility, 330 Kidder Peabody, 32, 127-28, 135-38 Kiev, Ari, 285, 288-97, 310, 313, 315 "Knowledge Based Retrieval on a Relational Database Machine" (Shaw), 258-59 Korea, Republic of (South Korea), 18 Kovner, Bruce, 191 Lancer Offshore, 31, 50 Lauer, Michael, 30-53 FarSight, 259 Hoik, Tim, 191-92 as financial analyst, 31-32, 48 fund managed by, 30, 31, 33, 36-37, 51 Federal Reserve Bank, 271, 277 fees, 35, 55, 275 Fidelity Magellan, 34-36, 38-39, 43, 67 financial statements, xiii, 57, 185, 255 home-equity loans, 12-14, 20 How to Trade in Stocks {Livermore), 176-77 losses of, 31,42-43, 46 Fletcher, Alphonse "Buddy," Jr., 127-47 background of, 129—31 as black, 127-28, 136-38 IBM, 37, 131-32, 133, 193, 227, 232, 269, 279, 308, 327 implied volatility, 330 index funds, 34-37, 39, 50, 52, 53, 217-18, 285, 321 as novice trader, 31-32, 48-49, 50 profits of, 30-31, 50 strategy of, 37-53, 229, 253, 301, 302, 306-7, 308-9, 31 1, 314, 315, 318, 319, 320, 321, 322, 326 LensCrafters, 242 INDEX Lescarbeau, Steve, 189-206 background of, 190-91 fund managed by, 189, 200-202 as novice trader, 190-94, 202-3 profits of, 189, 198,201 strategy of, 189, 194-206, 300, 302-3, 304, 311, 315,321 leverage, 47-48, 69-70, 117, 174, 204, 222, 314-15 as novice trader, 72, 170, 188 profits of, 170, 175, 188 strategy of, 169, 171-88,299,303,304,305,312, 314, 316,317, 318 Miramar Asset Management, 76, 85-89, 93 Molecular Simulations Inc., 259 money managers, 144-45, 176, 200-202, 205, 214, 226, 234, 235 INDEX primer on, 95re, 150n, 221n, 327-30 probability curve for, 221, 227-34, 236-37 cash flow vs., 44 discrepancies in, 254, 257-58 put, 84, 85, 102, 110, 116, 118, 159-60, 325, 327, earnings vs., 21, 22, 43-44, 52, 58, 59, 60, 62, 65, 328, 329 risk of, 101, 109-10, 112, 138-40,328 strike price for, 151, 159, 160, 167, 327-28, 329, 330 volatility of, 330 66, 72-73, 79, 81, 89, 92, 94, 149, 152, 154-55, 158, 164, 165, 166-67, 172, 216, 306, 320, 321, 325 entry and exit, 157-58, 162-63, 171, 217, 220, 231,232,257,305,308 recovery of, 25-26, 44 relative, 42, 51, 81, 1 72-73, 311 run-ups of, 17, 35-40, 43-44, 229, 277, 325 leveraged buyouts (LBOs), 248-50, 253 Levitt, Jim, 80, 81 Morgan Stanley, 256, 263 multiple regression, 1 30 Lewis, Michael, 246 Liars Poker (Lewis), 246 mutual funds, 34-36, 40, 55, 82, 189-206, 207, 214 Livermore, Jesse, 176-77 Liz Claibornc, 6, 10, 12 Nasdaq, 75, 110, 158 Network Associates, 89-90 Pairs Trading: Performance- of a Relative Value Arbitrage Lloyd's Bank, 4-6 Lo, Andrew, 265 Newman, Bill, 58 New Yorker, 229 Rule (Gatev, Goetzmann, and Rouwenhort), 256n parallel processors, 258-63 lognormal curve, 237n Long-Term Capital Management (LTCM), 19—20, New York Post, 278 Pearle Vision, 242 New York Stock Exchange, 107,312 Nintendo, 276 Novell, 22 Pegasystems, 91 Peil, Tom, 208 OEX, 84 phone cards, 146 pig-at-the-trough approach, 58, 73 locked-in, 255 PIMCO, 144 probable, 255 reinvestment of, 25 risk vs., 70-71, 76, 125, 127, 128-29, 132, 134-35, 166, 207, 222, 237, 249-50, 255, 265, 269, 299, 323, 326 sources of, xiii sales vs., 44 271, 306 Love, Richard, 171 LTXX, 63 Lynch, Peter, 34, 66-67, 157, 161, 316-17 oil prices, 81 Okumus, Ahmet, 148-68 background, 148-49, 160-61 fund managed by, 149, 153, 162-63 McGuinn, Ed, 59 macroeconomics, 15—16, 18, 26, 81 Magic Faith and Healing: Studies in Primitive Psychiatry Today (Kiev), 289 Mammis, Justin, 209 losses of, 149-53, 167 as novice trader, 148-49, 153-57 management, 25-26, 45,91, 141-42, 150, 156-57, 166,250-52 manufacturers, 65—66 Marcus, Michael, 191 margin calls, 103, 105 market makers, 234, 247^18 market timers, 196 Marlin fund, 207 Masters, Michael, 207-20 background of, 209-14 fund managed by, 207 as novice trader, 209 profits of, 207, 209, 218 strategy of, 211-12, 214-20, 301, 302, 304, 307, 308 profits of, 155-56, 162 strategy of, 155-68, 306, 308, 309-10, 318, 319, 325 Okumus Opportunity Fund, 148, 153, 162-63 Olympics, 288, 289-90, 291, 297, 310 Olympic Sports Medicine Committee, U.S., 288, 289-90 One Vf on Wall Street (Lynch), 66-67, 1 57 Oppenheimer & Co., 32 options: bid/ask differentials for, 134-35 box spread for, 134 call, 101-2, 103, 108-9, 113, 124, 150-51, 325, 327, 328, 329 cumulative tick indicator for, 107-11,114 Masters, Suzanne, 214 Masters Capital Management, 207 exercised, 103, 151n, 159, 325, 327, 328 expiration of, 121-22, 151,313, 329-30 M.B.A.s, 4, 5,31,209 Men's Wearhouse, 68 Meriwether, John, 271 index, 140 "in the money" vs.

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Traders, Guns & Money: Knowns and Unknowns in the Dazzling World of Derivatives by Satyajit Das

NatWest had a reputation for aggressive trading, especially for long maturities. A key input in valuing options is volatility, which refers to how much interest rates might be expected to move around in the future. The problem was that it was da future. Traders use implied volatility to mark-to-market 05_CH04.QXD 17/2/06 4:22 pm Page 145 4 S h o w m e t h e m o n e y – g re e d l o s t a n d re g a i n e d 145 positions, which is the level quoted by other traders as reflected in the prices of options being traded. The problem for NatWest was that implied volatility for interest rate options, especially long-dated ones, is not readily available. Few people trade these products and quote prices. NatWest was one of the few traders. Volatility also needs to be adjusted for the ‘smile’ effect. If an option is deep out-of-the-money (highly unlikely to be exercised) or deep in-themoney (highly likely to be exercised) then the volatility used is increased.

Stocks for the Long Run, 4th Edition: The Definitive Guide to Financial Market Returns & Long Term Investment Strategies by Jeremy J. Siegel

THE VOLATILITY INDEX (VIX) Measuring historical volatility is a simple matter, but it is far more important to measure the volatility that investors expect in the market. This is because expected volatility is a signal of the level of anxiety in the market, and periods of high anxiety have often marked turning points for stocks. By examining the prices of put and call options on the major stock market indexes, one can determine the volatility that is built into the market, which is called the implied volatility.11 In 1993, the Chicago Board Options Exchange (CBOE) introduced the CBOE Volatility Index, also called the VIX Index or the VIX, based on actual index options prices on the S&P 500 Index, and it calculated this index back to the mid-1980s.12 A weekly plot of the VIX Index from 1986 appears in Figure 16-4. In the short run, there is a strong negative correlation between the VIX and the level of the market.

., 59i, 61 Hertzberg, Daniel, 273n High-growth country stocks, 362 Hilscher, Jens, 158n Hirshleifer, David, 325n, 326n Historical prices, 117 Hitler, Adolph, 78 Holding period, risk and, 24–27, 25i, 26i Home Depot, 155, 156, 176 Honda Motor, 176 Hong Kong and Shanghai Bank, 184 Hoover, Herbert, 227 Houdaille, Maurice, 63 Houdaille Industries, 60i, 63 Household survey, 241 HSBC Holdings, 175, 184 Hsu, Jason, 159n, 355n, 356 Hu, Jian, 241n Humphrey-Hawkins Act of 1978, 194, 195n Hussein, Saddam, 85 Hwng, Chuan-Yang, 302n Ibbotson, Roger, 84 IBM, 56i, 57, 176i IDEX, 63n Imasco, 63 Implied volatility, 281 “In the Market We Trust” (Kaufman), 86–87 In-the-money puts, 266 Income: median, in United States, 189 national, ratio of corporate profits to, market valuation and, 115–116, 116i net, 102–104 (See also Earnings) Income taxes: on corporate earnings, failure of stocks as longterm inflation hedge and, 202–203 historical, 66, 67i total after-tax returns index and, 66, 68–69, 68i, 69i Index arbitrage, 257–258 Index futures (see Stock index futures) Index mutual funds, 342 choosing, 262–264, 263i no-load, 263 Index options, 264–268 buying, 266–267 importance of, 267–268 Index Index options (Cont.): selling, 267 Indexation: capitalization-weighted, 351–352, 352i, 353–355 fundamentally weighted, capitalization-weighted indexing versus, 353–355 India: global market share of, 179i, 180, 180i growing market share of, 178 Industrial and Commercial Bank of China (ICBC), 175, 184 Industrial sector: in GICS, 53 global shares in, 175i, 177 Inflation: after World War II, 9, 10i capital gains tax and, 70–72, 71i core, 245–246 economic downturns and, 30–31 impact on financial markets, 246 money supply and, 189–190, 191i during 1970s, 9 stocks as hedges against, 199, 200i, 201–204 Inflation hedges, 199, 200i, 201–204 long-term, failure of stocks as, 201–204 Inflation-indexed bonds, 35 Inflation reports, 244–246 Information cascade, 325 Information technology sector: in GICS, 53 global shares in, 175i, 177 Informed trading, 349–350 Initial public offerings (IPOs), 154–157 Insider holdings, in capitalization-weighted indexing, 353 Institute for Supply Management (ISM), 243 373 Integrys Energy Group, 48 Intel, 38, 155, 156, 158, 176i on Nasdaq, 44 Interest costs, inflationary biases in, failure of stocks as long-term inflation hedge and, 203–204 Interest rates: on bonds, 7–9, 9i failure of stocks as long-term inflation hedge and, 201–202 Fed actions and, 196, 197i, 198–199 federal funds rate, 196 on government bonds, above dividend yield on common stocks, 95–97 during Great Depression, 8 International Accounting Standards Board (IASB), 105, 174 International incorporations, 174 International stocks, 362 Inventory accounting methods, failure of stocks as long-term inflation hedge and, 203 Investing: bear market and its aftermath and, 89–91 beginning of great bull market and, 85–86 common stock theory of, 82 early views of, 79–82 postcrash view of returns and, 83–85 practical aspects of, 360 radical shift in sentiment and, 82–83 successful, guides to, 360–363 top of bubble and, 88–89 warnings of overspeculation and, 86–88 Investment advisors, 364 Investment strategy, 363–364 Investor sentiment, 333–334, 335i Iraq, defeat in Gulf War, 85 “It Pays to Be Contrary” (Neill), 333 Jagannathan, Ravi, 241n January effect, 306–311, 307i causes of, 309–310 weakening of, 310–311 Japan: global market share of, 179, 179i, 180i market bubble in, 165 sector allocation and, 175i, 176, 177 JCPenney, 64 JDS Uniphase, 89 Jegadeesh, Narasimhan, 302 Jensen, Michael, 99n, 345 Jobs and Growth Reconciliation Act of 2003, 69–70 Johnson, Lyndon B., 193, 226 Johnson & Johnson, 144, 176i, 177 Jones, Robert, 356 Jorion, Phillipe, 12n Journal of the American Statistical Association, 81 JPMorgan Chase, 176i Kahneman, Daniel, 322–323, 328, 330 Kansas City Board of Trade, 256 Kaufman, Henry, 86, 87n Keim, Donald, 306, 309n, 316–317 Kennedy, John F., 226 Keta Oil & Gas, in DJIA, 38–39 Keynes, John Maynard, 81, 278, 285, 287, 359q, 364 Knowles, Harvey, 147 Kohlberg Kravis Roberts & Co.

pages: 517 words: 139,477

Stocks for the Long Run 5/E: the Definitive Guide to Financial Market Returns & Long-Term Investment Strategies by Jeremy Siegel

THE VOLATILITY INDEX Measuring historical volatility is a simple matter, but it is far more important to measure the volatility that investors expect in the market. This is because expected volatility is a signal of the level of anxiety in the market, and periods of high anxiety have often marked turning points for stocks. By examining the prices of put and call options on the major stock market indexes, one can determine the volatility that is built into the market, which is called the implied volatility.13 In 1993, the Chicago Board Options Exchange introduced the CBOE Volatility Index, also called the VIX Index or the VIX (first mentioned in Chapter 3), based on actual index options prices on the S&P 500 Index, and it calculated this index back to the mid-1980s.14 A weekly plot of the VIX from 1986 appears in Figure 19-5. FIGURE 19-5 The Volatility Index (VIX) 1986–2012 In the short run, there is a strong negative correlation between the VIX and the level of the market.

See also Calendar anomalies, 334–335, 337 Home Depot, 188–189, 205 Hoover, President Herbert, 247–248 Horizons, 94–97 Hot hands, 364–365 How to Beat the Market , 365 HSBC, 44, 205 Hsu, Jason, 371 Humphrey-Hawkins Act, 216 Hussein, Saddam, 13 Ibbotson, Roger, 12, 76, 192 IBM internationally, 205 market price of, 108 performance of, 173–176, 193 in S&P 500 Index, 124 in technology sector, 122 ICE (Intercontinental Exchange), 112–113 Implied volatility, 303 In-the-money puts, 286 Income tax, 141–142 Index mutual funds, 282–284 Index options buying, 286 importance of, 287–288 introduction to, 267, 284–286 selling, 286–287 Indexing capitalization-weighted, 368–371 fund performance and. See Fund performance fundamentally weighted, 369–372, 376 introduction to, 347–349 India, 58, 64–66 Individual retirement accounts (IRAs), 284 Industrials.

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Smarter Investing by Tim Hale

It gets more tricky though The upside return is delivered via the purchase of a derivative contract linked to an asset class index, often an equity index, such as the FTSE 100 index which tracks the price level of the 100 largest UK listed companies (getting a bit more complex), usually in the form of a call option, although this could be a SWAP or a futures contract (oh dear), priced using an option pricing model such as Black-Scholes that takes into account factors such as the strike price of the option, the price today, the implied volatility of the index and the time to expiry (help!). Hopefully you get the point – the mechanics of this ‘simple’ product are actually quite sophisticated and complex. Investment banks employ bright, ambitious people on big packages to structure these products. Scope exists for some tricky pricing too, given that the average adviser, let alone retail investor, will have little chance of truly understanding the underlying costs involved.

Monte Carlo Simulation and Finance by Don L. McLeish

There is a more significant pricing error in the Black-Scholes formula now, more typical of the relative pricing error that is observed in practice. Although the graph can be shifted and tilted somewhat by choosing diﬀerent variance parameters, the shape appears to be a consequence of assuming a symmetric normal distribution for returns when the actual risk-neutral distribution is skewed. It should be noted that the practice of obtaining implied volatility parameters from options with similar strike prices and maturities is a partial, though not a compete, remedy to the substantial pricing errors caused by using a formula derived from a frequently ill-fitting GENERATING RANDOM NUMBERS FROM NON-UNIFORM CONTINUOUS DISTRIBUTIONS155 Figure 3.17: Relative Error in Black-Scholes formula when Asset returns follow extreme value Black_Scholes model.

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The Power of Passive Investing: More Wealth With Less Work by Richard A. Ferri

Consequently, all financial assets are priced based on their perceived risk. The greater the perceived risk, the greater the expected return. When the perceived risk of an asset class is low, the expected return is also low relative to more risky asset classes. Table 10.1 is a sample of long-term expected returns for various asset classes. Each year, I analyzed the primary drivers of asset class long-term returns, including risk as measured by implied volatility, expected earnings growth based on expected long-term GDP and foreign sales growth, an implied 3 percent inflation rate, and current cash payouts from interest and dividends on bond and stock indexes. These factors plus others are used in a valuation model to create an estimate for risk premiums over the next 30 years. Table 10.1 Asset Class Long-Term Expected Risk and Returns The risk in each asset class tends to be fairly stable over time relative to other asset classes, and that means that the risk premium in asset classes should be fairly stable relative to each other as well.

All About Asset Allocation, Second Edition by Richard Ferri

Daily and intraday returns are what we live and breathe, and this is when the terrifying moments occur. We can see this movement in measures of daily and intraday price volatility. Figure 14-2 illustrates the CBOE Volatility Index (VIX) level from 2007 through 2009 to provide a picture of this emotion-creating phenomenon. When to Change Your Asset Allocation FIGURE 297 14-2 High Market Volatility Signals Terrifying Moments in Investors’ Lives 100 90 80 Implied volatility 70 60 50 40 30 20 10 Jan-10 Oct-09 Jul-09 Apr-09 Jan-09 Oct-08 Jul-08 Apr-08 Jan-08 Oct-07 Jul-07 Apr-07 Jan-07 0 Spikes in price volatility create fear and uncertainty in the financial markets. Typically, a VIX reading above 30 means that investors are fearful. During late 2008 and early 2009, the VIX was consistently over 50 and hit 80 on more than one day. There are a few days during every bear market when we all wonder how low the market can go.

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Fault Lines: How Hidden Fractures Still Threaten the World Economy by Raghuram Rajan

There is evidence that this board pushed Citigroup into taking more of the risk that brought it to its knees.22 Although we cannot tell whether the board was independent or in management’s pocket, it apparently did not restrain the bank’s risk taking. Finally, equity markets were not entirely unaware of the risks. From the second quarter of 2005 to the second quarter of 2007, the two-year implied volatility of S&P 500 options prices—the market’s expectations of the volatility of share prices two years ahead—was 30 to 40 percent higher than the short-term one-month volatility.23 This figure suggests that the market expected the seeming calm would end, even though the high level of the market indicated it did not place a high probability on events turning out badly for shareholders. But this is precisely how we would expect the market to behave if it believed the banks were taking on subsidized tail risk.

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Nerds on Wall Street: Math, Machines and Wired Markets by David J. Leinweber

It should come as no surprise that when a thinly traded stock is suddenly brought to the attention of millions of people trading electronically, there is likely to be more action than on typical days where it remains well below everyone’s radar. Many options traders and market makers track the message activity as a textual indicator of risk, opportunity, and volatility. A recent survey of volatility prediction, by the distinguished econometricians Clive Granger and Ser-Huang Poon, concluded that the best and most elaborate quantitative models did not rival predictions based on implied volatilities. In their conclusion, they write: “A potentially useful area for future research is whether forecasting can be enhanced by using exogenous variables.”18 The line between manipulation and volatility-inducing events is gray. It is not unreasonable for us to expect to see textually based exogenous variables for volatility prediction in the future. Indeed, these are in use today. eAnalyst: “Can Computerized Language Analysis Predict the Market?”

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Democracy and Prosperity: Reinventing Capitalism Through a Turbulent Century by Torben Iversen, David Soskice

This is not simply a matter of demand for more complex financial products by firms to hedge against uncertainty in an economy that is simultaneously more decentralized and globalized; it is also a matter of educated workers demanding easier access to credit as they pursue increasingly “nonlinear” careers with more frequent changes in jobs, house purchases, flexible mortgages and savings, time off for retraining and additional schooling, and moves back and forth between work and family (especially as child birth is delayed among high-educated women), as well as complex retirement and partial retirement choices. Because of the implied volatility in income, access to credit markets serves an increasingly important income-smoothing function that is not adequately addressed by the social protection system. (5) Macroeconomic management: as discussed in chapter 3, there has been a widespread move to central bank independence combined with inflation targeting or membership of the Eurozone. In addition, apart from the prolonged zero lower bound in the post-financial-crisis world when fiscal policy activism replaced or at least augmented monetary policy, governments have adopted some form of “consistent fiscal framework.”

Commodity Trading Advisors: Risk, Performance Analysis, and Selection by Greg N. Gregoriou, Vassilios Karavas, François-Serge Lhabitant, Fabrice Douglas Rouah

Treasury, U.S. dollar [USD] versus major currency, USD versus Japanese yen, and Goldman Sachs Commodity Index [GSCI], the corresponding exposures turn out to be very different. The S&P index yields a beta of 1.49 with the Lehman Global U.S. Treasury while the beta is 0.67 for the Barclay index. In the same vein, the CSFB index has a −0.69 beta with the USD versus major currency while the beta is 0.18 for the Barclay index. Only two indices (CSFB and HF Net) appear to exhibit significant exposure to the S&P 500 and only one (HF Net) to the evolution of the VIX (implied volatility on the S&P 500). Since the choice of index may have a significant impact on the whole investment process (from strategic allocation through performance evalua- Benchmarking the Performance of CTAs 21 tion and attribution), investors should be aware of and tackle those differences in factor exposures. In what follows, we present an index construction methodology aimed at addressing this issue.

pages: 461 words: 128,421

The Myth of the Rational Market: A History of Risk, Reward, and Delusion on Wall Street by Justin Fox

The universe has only existed an estimated 12 billion years; the New York Stock Exchange was, as of October 1987, 170 years old.13 Either stock market investors were desperately, spectacularly, unimaginably unlucky that October day, or the bell curve did not come remotely close to representing the true nature of financial market risk. This realization came quickly to some options traders. After October 19, options prices displayed what came to be called a “volatility smile.” By turning the Black-Scholes equation around, one can calculate the implied volatility of any stock from the price of its options. Put options allow one to sell a share of stock at a preset price. After the 1987 crash, put options that were well out of the money (the stock was at \$40, say, and the put allowed one to sell it for \$10) traded at prices that, according to Black-Scholes, implied a similar crash every few years. Other options on the same stocks, though, continued to trade at prices that implied less extreme volatility.

pages: 611 words: 130,419

Narrative Economics: How Stories Go Viral and Drive Major Economic Events by Robert J. Shiller

In a succession of utterances by individual financers [sic] and at bankers’ conferences, the prediction has been publicly made that the end of the speculative infatuation cannot be far off and that an inflated market is riding for a fall.4 Clearly, evidence of speculation was available to the public, which read about it in the news and talked about it on train cars. For example, in the year before its 1929 peak, the US stock market’s actual volatility was relatively low. But the implied volatility, reflecting interest rates and initial margin demanded by brokers on stock market margin loans, was exceptionally high, suggesting that the brokers who offered margin loans were worried about a big decline in the stock market.5 So the evidence of danger was there in 1929 before the market peak, but it was controversial and inconclusive. A high price-earnings ratio for the stock market can predict a higher risk of stock market declines, but it is not like a professional weather forecast that indicates a dangerous storm is coming in a matter of hours.

pages: 586 words: 159,901

Wall Street: How It Works And for Whom by Doug Henwood

National Economic Research Associates Inc., White Plains, N.Y. Chen, Nai-Fu. (1991). "Financial Investment Opportunities and the Macroeconomy,"/our- nal of Finance 46. pp. 529-554. Chen, Nai-Fu, Richard Roll, and Stephen A. Ross (1986). "Economic Forces and the Stock Market," fournal of Business 59, pp. 383-403-Cherian, Joseph A., and Robert A. Jarrow (1994). "Options Markets, Self-Fulfilling Prophecies, and Implied Volatilities," mimeo, Boston University School of Management (October). BIBLIOGRAPHY Chick, Victoria (1976). Transnational Enterprises and the Evolution of the International Monetary System, University of Sydney, Transnational Corporations Research Project, Research Monograph No. 5. — (1983). Macroeconomics After Keynes: A Reconsideration of the General Theory (Deddington, U.K.: Philip Allan Publishers).

pages: 593 words: 189,857

Stress Test: Reflections on Financial Crises by Timothy F. Geithner

But either by netting out cash, a risk-free asset, or paying down \$500 billion in liabilities, the bank’s leverage ratio would be reduced to 20:1. 2 capital alone wouldn’t stop a run already in progress: We had no way of determining then how much capital would be enough, but we knew we wouldn’t have unlimited amounts of capital to deploy. That meant that guarantees were needed alongside capital to credibly backstop the system. 3 “fear index”: The Chicago Board Options Exchange Market Volatility Index, or VIX, is a measure of market volatility popularly referred to as the “fear index.” The measure is based on the implied volatility of options on the S&P 500 index of stocks. The VIX captures investor expectations of near-term stock market volatility—how uncertain investors are about whether and how far the S&P will rise or fall. 4 two complex new Maiden Lane vehicles: Among AIG’s major liquidity needs were their securities lending operations and the credit default swaps written by AIG Financial Products on collateralized debt obligations.

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Big Debt Crises by Ray Dalio

The report, published by the Mortgage Bankers Association, came as the Federal Reserve held a hearing on what regulators could do to address aggressive abusive lending practices.” –New York Times June 21, 2007 Bear Stearns Staves Off Collapse of 2 Hedge Funds -New York Times June 23, 2007 Bear Stearns to Bail Out Troubled Fund -New York Times July 3, 2007 Glancing at Implied Vols “Recent market action has begun to show a slight pickup in implied volatility across all markets, but these increases have come from levels that were as low as they have been in more than 10 years. Looking broadly across markets, we continue to see very low expected future currency, bond, and commodity volatility, while future expected volatility in the equities is low but closer to normal relative to history.” –BDO July 13, 2007 Fitch May Downgrade Bonds Tied to Subprime Mortgages -Bloomberg July 14, 2007 Dow and S.