Appendix 1: Option Valuation We have mentioned at several points in this text the importance of option pricing and valuation in the work of derivative product and credit profession­ als. The seminal work in the area of option valuation is the 1973 article by Black and Scholes entitled 'The Pricing of Options and Corporate Liabili­ ties', where the authors develop an analytic solution to the valuation of Euro­ pean options on non-dividend paying stocks. The simplifying assumptions used to develop the equation include unlimited borrow and lending at a constant risk-free rate, full access to short-selling, an underlying asset which follows a continuous time stochastic process, and frictionless markets with no taxes or transactions costs.' It is not our intent in this appendix to develop a detailed explanation of the mathematics of the Black-Scholes model; there are many works which do an excellent job of explaining the quantitative aspects of the model. Instead, we provide a very brief overview of the main points of the model and illustrate, with a few simple examples, the computation of option prices and comparative statics based on the model. To commence, we define in advance several of the terms used in the equa­ tions below (note that we make a few changes to the notation from earlier in the text; most notably, CMP becomes S, and SP becomes X): c is the value of a European call p is the value of a European put S is the stock price S(T) is the terminal stock price X is the strike price t is the time to maturity r is the risk-free rate J.L is the expected return of the stock a is the standard deviation of the return We begin our outline by defining the value of a European call at expiry as C(ST, X, 't) == max(O, ST - X) (Al.I) The value of the call today is simply the discounted value of the expected payoff at expiry: 342 Option Valuation C(ST. X. t) = e-rt E(max(O. ST - X» (A 1.2) As indicated above, stock prices under the Black-Scholes model follow a stochastic process; in this case prices follow a geometric Brownian motion, where the returns of the stock prices are lognormally distributed (see Chapter 2 for additional detail). Under the lognormal distribution the expected value of the stock at maturity of the option is: E(ST) = Se(p-!f)t (A 1.3) where the price of the stock, S, increases by the instantaneously compounded expected return J1 less half the variance of the stock price. In the equation above, 0 is the standard deviation of the expected return. Under risk neutral­ ity, one of the key elements of the Black-Scholes framework. the value of the expected return can be ignored. This occurs because an investor can cre­ ate a risk-free portfolio by combining a short position in a call option (-c) with a delta equivalent amount of stock (AS): the perfect correlation between the call and the stock over a very small time period means gains on the call are offset by losses on the stock and vice versa. Therefore, the value of the option is not influenced by the expected return. and investors' risk prefer­ ences need not be considered; such simplifies the pricing framework consid­ erably. Knowing that stock prices are distributed lognormally. we return to the equation presented in Chapter 2 (Le. equation to in note 6): In(ST) ~ • [In(S) + (J1 - ~ 't), <rftJ (AI.4) Under risk neutrality. J1 is replaced with the risk-free rate, r, yielding: In(ST) ~ • [In(S) + (r - 0; t). o-ft] (A 1.5) Returning to equation A 1.2, it is necessary to solve for the value of c (where g is the distribution of SeT) under risk-neutrality): C(ST, X. t) = e-rt i<ST - X)g(ST)dST (A 1.6) x After a rather extensive mathematical exercise. equation (A1.6) yields the familiar Black-Scholes call valuation equation: 343 Appendix J (Al.7) where N(dl) and N(d2), defined below, represent the cumulative probability distribution function for a standardized normal variable. The N(d) values can be obtained from standard tables of probability functions. The values of d 1 and d2 are shown as: (Al.B) and: d _ In( ~) + (r - ~)~ (Al.9) 2- a..f.t where d2 may also be expressed as: (Al.lO) Through this formula, one can value a European call option on a non-dividend paying stock, under the constraints detailed above. Knowing the above, and employing put-call parity, which effectively says that due to boundary conditions and arbitrage: c + Xe- rc == p + S (AUt) it is relatively easy to solve for the Black-Scholes equation for puts: (Al.12) Let us consider a numerical example in order to demonstrate the model. Assume we want to purchase a European call on a stock with a current price of 20 and a strike price of 22 (i.e. it is 9 per cent out-of-the-money); expiry is in three months (90 days). Recall that ~ is adjusted to a fraction of a year if the volatility measure is annualized; in this case ~ equals 0.25 years. The risk free rate is 4 per cent and the volatility of the stock is 20 per cent. The fair value of the call, based on the equations above, is: dl = On(20122) + (0.04 + (0.201\212) * 0.25»/(0.20 * "0.25) 344 Option Valuation = «-0.0953) + (0.015»/.10 = -0.803 d2 = -0.803 - 0.10 = -0.903 c = 20 '" N( -0.803) - 22 '" (0.99) '" N( -0.903) Using statistical tables we detennine the probabilities associated with N(dl) and N(d2): c = 20 '" (1 - 0.7967) - 22 '" 0.99 '" (l - 0.8238) = 4.066 - 3.837 = .229 Thus, one caIl on the stock is worth nearly $0.23. If we apply the same procedure to an in-the-money call (i.e. the price S is now $24, instead of $20, with the strike remaining at $22), the value of the call rises: dl = (In(24/22) + (0.04 + (0.2()A2/2) '" 0.25»/(0.20 '" ",,0.25) = «0.087) + (0.015»/0.10 = 1.020 d2 = 1.020 - 0.10 = 0.920 c = 24 '" N(1.020) - 22 '" (0.99) '" N(0.920) = 24 '" 0.8461 - 22 '" (0.99) '" 0.8212 = 20.306 - 17.885 = 2.421 As one might expect, the value of the in-the-money caIl is much higher than the out-of-the-money caIl (at $2.42 vs. $0.23) as it already has intr!nsic value. In Chapter 3 we have described the comparative statics delta, gamma, vega and theta; these statics measure an option's sensitivity to changes in the stock price, volatility and time. Utilizing the Black-Scholes model, we highlight the equations which measure these statics (note that we shaIl not consider rho, sensitivity of the option's value to a change in the risk-free rate, as the effects of rho on a stock option are generaIly negligible when compared with the other statics). It can be shown that delta, which is the change in the option's price for a unit change in the stock price, is equal to the first derivative of the option price with respect to the stock price: 345 Appendix J (A 1.13) The quantity we obtain from N(dl) above is precisely equal to delta. We shall not explore the mathematics behind this, but we may interpret the above as saying that a short call is delta hedged by an amount of long stock equal to N(dl); conversely, a long call is delta hedged by an amount of short stock equal to N(dl). The equation for puts is: (A1.l4) Returning to the second example above. we note that the result obtained from N(dl) is equal to 0.846. This means that a portfolio is delta hedged when a short call is offset by 0.846 shares of long stock; alternatively, a long call is hedged by shorting 0.846 shares of stock. This also means that if the stock rises by $1, the value of the call increases by $0.846. Gamma, which is the change in the delta of the option for a unit change in the stock price, is the first derivative of delta with respect to the stock price, or the second derivative of the option price with respect to the stock price. This may be shown as: (A1.15) where (AI.l6) The first derivative of N(dl) is the standard normal density function. The gamma of a put is: (AU7) which is precisely equal to the gamma of a call. Returning to the example above. we calculate the gamma of the call as r = 1/(24 * (0.20 * "0.25» * N'(dl) = 0.4167 * N'(dl) 346 Option Valuation N'(dl) = 11(2.507) * el\-(0.520) = 0.2371 = 0.4167 * 0.2371 = 0.0988 Thus, for a $1 increase in the stock price, the delta of the option rises by $0.099. Theta, which is the change in the option's price for a change in time, is simply the derivative of the option price with respect to time until maturity. This is shown as: (A 1.18) Where N'(dl) and N(d2) are defined above. Note the derivative of the call option price with respect to the price, if positive, yields a value greater than O.