# Black-Scholes formula

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A Primer for the Mathematics of Financial Engineering by Dan Stefanica

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From definition (3.6), we know that P(a :'S X :'S b) = Y~J-L P(a~X~b) Note that the constant random variable X = IL is a degenerate normal variable with mean IL and standard deviation O. h(x) = 93 FINANCIAL APPLICATIONS The Black-Scholes formula. CHAPTER 3. PROBABILITY. BLACK-SCHOLES FORMULA. 94 Assume that the price of the underlying asset has lognormal distribution and volatility a, that the asset pays dividends continuously at the rate q, and that the risk-free interest rate is constant and equal to r. Let C (S, t) be the value at time t of a call option with strike K and maturity T, and let P(S, t) be the value at time t of a put option with strike K and maturity T. Then, Implied volatility. The concept of hedging. ~-hedging and r-hedging for options. Implementation of the Black-Scholes formula. 0(8, t) The Black-Scholes formula P(8, t) The Black-Scholes formulas give the price of plain vanilla European call and 4 put options, under the assumption that the price of the underlying asset ~as lognormal distribution.

We call this routine cum_disLnormal(tt), and include its pseudocode in Table 3.1. The pseudocode for implementing the Black-Scholes formulas using the routine cum_disLnormal(t) for approximating N(t), is given in Table 3.2. d l = (In(*)+(r q+~2)(T-t))/((J"VT-t);d2=dl-(J"VT-t C = Se-q(T-t)cum_dist-llormal( dd - K e-r(T-t)cum_dist-llormal( d 2 ) P = Ke-r(T-t)cum_disLnormal( -d2 ) - Se-q(T-t)cum_dist-llormal( -dl ) Example: Use the Black-Scholes formula to price a six months European call option with strike 40, on an underlying asset with spot price 42 and volatility 30%, which pays dividends continuously, with dividend rate 3%. Assume that interest rates are constant at 5%. CHAPTER 3. PROBABILITY. BLACK-SCHOLES FORMULA. 110 Price a six months European put option with strike 40 on the same asset, using the Black-Scholes formula. Check whether the Put-Call parity is satisfied.

RISK-NEUTRAL DERIVATION OF BLACK-SCHOLES (4.47) While risk-neutral pricing does not hold for path-dependent options or American options, it can be used to price plain vanilla European Call and Put options, and is one way to derive the Black-Scholes formulas. CHAPTER 4. LOGNORMAL VARIABLES. RN PRICING. 134 Applying (4.45) to plain vanilla European options, we find that C(O) = e- rT ERN[max(8(T) - K,O)]; P(O) = e- rT ERN[max(K - 8(T), 0)], (4.48) (4.49) where the expected value is computed with respect to 8(T) given by (4.47). We derive the Black-Scholes formula for call options using (4.48). The Black-Scholes formula for put options can be obtained similarly from (4.49). By definition, max(8(T) _ K,O) _ - {8(T) - K, if 8(T) 2 K; 0, otherwise. d1 = d2 + ~VT = In ( S~) ) + (r _q + ~2) vfr ~ T T . (4.53) Note that the term d 1 from (4.53) is the same as the term d 1 from the BlackScholes formula given by (3.55) if t = o. Formula (4.52) becomes 1 = 8(0)e- qT . rn= v 27r 1 00 -d1 e- Ty2 dy - 1 00 Ke- rT 1 rn= V 27r 2 e- Tx dx

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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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Here, with a 3M = 90 days option, we have: and similarly: moneyness: strike: −3 60.819 −2 68.058 −1 75.296 —0 82.535 = F —1 89.774 —2 97.012 —3 104.251 Figure 10.5 Determination of a moneyness scale These values are shown in Figure 10.5 together with two strikes, K′′ and K′, respectively at: K′ = S − €10 = 73.06 corresponding to a moneyness of −1.31 K′′ = S + €10 = 93.06 corresponding to a moneyness of +1.45. The moneyness measure is mainly used with respect to the option smile, as developed in Chapter 12, Section 12.1.3. 10.2.5 Beyond the Black–Scholes formula The Black–Scholes formula is an answer to the diffusion equation (cf. Eq. 10.6, for call options) leading to an option valuation subject to the very specific assumptions as set in Section 10.2.1. This formula, and its variants, is called an “analytical” solution to option pricing, since if suffices to replace the variables of the formula by their values relating to the option to be priced. Moreover, the fact remains that the analytic – also called “close form” or “closed-form” – Black–Scholes formula presents the advantage of allowing a straightforward calculation of options sensitivities (cf. Section 10.5). However, in many instances, some of the Black–Scholes assumptions must be relaxed or modified, for example in the case of: incorporating dividend payments (options on equities); American options; options on interest rates, volatility, or other underlyings, that do not fit with the geometric general Wiener process (see Chapters 11 and 12); second generation options (cf.

They all pursue the same objective of modeling in a more or less realistic way how the underlying spot price will move over time, to compute what should be the option fair/theoretical price accordingly. 10.2 THE BLACK–SCHOLES FORMULA 10.2.1 Introduction F. Black and M. Scholes were the first to publish, 2 in 1973, a well-grounded formula for computing call and put options prices. The way their formula is established is useful to better understand the underlyings of option pricing. This formula is subject to rather restrictive hypotheses, which may be questioned in some circumstances but at least it constitutes a robust pricing tool, not necessarily the case for further, more complex, pricing models (cf. Chapter 15, Section 15.1), whose sophistication is also synonymous of real difficulties to properly assess correct values to their ingredients. The Black–Scholes formula is applicable to European options only, and provided the underlying financial instrument offers no return during the lifetime of the option: for example, a stock delivering no dividend during such period, or any non-financial commodity.

Let us compute a European call option maturing in 90 days (or = 90/365 = 0.2466 year) on L'OREAL stock quoting EUR 64.5 (data as of Jan 06), with an ATMS strike price of EUR 64.5; the risk-free interest rate is 2.514% p.a. and the stock volatility is 11.9% p.a. Equations 10.7–10.9 give: because ln(S/K) = ln(64.5/64.5) = ln1 = 0, d1 = (0.02514 + 0.1192/2) × 0.2466/0.119√0.246 = 0.134457 d2 = d1 − 0.119√0.2466 = 0.075363 hence, using the cumulative normal distribution N(0, 1), N(d1) = 0.553479, and N(d2) = 0.530037 → C = 64.5 × 0.553479 − 64.5 × e−0.02514×0.2466 × 0.530037 = €1.72 (rounded). 10.2.2 Variants of the Black–Scholes formula The Black–Scholes formula can be extended to European options on any kind of underlying offering a return ≠ 0, provided that its process can be reasonably modeled by a geometric Wiener process. This extension is valid if the underlying return can be considered as continuous in time.3 This will be the case of a LIBOR rate of return, for example. Let precise the above r return, by calling it rm, as the market rate of return, and calling ru the return of the underlying, both up to the maturity T of the option.

Monte Carlo Simulation and Finance by Don L. McLeish

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/(dif+BLSPRICE(So,strike,r,T,sigma,0))]; plot(strike/So,re) xlabel(’Strike price/initial price’) ylabel(’relative error in Black Scholes formula’) The relative error in the Black-Scholes formula obtained from a simulation of 100,000 is graphed in Figure 3.14. The logistic distribution diﬀers only slightly GENERATING RANDOM NUMBERS FROM NON-UNIFORM CONTINUOUS DISTRIBUTIONS151 Figure 3.14: Relative Error in Black-Scholes price when asset prices are loglogistic, σ = .4, T = .75, r = .05 from the standard normal, and the primary diﬀerence is in the larger kurtosis or weight in the tails. Indeed virtually any large financial data set will diﬀer from the normal in this fashion; there may be some skewness in the distribution but there is often substantial kurtosis. How much diﬀerence does this slightly increased weight in the tails make in the price of an option? Note that the Black-Scholes formula overprices all of the options considered by up to around 3%.

Assume that we simulated the asset prices under this model and then valued a call option, say. Then since ln(ZT /Z0 ) has a N ((c − σ2 )T, σ2 T ) distribution 2 we could use the Black-Scholes formula to determine the conditional expected value 268 CHAPTER 5. SIMULATING THE VALUE OF OPTIONS Z E0 [exp{− T rt dt}(ZT − K)+ |rs , 0 < s < T ] (5.18) 0 = EE0 [(S0 e(c−r)T eW − e−rT K)+ |rs , 0 < s < T ], where W has a N (−σ2 T /2, σ 2 T ) (c−r)T = E[BS(S0 e 1 , K, r, T, σ)], with r = T Z 0 Here, r is the average interest rate over the period and the function BS is the Black-Scholes formula (5.2). In other words by replacing the interest rate by its average over the period and the initial value of the stock by S0 e(c−r)T , the Black-Scholes formula provides the value for an option on an asset driven by (5.17) conditional on the value of r. The constant interest rate model is a useful control variate for the more general model (5.16).

It follows that the conditional distribution is lognormal with mean η = Xt q(t, T ) and volatility qR T 2 σ (u)du. parameter t We now derive the well-known Black-Scholes formula as a special case of 2.53. For a call option with exercise price E, the payoﬀ function is V0 (ST ) = max(ST − E, 0). Now it is helpful to use the fact that for a standard normal random variable Z and arbitrary σ > 0, −∞ < µ < ∞ we have the expected value of max(eσZ+µ , 0) is eµ+σ 2 /2 Φ( µ µ + σ) − Φ( ) σ σ (2.57) where Φ(.) denotes the standard normal cumulative distribution function. As a result, in the special case that r and σ are constants, (2.53) results in the famous Black-Scholes formula which can be written in the form V (S, t) = SΦ(d1 ) − Ee−r(T −t) Φ(d2 ) where d1 = √ log(S/E) + (r + σ 2 /2)(T − t) √ , d2 = d1 − σ T − t σ T −t (2.58) 90 CHAPTER 2.

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

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We then see how this discrete model can be used as an approximation to a continuous model, and we deduce the Black-Scholes formula for the price of a call option via a limiting argument. Having developed the Black-Scholes formula, we then discuss in Chapter 4 its flaws and how these flaws affect its use in practice. This chapter is very much a foretaste for chapters near the end of the book where we study alternative models of price evolution which try to compensate for the shortcomings of the BlackScholes model. In Chapter 5, we step up a mathematical gear and introduce the Ito calculus. With this calculus we introduce the geometric Brownian motion model of stock price evolution and deduce the Black-Scholes equation. We then show how the BlackScholes equation can be reduced to the heat equation. This yields a derivation of the Black-Scholes formula. In Chapter 6, we step up another mathematical gear and this is the most mathematically demanding chapter.

(6.113) The final expectation is just the probability that ST is greater than K in the S-numeraire martingale measure. Using the solution of the SDE for a Brownian motion with drift, this is equal to the probability that Soe(' +a212)T+a,ITN(o,1) > K. (6.114) A straightforward computation gives us the first term in the Black-Scholes formula. Risk neutrality and martingale measures 170 To get the second term in the Black-Scholes formula, it is easier to use B as numeraire. Note that this neatly explains the division of the Black-Scholes formula into two terms with coefficients So and e-',T K. Note also that the computation of the first term is made substantially easier by the use of the correct numeraire. O For each complete market, we now have multiple martingales measure each one associated to a choice of numeraire. How do they relate to each other?

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

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Then HBS,c(x, K, σ, T, r)=xΦ(d+)−Ke−rTΦ(d−), HBS,p(x, K, σ, T, r)=HBS,c(x, K, σ, T, r)−x+Ke−rT, (5.18) where and where (5.19) This is the celebrated Black-Scholes formula. Note that the formula for put follows from the formula for call from the put-call parity (Corollary 5.51). Numerical calculation via the Black-Scholes formula MATLAB code for Φ(·) function[f]=Phi(x) N=400; eps=abs(x+4)/N; f=0; pi=3.1415; for k=1:N; y=x-eps*(k-1); f=f+eps/sqrt(2*pi)*exp(-y^2/2); end; Here N=400 is the number of steps of integration that defines preciseness. One can try different N=10, 20, 100,…. (See also the MATLAB erf function.) © 2007 Nikolai Dokuchaev Mathematical Finance 96 MATLAB code for Black-Scholes formula (call) function[x]=call(x, K, v, T, r) x=max(0, s-K); if T>0.001 d=(log(s/K)+T*(r+v^2/2))/v/sqrt(T); d1=d-v*sqrt(T); x=s*Phi(d)-K*exp(-r*T)*Phi(dl); end; MATLAB code for Black-Scholes formula (put) function[x]=put(x,K,v,T,r) x=call(x,K,v,T,r)-s+K*exp(-r*T); end; Problem 5.57 Assume that r=0.05, σ=0.07, S(0)=1.

Here E* is the expectation defined by the risk-neutral probability measure (this measure gives the same probability distribution of P(·) as the original measure for the case when r=a). Find an analogue of the Black-Scholes formula for the call option. (Hint: (1) find the expectation via calculation of an integral with a certain (known) probability density; (2) for simplicity, you may take first r=0.) Black-Scholes formula Problem 5.79 Let HBS,c(s, K, r, T, σ) and HBS,p(s, K, r, T, σ) be the Black-Scholes prices for call and put options respectively. Here is the volatility, r≥0 is the bank interest rate, s=S(0) is the initial stock price, K is the strike price. (i) Are these functions increasing (decreasing) in s? Prove. (Hint: use the basic riskneutral valuation rule.) (ii) Find the limits for these functions as: (a) T→+∞; (b) σ→+∞; (c) T→+0; (d) σ→+0. (Hint: use the Black-Scholes formula.) Challenging problems Problem 5.80 Let ã(t)≡â≠0 not depend on time, and let a self-financing strategy be defined in closed-loop form such that where is the corresponding discounted wealth.

MATLAB code for the price of an option with payoff F(x)=1+cos(x) function [f]=option(s,r,T,v) N=800; eps=0.01; f=0; pi=3.1415; for k=1:800; x=-4+eps*(k-1); f=f+eps/sqrt(2*pi*T) *exp(-x^2/(2*T))*(1+cos(s*exp((r-v^2/2)*T+v*x))); end; f=exp(-r*T)*f; © 2007 Nikolai Dokuchaev Continuous Time Market Models 95 Problem 5.56 (i) Write your own code for calculation of the fair price for payoff F(S(T)) where F(x)=|sin(4x)|ex. (ii) Let (S(0), T, σ, r)=(2, 1, 0.2, 0.07). Find the option price with payoff F(S(T)). 5.9.5 Black-Scholes formula We saw already that the fair option price (Black-Scholes price) can be calculated explicitly for some cases. The corresponding explicit formula for the price of European put and call options is called the Black—Scholes formula. Let K>0, σ>0, r≥0, and T>0 be given. We shall consider two types of options: call and put, with payoff function ψ where ψ=(S(T)−K)+ or ψ=(K−S(T))+, respectively. Here K is the strike price. Let HBS,c(x, K, σ, T, r) and HBS,p(x, K, σ, T, r) denote the fair prices at time t=0 for call and put options with the payoff functions F(S(T)) described above given (K, σ, T, r) and under the assumption that S(0)=x.

Mathematics for Finance: An Introduction to Financial Engineering by Marek Capinski, Tomasz Zastawniak

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What we have just derived is the celebrated Black–Scholes formula for European call options. The choice of time 0 to compute the price of the option is arbitrary. In general, the option price can be computed at any time t < T , in which case the time remaining before the option is exercised will be T − t. Substituting t for 0 and T − t for T in the above formulae, we thus obtain the following result. Theorem 8.6 (Black–Scholes Formula) The time t price of a European call with strike price X and exercise time T , where t < T , is given by C E (t) = S(t)N (d1 ) − Xe−r(T −t) N (d2 ) 8. Option Pricing 189 with 1 2 ln S(0) (T − t) X + r + 2σ √ d1 = , σ T −t 1 2 ln S(0) (T − t) X + r − 2σ √ d2 = . (8.11) σ T −t Exercise 8.15 Derive the Black–Scholes formula P E (t) = Xe−r(T −t) N (−d2 ) − S(t)N (−d1 ), with d1 and d2 given by (8.11), for the price of a European put with strike X and exercise time T .

The precise relationship comes from a version of the Central Limit Theorem: It can be shown that the option price given by the Cox–Ross– Rubinstein formula tends to that in the Black–Scholes formula in the continuous time limit described in Chapter 3. Figure 8.2 Option price C E in continuous and discrete time models as a function of time T remaining before the option is exercised 190 Mathematics for Finance Rather than looking at the details of this limit, we just refer to Figure 8.2 for illustration. It shows the price C E of a European call with strike X = 100 on a stock with S(0) = 100, σ = 0.3 and m = 0.2. (Though m is irrelevant for the Black–Scholes formula, it still features in the discrete time approximation.) The continuous compounding interest rate is taken to be r = 0.2. The option price is computed in two ways, as a function of the time T remaining before the option is exercised: a) (solid line) from the Black–Scholes formula for T between 0 and 1; b) (dots) using the Cox–Ross–Rubinstein formula with T increasing from 0 to 1 in N = 10 steps of duration τ = 0.1 each; the discrete growth rates for each step are computed using formulae (3.7).

As a result, the price of the American call is higher than that of the European call. Exercise 8.12 Compute the prices of European and American puts with exercise and strike price X = 14 dollars expiring at time 2 on a stock with S(0) = 12 dollars in a binomial model with u = 0.1, d = −0.05 and r = 0.02, assuming that a dividend of 2 dollars is paid at time 1. 8.3 Black–Scholes Formula We shall present an outline of the main results for European call and put options in continuous time, culminating in the famous Black–Scholes formula. Our treatment of continuous time is a compromise lacking full mathematical rigour, which would require a systematic study of Stochastic Calculus, a topic 186 Mathematics for Finance treated in detail in more advanced texts. In place of this, we shall exploit an analogy with the discrete time case. As a starting point we take the continuous time model of stock prices developed in Chapter 3 as a limit of suitably scaled binomial models with time steps going to zero.

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Tools for Computational Finance by Rüdiger Seydel

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These may be elementary evaluations of functions like the logarithm or the square root such as in the Black-Scholes formula, or may consist of a subalgorithm like Newton’s method for zero ﬁnding. There is hardly a purely analytic method. The ﬁnite-diﬀerence approach, which approximates the surface V (S, t), requires intermediate values for 0 < t < T for the purpose of approximating V (S, 0). In the ﬁnancial practice one is basically interested in V (S, 0) only, intermediate values are rarely asked for. So the only temporal input parameter is the time to maturity T − t (or T in case the current time is set to zero, t = 0). Recall that also in the Black-Scholes formula, time only enters in the form T − t (−→ Appendix A4). So it makes sense to write the formula in terms of the time to maturity τ , τ := T − t , which leads to the compact version of the Black-Scholes formulas (A4.10), 4.8 Analytic Methods 1 S √ + r+ log K σ τ 1 S + r− d2 (S, τ ; K, r, σ) := √ log K σ τ d1 (S, τ ; K, r, σ) := # σ2 τ 2 # σ2 τ 2 167 (4.40) VPeur (S, τ ; K, r, σ) = −SF (−d1 ) + Ke−rτ F (−d2 ) VCeur (S, τ ; K, r, σ) = SF (d1 ) − Ke−rτ F (d2 ) (dividend-free case).

An immediate candidate for the lower bound V low is the value VPeur provided by the Black-Scholes formula. Thus, VPeur (S, τ ; K) ≤ VPam (S, τ ; K) ≤ VPam (S, τ ; Kerτ ) The latter of the inequalities is the monotonicity with respect to the strike, see Appendix D1. Following [Mar78], an American put with strike Kerτ rising exponentially with τ at the risk-free rate is not exercised, so VPam (S, τ ; Kerτ ) = VPeur (S, τ ; Kerτ ) , 168 Chapter 4 Standard Methods for Standard Options which serves as upper bound. This allows to apply the Black-Scholes formula (4.40) to the European option and provides the upper bound to VPam (S, t; K). The resulting approximation formula is V := αVPeur (S, τ ; Kerτ ) + (1 − α)VPeur (S, τ ; K) . (4.41) The parameter α will depend on S, τ, K, r, σ, so does V . Actually, the BlackScholes formula (4.40) suggests that α and V only depend on the three parameters S/K, rτ , and σ 2 τ .

To match the extreme cases, γ should vanish for τ = 0, and γ ≈ 1 for large values of τ . [Joh83] suggests σ2 τ , b0 + b1 b0 = 1.04083 , b1 = 0.00963 . γ := σ2 τ (4.44) The constants b0 and b1 are again obtained by a regression analysis. The analytic expressions of (4.43), (4.44) provide an approximation of Sf , and then by (4.42), (4.41) an approximation of VPam for S > Sf , based on the Black-Scholes formulas (4.40) for VPeur . 4.8 Analytic Methods Algorithm 4.16 169 (interpolation) For given S, τ, K, r, σ evaluate γ, S f , β, α . Evaluate the Black-Scholes formula for VPeur for the arguments in (4.41). Then V from (4.41) is an approximation to VPam . Numerical experiments show that the approximaton quality of S f is poor. But for S not too close to S f the approximation quality of V is quite good. As reported in [Joh83], the error is small for rτ ≤ 0.125, which is satisﬁed for average values of the risk-free rate r and time to maturity τ .

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

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Through his clever mathematical construct, Merton was able to relax some of the strongest assumptions that offended those most critical of the Black-Scholes formula. Consequently, he was able to increase the generality of the Black-Scholes formula. He demonstrated that dividends and early calls could also be accommodated by his enveloping portfolio. He managed to show that this enveloping portfolio was smooth and hence the differential equation for the enveloping portfolio could be solved. In fact, this enveloping portfolio of an option for an underlying stock that may exhibit jumps and dividends nonetheless follows the same differential equation defined by Black. Merton demonstrated that the Black-Scholes formula is accurate, even if the option may need to be continuously rebalanced at each point in time to accommodate jumps or ex-dividend repricing. 156 The Rise of the Quants We might imagine that there could be great profits to be had if Black, Scholes, and Merton kept their equation secret and started their own investment firm.

At the time, financial theorists, especially Merton, were increasingly skeptical of the static and backward-looking characteristics of the CAPM, and were seeking to create dynamic extensions of it. Merton was convinced that the Black-Scholes formula, which was a special case of Spreckle’s derivation, must be a further special case of a more general and dynamic CAPM. Black had already realized that it might be safer to interpret the formula as Scholes had done while wearing his Chicago arbitrage and efficient market lenses. The intuition Meanwhile, Merton managed to overcome his skepticism of any approach based on the static CAPM model.1 He reasoned that, even if he looked from the dynamic CAPM perspective at a portfolio that is readjusted at each period in time, he could mimic the option returns specified by the Black-Scholes formula by combining positions on the underlying stock with borrowing at the risk-free interest rate.

They were left with the Black-Scholes equation: C(S,t) = SN(d1 ) − Ke− r (t ∗ −t) N(d 2 ) where K is the strike price, d1 ⫽ (ln(S/C) ⫹ (t*⫺t)(r ⫹ v2/2))/(r(t*⫺t)1/2), d2 ⫽ d1 ⫺ v(t*⫺t)1/2, t* is the expiration date, and the optimal hedge ␦C/␦S is simply N(d1). As in Spreckle’s solution and, for that matter, Bachelier’s derivation 70 years earlier, N(d) is the normalized cumulative probability distribution function. Alternative derivations of the Black-Scholes formula Black and Scholes had made a few implicit assumptions in their analysis. First, they assumed that the number of shares outstanding does not change before the settlement date. If so, this would dilute the price of the stock and affect the option price. Similarly, they assumed that no dividends are paid and that the stock evolution follows a log-normal random walk with a constant drift and volatility.

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The Misbehavior of Markets by Benoit Mandelbrot

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He spends much of each day studying the fast-changing “volatility surface” of the options market—an imaginary 3-D graph of how price fluctuations widen and narrow as the terms of each option contract vary. By the Black-Scholes formula, there should be nothing of interest in such a surface; it should be flat as a pancake. In fact it is a wild, complex shape. Tracking it and predicting its next changes are fundamental ways in which Citigroup’s options traders make money. About 10 percent of the world FX options market is of a class called exotic. It has mind-numbing combinations of precise options terms tailor-made to pay off only under certain circumstances. These combinations are obscure to most people, but perhaps just what the CFO of GM needs to guard against one particular risk that worries him in his company’s yen-based cash-flow. None of this would exist if the original Black-Scholes formula were accurate. Of course, the formula remains important; it is the benchmark to which everyone in the market refers, much the way, say, people talk about the temperature in winter even though whether they actually feel cold also depends on the wind, the snow, the clouds, their clothing, and their health.

In the options industry, where mistakes can cost millions, that is exactly what they have received. Hundreds of scholarly papers, several textbooks, and scores of financial conferences have been devoted to studying the errors. A wide error range. This diagram, from Schoutens 2003, plots the volatility that the standard Black-Scholes formula would infer from the market prices for one family of options. All the curves here show the same type of option, but with different times, T, to maturity. The “strike” price at which each contract can be exercised is on the bottom scale; the volatility that the Black-Scholes formula infers from the data is on the vertical scale, in standard deviations. If the formula were right, there would be nothing much to see: just one flat line. Improving or replacing Black-Scholes is one of the liveliest subdisciplines in mathematical finance.

In hindsight, the newspaper appears to have underestimated, and thus under-played, the importance of the exchange’s opening. 72 “The answer came…” From the eulogy of Black, Scholes 1995. The account of their discovery given here is based on the published recollections of the participants, including Black 1989, Scholes 2001, and the autobiographical essays Merton and Scholes 1997. 73 “The Black-Scholes formula permitted…” The Black-Scholes formula looks complex, but working with it is a simple matter of plugging numbers into their proper places in a spreadsheet or calculator. The price of a call option to buy a stock at a specific price and time is: Here, C0 is the price of the call option; S0 is the current stock price; X is exercise price at which the option contract allows you to buy the stock; r is the risk-free interest rate; and T is the time to maturity.

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Frequently Asked Questions in Quantitative Finance by Paul Wilmott

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First let me reassure you that you won’t theoretically lose money in either case (or even if you hedge using a volatility somewhere in the 20 to 40 range) as long as you are right about the 40% and you hedge continuously. There will however be a big impact on your P&L depending on which volatility you input. If you use the actual volatility of 40% then you are guaranteed to make a profit that is the difference between the Black-Scholes formula using 40% and the Black- Scholes formula using 20%. V(S,t;σ) - V(S,t;σ̃), where V(S,t;σ) is the Black-Scholes formula for the call option and σ denotes actual volatility and σ̃ is implied volatility. That profit is realized in a stochastic manner, so that on a marked-to-market basis your profit will be random each day. This is not immediately obvious, nevertheless it is the case that each day you make a random profit or loss, both equally likely, but by expiration your total profit is a guaranteed number that was known at the outset.

The expectation being over the risk-neutral measure for the diffusion but the real measure for the jumps. There is a simple solution of this equation in the special case that the logarithm of J is Normally distributed. If the logarithm of J is Normally distributed with standard deviation σ ′ and if we writek = E [J − 1] then the price of a European non-path-dependent option can be written as In the above and and VBS is the Black-Scholes formula for the option value in the absence of jumps. This formula can be interpreted as the sum of individual Black-Scholes values each of which assumes that there have been n jumps, and they are weighted according to the probability that there will have been n jumps before expiry. Jump-diffusion models can do a good job of capturing steepness in volatility skews and smiles for short-dated option, something that other models, such as stochastic volatility, have difficulties in doing.

Because that is a statistical measure, necessarily backward looking, and because volatility seems to vary, and we want to know what it will be in the future, and because people have different views on what volatility will be in the future, things are not that simple. Example Actual volatility is the σ that goes into the Black-Scholes partial differential equation. Implied volatility is the number in the Black-Scholes formula that makes a theoretical price match a market price. Long Answer Actual volatility is a measure of the amount of randomness in a financial quantity at any point in time. It’s what Desmond Fitzgerald calls the ‘bouncy, bouncy.’ It’s difficult to measure, and even harder to forecast but it’s one of the main inputs into option pricing models. It’s difficult to measure since it is defined mathematically via standard deviations which requires historical data to calculate.

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Imagine that the option factory is up and running and selling its products in the market. By assuming that smart, aggressive traders like Meriwether would snap up any mispriced options and build their own factory to pick them apart again using the mathematical recipe, Black, Scholes, and Merton followed in Miller’s footsteps with a no-arbitrage rule. In other words, you’d better believe the math because, otherwise, traders will use it against you. That was how the famous Black-Scholes formula entered finance. When the formula was first published in the Journal of Political Economy in 1973, it was far from obvious that anyone would actually try to use its hedging recipe to extract money from arbitrage, although the Chicago Board Options Exchange (CBOE) did start offering equity option contracts that year. However, there was now an added incentive to play the arbitrage game because Black, Scholes, and Merton had shown that (subject to some assumptions) their formula exorcised the uncertainty in the returns on underlying assets.

He set up his own highly lucrative hedge fund, LTCM, which made \$5 billion from 1994 to 1997, earning annual returns of over 40 percent. By April 1998, Merton and Scholes were partners at LTCM and making millions of dollars per year, a nice bump from a professor’s salary. By the late 1990s, investment banks were supplanting exchanges as the favored market-making institution for options and other derivatives, but LTCM worked with both. The original mathematics behind the Black-Scholes formula had gone through several generations of upgrades and refinements since 1973 and was gathering acolytes daily. According to Black-Scholes, the cost of manufacturing options increased with market volatility. Traders learned to use the option price as a kind of “fear gauge,” measuring what the market expected future volatility to be. (In 2005 the CBOE would adopt this fear gauge in the form of a new index called the VIX.)

How could arbitrage trades that were immunized from swings in fundamental markets such as equities or interest rates lose \$4 billion in a matter of months? How come VAR, the tool that LTCM and all other big trading banks used to control their exposures, broke down, when it had worked like a dream in 1994? These trades were supposedly safe bets because of the no-arbitrage principle. For example, the Black-Scholes formula suggested that buyers of options were being overcharged compared with the replication cost over time (which tracked underlying market volatility). So LTCM sold options and paid the replication costs, earning a profit as the option price converged on the replication cost, as the quants’ calculations said it would. Because of smart organizations like LTCM (and the fund attracted lots of imitators) the “ought” would become the “is.”

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Derivatives Markets by David Goldenberg

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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. Why not let the model generate estimates of that are consistent with the assumption that the market prices options using the Black–Scholes formula? This is a good idea. In order to implement it, all we have to do is plug all the parameters, except , into the Black–Scholes formula.

This is interesting, because it is an economic rationale for level-dependent volatility, and not just a statistical, or purely mathematical generalization of constant . 16.8.3 Modeling Changing Volatility, the Deterministic Volatility Model There are three initial ways to alter , the instantaneous volatility of percentage returns, that appears in the Black–Scholes formula. 1. =(t)=t , meaning that is not constant but is a deterministic function of time. Black–Scholes can be easily modiﬁed to accommodate this case simply by averaging t . A modiﬁcation of the Black–Scholes formula holds even if the instantaneous variance of percentage returns, t , depends on time. One obtains it by substituting ∫ T t s2ds for 2 (T − t ) in the Black–Scholes formula. OPTION PRICING IN CONTINUOUS TIME 587 Of course, this assumes that the functional dependence of on time is known, or can be estimated. 2. =(St ), meaning that depends upon the current stock price.

OPTION PRICING IN CONTINUOUS TIME 581 Therefore (GBM 16) is equal to, e rT Y0 ∫ e ∞ 1 - (z ′ )2 2 z ′<−(z (K )− T ) 2 ( dz′ ) = erT Y0 N ⎡⎣− z(K ) − T ⎤⎦ (GBM 17) Now all we have to do is to calculate, ( ) − z(K ) − T = − ⎡ ⎛ K ⎞ ⎛ 2 ⎞ ⎤ ⎢ln ⎜ ⎟ − ⎜r − ⎟T ⎥ 2 ⎠ ⎦⎥ ⎢⎣ ⎝ Y0 ⎠ ⎝ + T 2T ⎡ ⎛Y ⎞ ⎛ 2 ⎞ ⎤ ⎢ln ⎜ 0 ⎟ + ⎜r − ⎟T ⎥ 2 ⎠ ⎦ ⎣ ⎝K ⎠ ⎝ = + T T ⎡ ⎛Y ⎞ ⎛ 2 ⎞ ⎤ ⎢ln ⎜ 0 ⎟ + ⎜r − ⎟T + 2T ⎥ 2 ⎠ ⎣ ⎝K ⎠ ⎝ ⎦ = T = ⎡ ⎛Y ⎞ ⎛ 2 ⎞ ⎤ ⎢ ln ⎜ 0 ⎟ + ⎜ r + ⎟T ⎥ 2 ⎠ ⎦ ⎣ ⎝K ⎠ ⎝ T ≡ d1 This is the deﬁnition of d1 in the Black–Scholes formula, where we again used the fact that, ⎛K ⎞ ⎛Y ⎞ − ln ⎜ ⎟ = ln ⎜ 0 ⎟ ⎝K ⎠ ⎝ Y0 ⎠ Hence, ∞ e rT Y0 ∫ z′<−(z(K )− = erT Y0 N (d1 ) ⎡ ⎛ 1 2⎞ ⎢exp ⎜ − ( z′) ⎟ T) ⎠ ⎣ ⎝ 2 ⎤ 2 ⎥dz′ ⎦ This completes the derivation of the integral in GBM (8a). (GBM 18) 582 OPTIONS To get the full European call option price, we have to remember to discount by B(0,T). When we do so, we obtain that our entire European call option formula reduces to, C (Y0 ,T ;K ) = e −rT ⎡⎣e rT Y0 N (d1 ) − K N (d2 )⎤⎦ = Y0 N (d1 ) − e−rT K N (d2 ) Therefore, the Black–Scholes formula is given by, C (Y0 ,T ;K ) = Y0 N (d1 ) − e −rT K N (d 2 ) d1 = ⎡ ⎛Y ⎞ ⎛ 2 ⎞ ⎤ ⎢ln ⎜ 0 ⎟ + ⎜r + ⎟T ⎥ 2 ⎠ ⎦ ⎣ ⎝K ⎠ ⎝ T (Black–Scholes, [0,T]) and, d2 = d1 − T N(di ) is the cumulative normal distribution up to di , i=1,2.

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Automate This: How Algorithms Came to Rule Our World by Christopher Steiner

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About a year after the men had put their algorithm to work, a thunderclap sounded above Wall Street. In 1973 Fischer Black and Myron Scholes, both professors at the University of Chicago, published a paper that included what would become known as the Black-Scholes formula, which told its users exactly how much an option was worth. Algorithms based on Black-Scholes would over the course of decades reshape Wall Street and bring a flock of like-minded men—mathematicians and engineers—to the front lines of the financial world. The Black-Scholes solution, quite similar to Peterffy’s, earned Myron Scholes a Nobel Prize in 1997 (Black had died in 1995). Change didn’t happen overnight. The Black-Scholes formula, a partial differential equation, was brilliant. But most traders didn’t peruse academic journals. Even if they did, employing the formula wasn’t simple; it took significant math skills to wield.

“You know, you still have our Nobel Prize,” Jarecki said to Scholes. The remark elicited a dry grimace. “He was not amused,” Jarecki says. For traders who understood it, Black-Scholes gave them a way to calculate the exact price at which options should be traded. It was like having a cheat sheet for the market. There was money to be made by anybody who could accurately calculate each factor within the Black-Scholes formula and apply it to options prices in real time. Traders using the formula would sell options that were priced higher than the formula stipulated and buy ones that were priced lower than their fair price. Do this enough times with enough securities and a healthy profit was virtually guaranteed. TO BE A WALL STREET HACKER IN 1980: PERFECT PLACE, PERFECT TIME The late 1970s marked the faint dawn of the hacker era on Wall Street, when algorithms began to step in front of humans, a trend that has come to dominate all financial markets in every corner of the world.

“I instantly calculated how much money I could save in twenty years by no longer smoking,” he explains. “I needed everything.” Peterffy returned to the pits with a renewed focus. He stuck to his sheets, as always, but with DuPont haunting him, he didn’t make what he called “cowboy bets.” He slowly rebuilt his capital, one grinding day at a time. Sticking to his algorithmic system, he rarely experienced days with substantial losses. Even though the Black-Scholes formula had been published seven years before, it wasn’t moving the markets enough to bother Peterffy or others who were cashing in on its genius. As effective as his algorithms and sheets were, Peterffy was only one man. He needed more people in the pits. So he slowly hired more traders. To prevent losses and keep control of how his traders operated, he trained them to bid and offer only off of values on his sheets, which he would update with fresh numbers from his algorithm every night.

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

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In a lesser but similar way, I wanted to find a set of consistent rules that would let you convince a poor trader that an option formula was right without resorting to the advanced mathematics behind dynamic options replication. I thought about what the Black-Scholes formula really tells you. In principle, you can derive the formula from the Merton strategy of dynamic replication; from this point of view, the formula dictates in exquisite detail exactly how to synthesize a stock option out of a chang ing mixture of stock and riskless bonds. But looked at more naively, the formula gives you the fair price of the option in terms of the current price of the stock and the current price of a riskless bond. Its key insight is that the option is a mixture. Like the ancient Greeks' mythological centaur, part horse and part man, a call option is a hybrid, too-part stock and part bond. From this point of view, I came to regard the BlackScholes formula as a simple and sensible way of interpolating from the known market prices of a stock and a bond to the fair value of the hybrid.

In fruit salad terms, you might start with 50 percent apples and 50 percent oranges, and then, as apples increase in price, move to 40 percent apples and 60 percent oranges; a similar decrease in the price of apples might dictate a move to 70 percent apples and 30 percent oranges. In a sense, you are always trying to keep the price of the mixture constant as the ingredients' prices change and time passes.The exact recipe you need to follow is generated by the Black-Scholes equation. Its solution, the Black-Scholes formula, tells you the cost of following the recipe. Before Black and Scholes, no one even guessed that you could manufacture an option out of simpler ingredients, and so there was no way to figure out its fair price. This discovery revolutionized modern finance. With their insight, Black and Scholes made formerly gourmet options into standard fare. Dealers could now manufacture and sell options on all sorts of underlying securities, creating the precise riskiness clients wanted without taking on the risk themselves.

To our amazement, we discovered that even for 10,000 rehedgings on a one-year option-that is, for more than thirty rebalancings in a day-we still couldn't obtain the exact Black-Scholes value. There was always a residual discrepancy This seemed wrong, so I wrote my own version of the program and found the same small but significant discrepancy. This was very puzzling; it suggested that the Black-Scholes formula was less applicable to the conditions of actual markets than we had expected. I was perturbed enough to want to speak to Fischer about this, and went over to his office in another building on Goldman's growing campus. When I explain what I had found, he briefly became quite excited at the apparent inability of Merton's replication method to produce the exact Black-Scholes value, and said something like, "You know, I always thought there was something wrong with the replication method."

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The Quants by Scott Patterson

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He programmed the formula into his HP computer, and it quickly produced a graph showing the price of a stock option that closely matched the price spat out by his own formula. The Black-Scholes formula was destined to revolutionize Wall Street and usher in a wave of quants who would change the way the financial system worked forever. Just as Einstein’s discovery of relativity theory in 1905 would lead to a new way of understanding the universe, as well as the creation of the atomic bomb, the Black-Scholes formula dramatically altered the way people would view the vast world of money and investing. It would also give birth to its own destructive forces and pave the way to a series of financial catastrophes, culminating in an earthshaking collapse that erupted in August 2007. Like Thorp’s methodology for pricing warrants, an essential component of the Black-Scholes formula was the assumption that stocks moved in a random walk.

“I realized that the existence of the smile was completely at odds with Black and Scholes’s 20-year-old foundation of options theory,” wrote Emanuel Derman, a longtime financial engineer who worked alongside Fischer Black at Goldman Sachs, in his book My Life as a Quant. “And, if the Black-Scholes formula was wrong, then so was the predicted sensitivity of an option’s price to movements in its underlying index. … The smile, therefore, poked a small hole deep into the dike of theory that sheltered options trading.” Black Monday did more than that. It poked a hole not only in the Black-Scholes formula but in the foundations underlying the quantitative revolution itself. Stocks didn’t move in the tiny incremental ticks predicted by Brownian motion and the random walk theory. They leapt around like Mexican jumping beans. Investors weren’t rational, as quant theory assumed they were; they panicked like rats on a sinking ship.

A group of economists at the University of Chicago, led by free market guru Milton Friedman, were trying to establish an options exchange in the city. The breakthrough formula for pricing options spurred on their plans. On April 26, 1973, one month before the Black-Scholes paper appeared in print, the Chicago Board Options Exchange opened for business. And soon after, Texas Instruments introduced a handheld calculator that could price options using the Black-Scholes formula. With the creation and rapid adoption of the formula on Wall Street, the quant revolution had officially begun. Years later, Scholes and Robert Merton, an MIT professor whose ingenious use of stochastic calculus had further validated the Black-Scholes model, would win the Nobel Prize for their work on option pricing. (Black had passed away a few years before, excluding him from Nobel consideration.)

Analysis of Financial Time Series by Ruey S. Tsay

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The unobservability of volatility makes it difficult to evaluate the forecasting performance of conditional heteroscedastic models. We discuss this issue later. 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.

Continuous-Time Models and Their Applications 6.1 6.2 6.3 6.4 6.5 Options, 222 Some Continuous-Time Stochastic Processes, 222 Ito’s Lemma, 226 Distributions of Stock Prices and Log Returns, 231 Derivation of Black–Scholes Differential Equation, 232 221 ix CONTENTS 6.6 Black–Scholes Pricing Formulas, 234 6.7 An Extension of Ito’s Lemma, 240 6.8 Stochastic Integral, 242 6.9 Jump Diffusion Models, 244 6.10 Estimation of Continuous-Time Models, 251 Appendix A. Integration of Black–Scholes Formula, 251 Appendix B. Approximation to Standard Normal Probability, 253 7. Extreme Values, Quantile Estimation, and Value at Risk 7.1 7.2 7.3 7.4 7.5 7.6 7.7 8. 256 Value at Risk, 256 RiskMetrics, 259 An Econometric Approach to VaR Calculation, 262 Quantile Estimation, 267 Extreme Value Theory, 270 An Extreme Value Approach to VaR, 279 A New Approach Based on the Extreme Value Theory, 284 Multivariate Time Series Analysis and Its Applications 299 8.1 Weak Stationarity and Cross-Correlation Matrixes, 300 8.2 Vector Autoregressive Models, 309 8.3 Vector Moving-Average Models, 318 8.4 Vector ARMA Models, 322 8.5 Unit-Root Nonstationarity and Co-Integration, 328 8.6 Threshold Co-Integration and Arbitrage, 332 8.7 Principal Component Analysis, 335 8.8 Factor Analysis, 341 Appendix A.

We begin the chapter with some terminologies of stock options used in the chapter. In Section 6.2, we provide a brief introduction of Brownian motion, which is also known as a Wiener process. We then discuss some diffusion equations and stochastic calculus, including the well-known Ito’s lemma. Most option pricing formulas are derived under the assumption that the 221 222 CONTINUOUS - TIME MODELS price of an asset follows a diffusion equation. We use the Black–Scholes formula to demonstrate the derivation. Finally, to handle the price variations caused by rare events (e.g., a profit warning), we also study some simple diffusion models with jumps. If the price of an asset follows a diffusion equation, then the price of an option contingent to the asset can be derived by using hedging methods. However, with jumps the market becomes incomplete and there is no perfect hedging of options.

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Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street by William Poundstone

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They were the same except for an exponential factor incorporating the risk-free interest rate. Thorp had not included this because the over-the-counter options he traded did not credit the trader with the short-sale proceeds. The rules were changed when options began trading on the Chicago Board of Exchange. Black and Scholes accounted for this. Otherwise, the formulas were equivalent. The Black-Scholes formula, as it was quickly christened, was published in 1973. That name deprived both Merton and Thorp of credit. In Merton’s case, it was a matter of courtesy. Because he had built on Black and Scholes’s work, he delayed publishing his derivation until their article appeared. Merton published his paper in a new journal that was being started by AT&T, the Bell Journal of Economics and Management Science.

Thorp considers the Merton paper “a masterpiece.” “I never thought about credit, actually,” Thorp said, “and the reason is that I came from outside the economics and finance profession. The great importance that was attached to this problem wasn’t part of my thinking. What I saw was a way to make a lot of money.” Man vs. Machine FEW THEORETICAL FINDINGS changed finance so greatly as the Black-Scholes formula. Texas Instruments soon offered a handheld calculator with the formula programmed in. The market in options, warrants, and convertible bonds became more efficient. This made it harder for people like Thorp to find arbitrage opportunities. Of necessity, Thorp was constantly moving from one type of trade to another. In 1974 Thorp and Regan changed the name of their fund to Princeton-Newport Partners, a name steeped in the Ivy League and East Coast old money.

In some cases, the funds’ trading is dictated completely by computer printouts, which not only suggest the proper position but also estimate its probable annual return. “The more we can run the money by remote control the better,” Mr. Thorp declares. The Journal linked Thorp’s operation to “an incipient but growing switch in money management to a quantitative, mechanistic approach.” It mentioned that the Black-Scholes formula was being used by at least two big Wall Street houses (Goldman Sachs and Donaldson, Lufkin & Jenrette). The latter’s Mike Gladstein offered the defensive comment that the brainy formula was “just one of many tools” they used. “The whole computer-model bit is ridiculous because the real investment world is too complicated to be reduced to a model,” an unnamed mutual fund manager was quoted as saying.

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

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If the portfolio declines in value, the hedge is increased, so that finally, if the portfolio value falls well below the floor price, the portfolio is completely hedged. Thus the portfolio is hedged when it needs it and is free to take market exposure when there is a buffer between its value and the floor value. Because the hedge increases and decreases over time, it is called a dynamic hedge. The hedging method of portfolio insurance is based on the theoretical work of Fischer Black, Robert Merton, and Myron Scholes. Their work is encapsulated in the Black-Scholes formula, which makes it possible to set a price on an option. No other formula in economics has had as much impact on the world of finance. Merton and Scholes both received the Nobel Prize for it. (Fischer Black had died a few years before the award was made.) The theory and mathematics behind it were readily embraced by the academic community. Adopted from the mathematics of the heat transfer differential equation of physics and employing the new tools of stochastic calculus, it appealed to an academic core that seemed to derive a twisted pleasure from the mathematically arcane.

Adopted from the mathematics of the heat transfer differential equation of physics and employing the new tools of stochastic calculus, it appealed to an academic core that seemed to derive a twisted pleasure from the mathematically arcane. Despite its esoteric derivation, the formula was timely and—a rarity for work on the mathematical edge of economics—was immediately applicable. First, there was a ready market that required such a pricing tool: the Chicago Board Options Exchange (CBOE) opened for business in 1973, the same year both the paper presenting the Black-Scholes formula and a 9 ccc_demon_007-032_ch02.qxd 2/13/07 A DEMON 1:44 PM OF Page 10 OUR OWN DESIGN more complete exposition on option pricing by Merton were published.1 Second, although the formula required advanced mathematics and computing power, it really worked, and it worked in a mechanistic way. The formula gave rise to portfolio insurance through the work of two University of California at Berkeley finance professors, Hayne Leland and Mark Rubinstein.2 With John O’Brien, their marketing partner, they founded a management company, Leland O’Brien Rubinstein Associates (LOR), in 1981 to sell their technique.

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

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Fischer Black had been stuck on the options problem for “many, many months” when he started working with Myron Scholes. Scholes collaborated with Black to unlock the puzzles of option pricing. Their article on the subject was rejected at first as excessively specialized, but thanks to Merton Miller’s intervention it finally appeared just as the Chicago Board Options Exchange opened for business in 1973. The Black-Scholes formula was soon in general use there and has subsequently formed the basis for many investment, trading, and corporate finance strategies. (©1990 photography by Andy Feldman) In 1968, MIT was the only graduate school that would accept Robert Merton, now of Harvard Business School, when he decided to abandon math for economics. Paul Samuelson immediately selected Merton as an assistant and collaborator and stimulated his interest in option pricing.

If the olive harvest is about the same every year, there is little risk that press capacity and olive production will be badly matched. If the harvest is unexpectedly large, the olive grower will want to hedge against the possibility that he will have no access to the presses when his crop comes in. In light of all these considerations, how does an investor determine whether an option is cheap, expensive, or priced about right? The answer is to use the Black-Scholes formula. The investor knows the current prices of the stock and the option, the price at which the option can be exercised, the time to expiration, and the going rate of interest. With this information, the model will provide an estimate of the stock’s volatility that is implied in the price of the option. Then it is up to the investor to judge whether the market’s expectations about volatility look too low, too high, or about right.

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Models. Behaving. Badly.: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life by Emanuel Derman

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Black-Scholes provides a recipe for manufacturing a call by borrowing money to buy shares of the stock. The model tells you exactly how many shares to buy initially and then, at every future instant of time and at every future stock price, how much additional stock to buy or sell so that the stock you own will replicate the payoff of the option contract. The value of the option is the total cost of its manufacture, the cost of all the required trading with borrowed money. The Black-Scholes formula explains how the option value—the estimated cost of trading— depends on the stock price, the interest charged for borrowing, and the riskiness of the stock itself. Just as a weather model makes assumptions about how fluids flow and how heat undergoes convection, just as a soufflé recipe makes assumptions about what happens when you whip egg whites, so the Black-Scholes Model makes assumptions about the riskiness of stock prices, that is, about how stock prices fluctuate.

It is better to ensure that one owns a portfolio that will not suffer too badly under disastrous scenarios than it is to try to estimate the probability of destruction. So die the dreams of financial theories. Only imperfect models remain. The movements of stock prices are more like the movements of humans than of molecules. It is irresponsible to pretend otherwise. aMyron Scholes together with Robert C. Merton, who derived a different proof of the Black-Scholes formula and developed much of the elegant mathematics associated with options pricing, received the 1997 Nobel Memorial Prize in Economics for their work on the model. Fischer Black died two years earlier. Chapter 6 Breaking The Cycle Caught in a fiendish cage • Avoiding pragmamorphism • The great financial crisis and the abandonment of principle • The point of financial models • Be sophisticatedly vulgar • Let the dirt be visible • Beware of idolatry • The modelers’ Hippocratic oath • We need free markets, but we need them to be principled • Once in a blue moon, people stop behaving mechanically “Alas” said the mouse, “the world is growing smaller every day.

Markets are by definition vulgar, and the most useful models are vulgar too, using variables (such as price per square foot) that crowds use to describe the value of the assets they trade. One should build vulgar models in a sophisticated way. Some of the best and most practical models involve interpolation, not in prices but rather in the intuitive variables sophisticated users employ to estimate value, for example, volatility. Of course over time crowds and markets get smarter, and yesterday’s High Dutch becomes tomorrow’s patois. The Black-Scholes formula, which translate estimates of volatility into option prices, seemed so arcane when it burst upon the world that Black and Scholes had great difficulty getting their paper accepted for publication. Then, as users of the model grew more experienced, volatility became common currency. Nowadays traders and quants have grown so sophisticated that they talk fluently about models with stochastic volatility, a volatility that is itself volatile.

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I.O.U.: Why Everyone Owes Everyone and No One Can Pay by John Lanchester

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The interacting factors of time, risk, interest rates, and price volatility were so complex that they defeated mathematicians until Fischer Black and Myron Scholes published their paper in 1973, one month after the Chicago Board Options Exchange had opened for business. The revolutionary aspect of Black and Scholes’s paper was an equation enabling people to calculate the price of financial derivatives based on the value of the underlying assets. The Black-Scholes formula opened up a whole new area of derivatives trading. It was a defining moment in the mathematization of the market. Within months, traders were using equations and vocabulary straight out of Black-Scholes (as it is now universally known) and the worldwide derivatives business took off like a rocket. The total market in derivative products around the world is today counted in the hundreds of trillions of dollars.

., 77, 100, 204 regulation and, 184–86 risk and, 142–43, 164–66 conservatism, housing and, 98 correlation, correlations: CDOs and, 115–16, 158, 167 risk and, 74, 148–49, 158–59, 165, 167 credit, 8, 169–73 banks and, 24–26, 37, 41, 43, 209, 211 bubbles in, 42, 60, 109, 170, 176, 216–17, 221, 223 CDOs and, 114–15, 119–20, 172 crunch in, 37, 41, 43, 54n, 77, 84–86, 92–93, 94n, 136, 163–64, 169, 171–73, 182, 193, 201–2, 215–16, 218–19 histories and ratings on, 85, 100, 123–26, 158, 163, 165, 208–11 housing and, 84–86, 92–93, 94n, 100, 109, 112, 125, 129–30, 132, 163–64 Iceland’s economic crisis and, 10–12 interest rates and, 172–73, 175, 209 risk and, 136, 158, 165 see also banking-and-credit crisis Crédit Agricole, 36 credit cards, 27, 217 credit ratings and, 123–24 Iceland’s economic crisis and, 9, 11–12 risk and, 158–59, 163 credit default swaps (CDSs), 20, 63, 65–80, 117, 158–59, 183–86 AIG and, 75–78, 201 attractive aspects of, 72–74 examples of, 57–58 Exxon deal and, 67–70, 121 over-the-counter trading of, 184–85, 201 regulation and, 68, 70, 73, 184–86 risk and, 58, 66–70, 72–75, 78–80, 212 securitized bundles of, 69–70, 74 streamlining and industrializing of, 68–69 unfortunate side effect of, 74–75 Credit Suisse, 36, 227 Cuomo, Andrew, 99 Cutter family, 126–27 Darling, Alistair, 172, 220 debt, debts, 27–29, 34, 59–63, 118, 172n, 179, 216, 229 in balance sheets, 27–28, 30–31 benefits of, 59–61 bonds and, 59, 61–63, 208, 210 credit and, 123–26, 221 derivatives and, 52, 67, 69–72 housing and, 93, 100, 132, 176 paying the bill and, 220–22 personal, 221–22 regulation and, 181, 190 Russian default on, 55–56, 162, 164–65 see also collateralized debt obligations default, defaults, default rates, 162–65 CDOs and, 114–15 on mortgages, 159–60, 163, 165, 229 risk and, 154, 159–60, 163 of Russia, 55–56, 162, 164–65 see also credit default swaps Demchak, William, 69 democracy, democracies, 15–18, 108–9, 179, 213 free-market capitalism and, 15, 17, 23 housing and, 87, 98 DePastina, Anthony, 85 Depository Institutions Deregulation and Monetary Control Act (DIDMCA), 100 deregulation, see regulation, deregulation derivatives, 45–58, 63–80, 86, 210–12 in balance sheets, 30–31, 70 banks and, 20, 51–54, 57–58, 63–71, 74–75, 77, 79, 115–17, 120–21, 132, 183–84, 200, 205–6, 211 Black-Scholes formula and, 48, 54, 116–17, 151 bonds and, 58, 63–67, 112, 114, 118–19, 210–11 Buffett and, 56–57, 78 and City of London, 56–57, 79, 201 complexity of, 52–54, 56–57 Enron and, 56, 105–6, 185 futures and, 46–47, 49n, 51–52, 54, 184 Greenspan on, 166, 183–84 in history, 45–48, 147 mathematics and, 47–48, 52–54, 115–17, 166 offshore companies and, 70, 72 options and, 46–47, 50–52, 151, 174, 184 over-the-counter trading of, 184–85, 201, 205–6 prices and, 38, 46–52, 54, 56, 75, 158–59, 166 regulation and, 68, 70, 73, 153, 183–86, 200–201 risk and, 46–47, 49–52, 54–55, 57–58, 66–75, 78–80, 114–15, 117–22, 151, 153, 158–60, 163, 166–67, 184–85, 205, 212 size of market in, 48, 56, 80, 117, 201 see also collateralized debt obligations; credit default swaps Detroit, Mich., 81–82 Deutsche Bank, 36, 77, 83, 227 diversification, 146–48, 177 dividends, 101, 147–48 Doctorow, E.

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The Ascent of Money: A Financial History of the World by Niall Ferguson

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Markowitz, a Chicago-trained economist at the Rand Corporation, in the early 1950s, and further developed in William Sharpe’s Capital Asset Pricing Model (CAPM).83 Long-Term made money by exploiting price discrepancies in multiple markets: in the fixed-rate residential mortgage market; in the US, Japanese and European government bond markets; in the more complex market for interest rate swapsbf - anywhere, in fact, where their models spotted a pricing anomaly, whereby two fundamentally identical assets or options had fractionally different prices. But the biggest bet the firm put on, and the one most obviously based on the Black-Scholes formula, was selling long-dated options on American and European stock markets; in other words giving other people options which they would exercise if there were big future stock price movements. The prices these options were fetching in 1998 implied, according to the Black-Scholes formula, an abnormally high future volatility of around 22 per cent per year. In the belief that volatility would actually move towards its recent average of 10-13 per cent, Long-Term piled these options high and sold them cheap. Banks wanting to protect themselves against higher volatility - for example, another 1987-style stock market sell-off - were happy buyers.

The sixteen partners were left with \$30 million between them, a fraction of the fortune they had anticipated. What had happened? Why was Soros so right and the giant brains at Long-Term so wrong? Part of the problem was precisely that LTCM’s extraterrestrial founders had come back down to Planet Earth with a bang. Remember the assumptions underlying the Black-Scholes formula? Markets are efficient, meaning that the movement of stock prices cannot be predicted; they are continuous, frictionless and completely liquid; and returns on stocks follow the normal, bell-curve distribution. Arguably, the more traders learned to employ the Black-Scholes formula, the more efficient financial markets would become.97 But, as John Maynard Keynes once observed, in a crisis ‘markets can remain irrational longer than you can remain solvent’. In the long term, it might be true that the world would become more like Planet Finance, always coolly logical.

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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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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. Please note that the most important step in the estimation is the choice of proxy for the current stochastic volatility distribution.

He assumes that the portfolio is rebalanced at discrete time δt ﬁxed and transaction costs are proportional to the value of the underlying; that is, the costs incurred at each step is κ|ν|S, where ν is the number of shares of the underlying bought (ν > 0) or sold (ν < 0) at price S and κ is a constant depending on individual investors. Leland derived an option price formula which is the same as the Black–Scholes formula for European calls and puts with an adjusted volatility σ̂ = σ 1 + 2 κ √ π σ δt 1/2 . Following Leland’s idea, Hoggard et al. [3] derive a nonlinear PDE (partial differential equation) for the option price value in the presence of transaction costs. We outline the steps used in the next section. 14.1.1 OPTION PRICE VALUATION IN THE GEOMETRIC BROWNIAN MOTION CASE WITH TRANSACTION COSTS Let C(S, t) be the value of the option and be the value of the hedge portfolio.

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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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As the market sputtered, and hedge fund after hedge fund closed, Princeton-Newport generated positive, usually double-digit gains every year. After several years of this, Thorp got the notice in the mail that his secret formula was about to become public. It was a preprint of the Black-Scholes article, sent by Fischer Black, who professed in an introductory letter to be a “great admirer” of Thorp’s work. After some initial puzzlement, Thorp realized that the Black-Scholes formula was the same as his. Not long after that, the easy options money had mostly disappeared. But Thorp displayed an uncanny ability to keep finding new sources of profit—and get out of them before they stopped working. He was also willing to discuss his trades, at least after he’d made his money on them—something few of his black-box imitators have done since. This trade-and-tell act started with a Wall Street Journal front-page story in 1974, in which Thorp laid out in detail how he’d made an 8.5 percent profit in three weeks on underpriced Upjohn Co. options.

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. That was the smile—flat in the middle, rising at the edge. The Black-Scholes formula assumed that volatility would be constant, consistent, and normally distributed. That clearly wasn’t the case, and the search for better models of volatility was now on in earnest. One starting point was the statistical framework assembled twenty-five years before by Benoit Mandelbrot. Mandelbrot hadn’t predicted black Monday. He hadn’t written anything about finance in years. But anyone who had studied his market writings from the 1960s was far less surprised by events on Wall Street than those who had restricted their reading to standard finance textbooks.

., 154, 160, 165–66, 274, 352n. 3 Bernanke, Ben, xiii, 183 Bernoulli, Daniel, 51 beta, 87, 122–23, 126–27, 139, 152, 205, 207–8, 248–49, 345–46n. 30 Black, Fischer and asset pricing, 141, 149, 248, 322, 346–47n. 30 death, 235 and efficiency, 224 and Goldman Sachs, 224 and “joint hypothesis,” 105 and market crashes, 231 and market volatility, 138 and options, 144–47, 149, 277–78 and Ross, 150 and Shefrin and Statman, 358n. 25 and Summers, 199–201 and Thorp, 219 and Wells Fargo, 127 Black-Scholes formula, 147, 218–19, 233–34,237, 278–79, 280, 320 Bogle, John C., 112–13, 115, 122, 128–30, 306, 322 Bok, Derek, 169 bonds and bond markets, xi–xii, 13–14, 16, 21–22, 29, 38–39, 140–41, 167–68 book-to-price ratio, 208–9 Booth, David, 225 Bork, Robert, 158 Born, Brooksley, 244 Bossaerts, Peter, 297 Brealey, Richard, 355n. 38 Brennan, Michael, 284 Bretton Woods system, 92 Brinegar, Claude, 43–44 Brooks, John, 68, 118 Brown, Kathleen, 278 Brownian motion, 7, 13, 41, 65–69, 73 Bryan, William Jennings, 11 “Bubble Logic: Or, How I Learn to Stop Worrying and Love the Bull” (Asness), 261 Buchanan, James M., 159 Buffet, Warren, 118, 211–14, 214–16, 221–23, 229, 260, 271, 278–79, 323, 366n. 29 bull markets, 18, 61–62, 255, 279, 291 Burns, Arthur, 76, 217, 258 Bush, George W., 295 business cycle, 19–20, 28, 81, 309–10 Business Cycle Institute, 41 Business Week, 97–98 California Institute of Technology, 147 California Public Employees Retirement System (Calpers), 272, 273–74 Cambridge, University, 64 Camerer, Colin, 188 Cameron, David, 295 Campaign GM, 159 Campbell, John, 257 capital asset pricing model (CAPM).

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A Man for All Markets by Edward O. Thorp

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Then the CBOE announced it would start trading put options sometime in the following year, 1974. These options, like the call options we were already trading, were called American options, as distinguished from European options. European options can be exercised only during a short settlement period just prior to expiration, whereas American options can be exercised anytime during their life. If the underlying stock pays no dividends, the Black-Scholes formula, which is for the European call option, turns out to coincide with the formula for the American call option, which is the type that trades on the CBOE. A formula for the European put option can be obtained using the formula for the European call option. But the math for American put options differs from that for European put options, and—even now—no general formula has ever been found. I realized that I could use a computer and my undisclosed “integral method” for valuing options to get numerical results to any desired degree of accuracy for this as-yet-unsolved “American put problem.”

For most academic theorists, this was as close to impossible as anything can be. It was as though the sun suddenly winked out or the earth stopped spinning. They described stock prices using a distribution of probabilities with the esoteric name lognormal. This did a good job of fitting historical price changes that ranged from small to rather large, but greatly underestimated the likelihood of very large changes. Financial models like the Black-Scholes formula for option prices were built using the lognormal. Aware of this limitation in academia’s model of stock prices, as part of the indicators project we had found a much better fit to the historical stock price data, especially for the relatively rare large changes in price. So even though I was surprised by the giant drop, I wasn’t nearly as shocked as most. Though there was no major outside event to explain this one-day collapse, when I thought it through that evening I asked myself, Why did this happen?

Follow logic and analysis rather than sales pitches, whims, or emotion. Assume you may have an edge only when you can make a rational affirmative case that withstands your attempts to tear it down. Don’t gamble unless you are highly confident you have the edge. As Buffett says, “Only swing at the fat pitches.” 3. Find a superior method of analysis. Ones that you have seen pay off for me include statistical arbitrage, convertible hedging, the Black-Scholes formula, and card counting at blackjack. Other winning strategies include superior security analysis by the gifted few and the methods of the better hedge funds. 4. When securities are known to be mispriced and people take advantage of this, their trading tends to eliminate the mispricing. This means the earliest traders gain the most and their continued trading tends to reduce or eliminate the mispricing.

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Money Changes Everything: How Finance Made Civilization Possible by William N. Goetzmann

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Time only goes in one direction, and with it, the universe tends toward less organization, not more. The option pricing model is based on the principle of forecasting the range of future outcomes of the stock price by assuming it will follow a random walk that conforms to Regnault’s square-root of time insight. However, the Black-Scholes formula gives a solution to the option price today by mathematically rolling time backward. It reverses entropy. In this, it echoes the most basic trait of finance—it uses mathematics to transcend time. THERMODYNAMICS The Black-Scholes formula was published in 1973, just around the time that the Chicago Board Option Exchange began to trade standardized option contracts. Like Bachelier’s thesis, the path-breaking paper was not at first well received. The Journal of Political Economy, where it ultimately was published, needed serious urging from Chicago Professor Merton Miller to be convinced of its contribution.

Evidently none knew of Bachelier, and thus they had to retrace the mathematical logic of fair prices and random walks when they began work on the problem of option pricing in the late 1960s. Like Bachelier, they relied on a model of variation in prices—Brownian motion—although unlike Bachelier, they chose one that did not allow prices to become negative—a limitation of Bachelier’s work. The Black-Scholes formula, as it is now referred to, was mathematically sophisticated, but at its heart it contained a novel economic—as opposed to mathematical—insight. They discovered that the invisible hand setting option prices was risk-neutral. Option payoffs could be replicated risklessly, provided one could trade in an ideal, frictionless market in which stocks behaved according to Brownian motion. Later researchers4 developed a simple framework called a “binomial model” that was able to match the payoff of a put or a call by trading just the stock and a bond through time.

See also stock market, US Nicholas, Tom, 482–83 Nicholas de Anglia, 236 Nicholson, John, 394, 396–97, 399 Nippur, 65, 67 non-normality of security prices, 286 Northwest passage to Cathay, 311–12, 313–14 Norwegian Pension Fund Global, 512, 513, 515 number system: first evidence for, 28. See also Arabic numerals Objectivism, of Ayn Rand, 452 Ohio Company, 388–89 oil income, government investment of, 512 O’Keeffe, Georgia, 468, 475–76, 481 Old Babylonian period, 46, 49, 55–57, 65 Onslow’s Insurance, 370 operations research, 504, 507 opium trade, Chinese, 423, 425–26, 427, 441 Opium Wars, 425–26, 437, 441 option pricing: Bachelier on, 282–83; Black-Scholes formula for, 283–84; Brownian motion and, 276; fractal-based, 287; Lefèvre on, 279–82 options: defined, 280; on Law’s Mississippi Company shares, 357; in seventeenth-century Amsterdam stock market, 317; stock options as compensation, 171 oracle bones, Chinese, 146–47, 271 Ott, Julia, 469–70 owl coins, Athenian, 96–98, 101 Pacioli, Lucca, 246–47 paghe, 291–92 paper instruments, Chinese, 174–75; pawn tickets, 178–79; of Song dynasty, 186–89, 199 paper making and printing, 181–82 paper money: in American colonies, 386–88, 390, 400; Chinese invention of, 139, 168, 174–75, 183–84, 201–2, 400; Chinese nationalization of printing of, 185–86; eighteenth-century comeback of, 382, 399–400; of French revolutionary government, 391–92; of Law’s proposed land bank, 352–53 (see also land banks); Marco Polo’s account of, 191–93; Song dynasty collateral problem with, 387–88; Song dynasty color printing of, 182; of Song dynasty in military crisis, 199.

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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 1973 a trio of American academics—Fischer Black, Myron Scholes, and Robert Merton—cracked the problem of what to pay for an option. The answer they came up with, expressed as what is now known as the Black-Scholes equation, was based on a simple idea: two things that had identical outcomes ought to cost the same. The price of the option ought to be the same as whatever it cost to construct an investment portfolio that achieved the same end. The Black-Scholes formula enabled the rapid pricing of options and paved the way for explosive growth in derivatives markets. Greek academics have even used it to work out what Thales should have paid for his olive-oil option more than fifteen hundred years ago.25 The third driver was technology. We have seen how a new technology like the railways required finance to adapt in order to provide appropriate financing and screening mechanisms.

Together the three men cracked the problem of how to price an option, a financial instrument that gives the buyer the right, but not the obligation, to buy or sell an underlying asset. The question of what price to pay for an option was one to which there was no rigorous answer until Black, Scholes, and Merton came along. The answer they came up with, expressed as what is now known as the Black-Scholes equation, was based on the idea that the price of the option ought to be the same as the cost of constructing a perfect hedge for the underlying asset. The Black-Scholes formula, which coincided with the computerization of trading, enabled the rapid pricing of options and paved the way for huge growth in derivatives markets.7 At a time when financial innovation and derivatives have become dirty words, Merton has become practiced at answering the criticisms thrown their way. “When you get asked, ‘What is it like to be an ax murderer?’ you tend to question the premise” is how he puts it.

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

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In the midst of reading about Black-Scholes, I was also deeply involved with writing the book with Victor Ginzburg from the University of Chicago. Ginzburg was a first-rate mathematician and a terrific writer. He was also a perfectionist to the highest degree. We would write and rewrite chapters endlessly; and each new theorem we added seemed to inspire the need for new chapters. The book went from a proposed hundred-page set of lecture notes into a long, involved project.15 As I was learning about the Black-Scholes formula I was growing increasingly frustrated with the Ginzburg book project. I needed relief. I decided to take everything I was learning about options pricing and write a book on the subject. Now why would I do that? By working with Ginzburg, I had learned to take extremely complicated ideas and explain them clearly and concisely. Also, since I was learning BlackScholes from scratch, I thought I could bring a fresh perspective to it.

In Toronto, I continued to work on the book with Ginzburg—which we had named Representation Theory and Complex Geometry—and work on mathematics research, but I also began to write a paper on options pricing. Thus, my career as a quant slowly began in Toronto in 1994. I never published that first paper, but I did post it on the Social Sciences Research Network (SSRN.com). It was called “An Options Pricing Formula with Volume as a Variable.” The idea was that the Black-Scholes formula relies on perfect dynamic replication of an option with a portfolio of the underlying stock and a riskless security. My idea was to ask, what if instead of perfect replication you can only replicate with a certain probability? What I did was show that if you could replicate a security with another with an arbitrarily high degree of probability, then you could obtain pricing formulas that had all the good properties associated with perfect replication.

When I discovered that the counterparty for most of his trades was a subsidiary of Goldman Sachs (such was the trader’s faith in his own models he neither knew nor cared), I put my foot down and got trading halted. The problem was that in order to derive closed-form solutions, one generally has to work in the Black-Scholes framework. Everyone knows that, due to the fatter than lognormal tails in most asset returns, farfrom-the-money options should generally be priced significantly above the Black-Scholes formula price. But these barrier and double barrier options were close to the money, so this problem doesn’t apply, right? Wrong; if a distribution has fat tails then it must have a taller thinner peak to compensate. Thus a formula derived in the Black-Scholes framework must price near-the-money barrier options too expensively. This is the crucial intuition that the trader had missed; departures from log-normality are important for pricing barrier options, but he had left them out because of his devotion to closed-form formulae and the assumptions that they entail.

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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But the theory of options pricing was given a big boost in the 1970s when two academic economists, Fischer Black and Myron Scholes, developed the first mathematical for12 Chapter 16 will discuss a valuable index of option volatility called VIX. 266 PART 4 Stock Fluctuations in the Short Run mula to price options. The Black-Scholes formula was an instant success. It gave traders a benchmark for valuation where previously they used only their intuition. The formula was programmed on traders’ handheld calculators and PCs around the world. Although there are conditions when the formula must be modified, empirical research has shown that the Black-Scholes formula closely approximates the price of traded options. Myron Scholes won the Nobel Prize in Economics in 1997 for his discovery.13 Buying Index Options Options are actually more basic instruments than futures or ETFs. You can replicate any future or ETF with options, but the reverse is not true.

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Hedge Fund Market Wizards by Jack D. Schwager

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In 1967, I took some of the ideas about how to price warrants in the Random Character of Stock Prices by Paul Cootner and thought I could derive a formula if I made the simplifying assumption that all investments grew at the risk-free rate. Since the purchase or sale of warrants combined with delta neutral hedging led to a portfolio with very little risk, it seemed very plausible to me that the risk-free assumption would lead to the correct formula. The result was an equation that was equivalent to the future Black-Scholes formula. I started using this formula in 1967. Did you apply your formula (that is, the future Black-Scholes formula) to identify overpriced warrants and then delta hedge those positions? I didn’t have enough money to have a diversified warrant portfolio and to also place the hedge, since each side of a hedged position required separate margin. I used the formula to identify the most extremely overpriced warrants. I found warrants that were selling at two or three times what my formula said they should be priced at.

Small cap stocks were up 84 percent in 1967 and 36 percent in 1968. It was a terrible time to have net short exposure. However, the formula was good enough and the warrants were so overpriced that I still broke even on the naked short positions. The formula really proved itself under the most adverse circumstances. As far as I know, the short warrant positions I implemented during 1967 to 1968 were the first actual application of the Black-Scholes formula in the markets. When did Black-Scholes publish their formula? I believe they discovered it in 1969 and published it 1972 or 1973. Did you consider publishing your formula? The option-pricing formula seemed to me to be a big edge on everybody else. So I was happy just to use it. By 1969, I had started my first hedge fund, Princeton Newport Partners, and I thought that if I published the formula, I would lose the edge that was helping my investors.

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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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A call and put option give the holder the right to buy or sell a security at a given price, so the higher the security’s volatility, the greater the chance that the security’s price may move above or below the strike price, letting the investor make a profit. That’s why higher volatility means a higher option price. 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.

See also Investment banks; specific banks bailouts of basic operations of fees charged by Greek debt exposure as hedge funds housing bubble and leverage and Maughan on provision of emergency credit by runs on trust between trust in Barclays Barclays Global Investors (BGI) Basel Committee: Basel I document Basel II document financial crisis and guidelines of overview of Bear Stearns: bank run on collapse of failure of hedge funds of history and reputation of J.P. Morgan and leverage of LTCM and near-collapse of repo system and window dressing by Begleiter, Steve Benn, Orson Berkshire Hathaway. See also Buffett, Warren Bernanke, Ben Black, Fischer Black-Scholes formula Blankfein, Lloyd Blasnik, Steve Bond arbitrage Born, Brooksley Box trade Brady Plan Brazilian C bonds Brendsel, Leland Broker-dealers Buffett, Warren Buoni del Tesoro Poliennali Buoni Ordinari del Tesoro Bush, George Butler, Angus Butterfly yield curve trades Callan, Erin Capital, contingency Capital adequacy ratio (CAR) Capital markets Capital ratio and leverage Capital-to-asset ratio Carhart, Mark Cash business Cassano, Joseph Caxton macro hedge fund Cayne, James E.

pages: 349 words: 134,041

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

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DAS_C07.QXP 8/7/06 4:45 PM Page 195 6 N Super models – derivative algorithms 195 The price of a European put option can be derived as follows: Ppe = K e-Rf.T . N(– d2) – S. N(– d1) Where Ppe = price of European put option Despite the formidable appearance of the equation, you need only high school maths to derive the option values. The papers would have remained obscure had it not been for the confluence of events. In 1973, the Chicago Board Options Exchange started trading options on leading stocks and the Black–Scholes formula quickly became the market standard for pricing and trading options. HewlettPackard calculators with preprogrammed Black–Scholes option pricing model became available – the age of the super model had arrived. In 1997, Scholes and Merton received the Nobel Prize for economics in recognition of their work. Black did not receive the award, having died of throat cancer in 1995. The Nobel Prize rewards longevity as much as intellectual achievement.

pages: 543 words: 157,991

All the Devils Are Here by Bethany McLean

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But as a risk manager, Guldimann was often confronted with the problem of what to do when a trader wanted to increase his limit. “How should I know if he should get his increase?” Guldimann says. “All I could do is ask around. Is he a good guy? Does he know what he’s doing? It was ridiculous.” There was never any question about how Guldimann and his team would approach this task. They would use statistics and probability theories that had long been popular on Wall Street. (The Black-Scholes formula, for example, developed in the early 1970s for pricing options, had become one of the linchpins of modern Wall Street.) The quants swarming Wall Street were all steeped in those theories—this was the essential building block of virtually everything they did. They knew no other way to approach the subject. Sure enough, Value at Risk, or VaR, the model the J.P. Morgan quants came up with after years of trial and error, was built on a key tenet of the mathematics of probability, called Gaussian distribution.

See Federal bailouts Bair, Sheila Baker, Richard Bankers Trust, swap deal lawsuit Bank of America Countrywide acquired by Merrill Lynch acquired by subprime branches, closing Barnes, Roy Bartiromo, Maria Basel Committee on Banking Supervision, capital reserves rule Basis Yield Alpha Fund Bear Stearns ABS index Bank of America lawsuit CDOs foreclosures, plan to prevent hedge funds, collapse of High-Grade Structured Credit Fund High-Grade Structured Credit Strategies Enhanced Leverage Limited Partnership J.P. Morgan acquisition of Beattie, Richard Behavioral economics Beneficial Bensinger, Steve Bernanke, Ben Berson, David Birnbaum, Josh BlackRock Black-Scholes formula Black swans Blankfein, Lloyd during collapse compensation from Goldman Goldman Sachs under Blue sky laws, MBSs exemption from Blum, Michael Blumenthal Stephen BNC Mortgage BNP Paribas Bolten, Joshua Bomchill, Mark Bond, Kit Bond ratings CDOs and credit enhancements failures, examples of public trust in ratings shopping system of for tranches Value at Risk (VaR) applied to See also Moody’s; Standard & Poor’s Bonuses, post-TARP Born, Brooksley biographical information derivatives, regulatory efforts style/personality of Bothwell, James Bowsher, Charles Bradbury, Darcy Brandt, Amy Breit, John Brennan, Mary Elizabeth Brickell, Mark Brightpoint fraud Broad Index Secured Trust Offering (BISTRO) AIG FP credit protection features of Broderick, Craig Bronfman, Edward and Peter Bruce, Kenneth Buffett, Warren Burry, Michael Bush, George W.

pages: 545 words: 137,789

How Markets Fail: The Logic of Economic Calamities by John Cassidy

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Finally, in the early 1970s, Black, a man of few words, and Scholes, a voluble Canadian, derived a simple formula that related the price of an option to the volatility of the underlying stock. By coincidence, the paper that contained the Black-Scholes option pricing formula was published in May 1973, a month after the opening of the Chicago Board Options Exchange. To compute the value of an option using the Black-Scholes formula all you needed, in addition to the strike price, the current price, and the duration of the option, was the interest rate on government bonds, the standard deviation of the stock, and a table of the normal distribution. By the end of 1973, you didn’t even need a pen and paper to do the calculation: Texas Instruments had introduced a calculator that did it for you. That was only the beginning.

Mutual funds were able to insure themselves against the risk of corporations defaulting on their bonds, banks could insure themselves against some of their lenders defaulting, and insurance companies could insure against the chances of a freak hurricane leaving them with enormous claims from their policyholders. In each of these areas, the key was the development of mathematical methods to price risk. Almost all of these methods relied, to some extent, on the Black-Scholes formula and the bell curve. Simply by invoking the ghost of Louis Bachelier, it was possible to take much of the danger out of finance. Or was it? As far back as the 1960s and ’70s, some academics and Wall Street practitioners didn’t buy into the coin-tossing view of finance. Many old-school bankers and traders were put off by the mathematical demands it came with, but numbered among the skeptics were also some technically adept economists, including Sanford Grossman, of Wharton, and Joseph Stiglitz, who is now at Columbia.

pages: 504 words: 139,137

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

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It’s one thing to look at the water from afar; it’s another thing to look at it right up close. From afar, it looks pretty calm; up close, it looks chaotic. I felt that the experience of the chaotic world, married to my theoretical abilities, would allow me to gain unique perspectives. For that reason, I gravitated to work for a while at Salomon Brothers. LHP: When most people think about the Black–Scholes formula, they think first about equity options, but you focused on fixed income arbitrage—why? MS: Right. The fascination with fixed-income arbitrage came about after many years of thinking about the idea that there are natural segmented clienteles. Insurance companies and pension funds tend to be at the longer end of the interest rate curve. Macro hedge funds try to express their macro views at the 10-year part of the curve, and other issuers—mortgage issuers and so forth—are also at the 10–15 year part of the curve.

For each bond they provide the coupon, the conversion ratio, the conversion value—all the salient terms of the instrument. Based on the market prices in that book, there appeared to be some bonds that were mispriced. I made it my mission to educate myself on these instruments and to understand the pricing and trading of convertibles. LHP: Was the insight just based on some back of the envelope calculations, or did you need to already appreciate something like the Black–Scholes Formula or the binomial option pricing model at that time? KG: Back of the envelope, some common sense, and a bit of naïveté as to the dynamics around why these mispricings might exist. Many mispricings were driven by the inability to borrow the underlying common stock and therefore the convertible bond traded close to conversion value because the arbitrage was difficult. Nonetheless, I didn’t understand these dynamics at the moment.

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

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The problem is that the Almighty is not giving me or anyone else the probability distribution for the price of IBM a month from now. The standard approach, which is based on the Black-Scholes formula, assumes that the probability distribution will conform to a normal curve [the familiar bell-shaped curve frequently used to depict probabilities, such as the probability distribution of IQ scores among the population]. The critical statement is that it "assumes a normal probability distribution." Who ran out and told these guys that was the correct probability distribution? Where did they get this idea? [The Black-Scholes formula (or one of its variations) is the widely used equation for deriving an option's theoretical value. An implicit assump*A probability distribution is simply a curve that shows the probabilities of some event occurring—in this case, the probabilities of a given stock being at any price on the option expiration date.

pages: 261 words: 10,785

The Lights in the Tunnel by Martin Ford

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Stock options, which represent the right to buy or sell a stock at a given price at some point in the future, had been traded on markets for some time, but no one knew how to calculate a precise value for them. In the years that followed, and especially during the 1980s, a large number of people originally trained as physicists or mathematicians began to take much higher paying jobs on Wall Street. These guys (they were virtually all men) were referred to as “quants.” The quants started working with the Black-Scholes formula and expanded it in new ways. They turned their formulas into computer programs and gradually began to create new types of derivatives based on stocks, bonds, indexes and many other securities or combinations of securities.14 Copyrighted Material – Paperback/Kindle available @ Amazon Acceleration / 45 As their computers got faster and faster, the quants were able to do more and more. They created new exotic derivatives based on strange combinations of things.

pages: 204 words: 58,565

Keeping Up With the Quants: Your Guide to Understanding and Using Analytics by Thomas H. Davenport, Jinho Kim

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However, Black and Scholes performed empirical tests of their theoretically derived model on a large body of call-option data in their paper “The Pricing of Options and Corporate Liabilities.”16 DATA ANALYSIS. Black and Scholes could derive a partial differential equation based on some arguments and technical assumptions (a model from calculus, not statistics). The solution to this equation was the Black-Scholes formula, which suggested how the price of a call option might be calculated as a function of a risk-free interest rate, the price variance of the asset on which the option was written, and the parameters of the option (strike price, term, and the market price of the underlying asset). The formula introduces the concept that, the higher the share price today, the higher the volatility of the share price, the higher the risk-free interest rate, the longer the time to maturity, and the lower the exercise price, then the higher the option value.

pages: 295 words: 66,824

A Mathematician Plays the Stock Market by John Allen Paulos

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And options on a stock whose volatility is high will cost more than options on stocks that barely move from quarter to quarter (just as a short man on a pogo stick is more likely to be able to peek over a nine-foot fence than a tall man who can’t jump). Less intuitive is the fact that the cost of a call option also rises with the interest rate, assuming all other parameters remain unchanged. Although there are any number of books and websites on the Black-Scholes formula, it and its variants are more likely to be used by professional traders than by gamblers, who rely on commonsense considerations and gut feel. Viewing options as pure bets, gamblers are generally as interested in carefully pricing them as casino-goers are in the payoff ratios of slot machines. The Lure of Illegal Leverage Because of the leverage possible with the purchase, sale, or mere possession of options, they sometimes attract people who aren’t content to merely play the slots but wish to stick their thumbs onto the spinning disks and directly affect the outcomes.

pages: 311 words: 99,699

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After he left J.P. Morgan back in 2000, he had created a consultancy group that advised governments and companies on how to use innovative financial products, such as derivatives, to their benefit. The other founding members of the group were Roberto Mendoza, another former J.P. Morgan banker, and Robert Merton, the Nobel Prize–winning economist who had helped to create the pathbreaking Black-Scholes formula that had played a crucial role in the development of derivatives. For seven long years, Hancock had extolled the virtues of financial innovation, often in the face of client skepticism. Even as the banking world reeled in shock in late 2007, he remained committed to the cause. “A lot of the problems in structured finance have not been due to too much innovation, but a failure to innovate sufficiently,” Hancock observed.

pages: 338 words: 106,936

The Physics of Wall Street: A Brief History of Predicting the Unpredictable by James Owen Weatherall

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Throughout the text, however, I will mean something very specific: a strategy by which one constantly updates the proportions of stocks and options in one’s portfolio so that the portfolio as a whole is risk-free. “. . . successfully urged the Journal of Political Economy to reconsider . . .”: The article was published as Black and Scholes (1973). See also Merton (1973) and Black and Scholes (1972, 1974). For more on the Black-Scholes formula and its generalizations and extensions, see Hull (2011) and Cox and Rubinstein (1985). “The head of that committee was James Lorie . . .”: For more on the history of the CBOE, see Markham (2002) and MacKenzie (2006). “On the first day of trading . . .”: These numbers are from Markham (2002, vol. 3, p. 52). “But volume grew at an astonishing rate . . .”: These numbers are from Ansbacher (2000, p. xii).

pages: 437 words: 132,041

Alex's Adventures in Numberland by Alex Bellos

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‘The family went on a trip to the World’s Fair in Spokane and on the way back we stopped at Harrah’s [casino] and I told my kids to give me a couple of hours because I wanted to pay for the trip – which I did.’ Beat the Dealer is not just a gambling classic. It also reverberated through the worlds of economics and finance. A generation of mathematicians inspired by Thorp’s book began to create models of the financial markets and apply betting strategies to them. Two of them, Fischer Black and Myron Scholes, created the Black-Scholes formula indicating how to price financial derivatives – Wall Street’s most famous (and infamous) equation. Thorp ushered in an era when the quantitative analyst, the ‘quant’ – the name given to the mathematicians relied on by banks to find clever ways of investing – was king. ‘Beat the Dealer was kind of the first quant book out there and it led fairly directly to quite a revolution,’ said Thorp, who can claim – with some justification – to being the first-ever quant.

pages: 512 words: 162,977

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

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When I think about pricing an option, I may not know calculus, but in my mind I can draw a picture of how you would price an option that looks exactly like the theoretical pricing models in the textbooks. When did you first get involved in trading options? I did a little dabbling with stock options back in 1975-76 on the Chicago Board of Options Exchange, but I didn’t stay with it. I first got involved with options in a serious way with the initiation of trading in futures options. By the way, in 1975 I crammed the Black-Scholes formula into a TI52 hand-held calculator, which was capable of giving me one option price in about thirteen seconds, after I hand-inserted all the other variables. It was pretty crude, but in the land of the blind, I was the guy with one eye. When the market was in its embryonic stage, were the options seriously mispriced, and was your basic strategy aimed at taking advantage of these mispricings?

pages: 500 words: 145,005

Misbehaving: The Making of Behavioral Economics by Richard H. Thaler

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Where should this leave us regarding the validity of the Becker conjecture—that the 10% of people who can do probabilities will end up in the jobs where such skills matter? At some level we might expect this conjecture to be true. All NFL players are really good at football; all copyeditors are good at spelling and grammar; all option traders can at least find the button on their calculators that can compute the Black–Scholes formula, and so forth. A competitive labor market does do a pretty good job of channeling people into jobs that suit them. But ironically, this logic may become less compelling as we move up the managerial ladder. All economists are at least pretty good at economics, but many who are chosen to be department chair fail miserably at that job. This is the famous Peter Principle: people keep getting promoted until they reach their level of incompetence.