# implied volatility

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Python for Finance by Yuxing Yan

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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

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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

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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.

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

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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

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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

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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.

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Market Risk Analysis, Quantitative Methods in Finance by Carol Alexander

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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 = 11613 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

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© 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

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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.

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

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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.

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

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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.

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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

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

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

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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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Derivatives Markets by David Goldenberg

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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. OPTION PRICING IN CONTINUOUS TIME 585 B. The Implied Volatility Estimator Method Volatility is one of the key parameters in the Black–Scholes formula, but it is unobservable.

In order to implement it, all we have to do is plug all the parameters, except , 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 that can be iteratively solved for the implied volatility estimator, which we will denote by IV. The brief version of this procedure is, Ct,Black–Scholes=Ct,Market implies IV. End of Chapter Exercises 4 and 5 implement this procedure. The IV estimator turns out empirically to be a better estimator than the historical , 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 assumption. If is constant, then it is also constant across options with different exercise prices and IV 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 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 lognormal 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 OPTION PRICING IN CONTINUOUS TIME 589 other than a standard GBM.

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Tools for Computational Finance by Rüdiger Seydel

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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.

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

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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

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

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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.

Analysis of Financial Time Series by Ruey S. Tsay

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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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Hedge Fund Market Wizards by Jack D. Schwager

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

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

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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.

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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

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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.

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Market Sense and Nonsense by Jack D. Schwager

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

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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 σR τ St φ(d1 ) − Ke−rτ φ(d2 ) 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 Rise of the Quants: Marschak, Sharpe, Black, Scholes and Merton by Colin Read

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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.

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

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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

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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 by Benoit Mandelbrot

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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.

pages: 350 words: 103,270

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

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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.

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

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

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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.

pages: 312 words: 91,538

The Fear Index by Robert Harris

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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.

pages: 349 words: 134,041

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

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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.

pages: 289 words: 113,211

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

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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.

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

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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.

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

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(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.

pages: 358 words: 106,729

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

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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.

Monte Carlo Simulation and Finance by Don L. McLeish

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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.

All About Asset Allocation, Second Edition by Richard Ferri

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

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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.

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

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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).

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

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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?”

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

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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.

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

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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.

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

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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.