PART Two Technical Background of the Binomial Lattice and Black-Scholes Models ccc_mun_pt2_75-76.qxd 8/20/04 9:22 AM Page 75 ccc_mun_pt2_75-76.qxd 8/20/04 9:22 AM Page 76 CHAPTER 7 Brief Technical Background BLACK-SCHOLES MODEL The basic BSM is summarized as follows: where Φ is the cumulative standard-normal distribution function S is the value of the forecast stock price at grant date X is the option’s contractual strike price rf is the nominal risk-free rate σ is the annualized volatility T is the time to expiration of the option To illustrate its use, let us assume that an option exists such that both the stock price (S) and the strike price (X) are $100, the time to expiration (T) is one year with a 5 percent annualized risk-free rate (rf) for the same dura- tion, while the annualized volatility (σ) of the underlying asset is 25 per- cent. The BSM calculation yields $12.3360: Call S S X rf T T Xe S X rf T T Put Xe S X rf T T S S X rf T rf T rf T = ++ − +− =− +− −− ++ − − ΦΦ ΦΦ ln( / ) ( / ) ln( / ) ( / ) ln( / ) ( / ) ln( / ) ( / ) () () σ σ σ σ σ σ σ 22 22 22 22 σσ T 77 ccc_mun_ch07_77-82.qxd 8/20/04 9:23 AM Page 77 The cumulative standard-normal distribution function can be solved in Ex- cel by using its “NORMSDIST( )” function. In addition, you can create calculation codes within Excel’s Visual Basic for Applications (VBA) environment. Following are the VBA codes for the BSM for estimating call and put options. The equations for the BSM are simplified to functions in Excel named “BlackScholesCall” and “BlackScholesPut.” Public Function BlackScholesCall(Stock As Double, Strike As Double, Time As Double, Riskfree _ As Double, Volatility As Double) As Double Dim D1 As Double, D2 As Double D1 = (Log(Stock / Strike) + (Riskfree + 0.5 * Volatility ^ 2 ) * Time) / (Volatility * Sqr(Time)) D2 = D1 – Volatility * Sqr(Time) BlackScholesCall = Stock * Application.NormSDist(D1) – Strike * Exp(–Time * Riskfree) * _ Application.NormSDist(D2) End Function Public Function BlackScholesPut(Stock As Double, Strike As Double, Time As Double, Riskfree _ As Double, Volatility As Double) As Double Dim D1 As Double, D2 As Double D1 = (Log(Stock / Strike) + (Riskfree + 0.5 * Volatility ^ 2 ) * Time) / (Volatility * Sqr(Time)) D2 = D1 – Volatility * Sqr(Time) BlackScholesPut = Strike * Exp(–Time * Riskfree) * Application.NormSDist(–D2) – Stock * _ Application.NormSDist(–D1) End Function 78 BACKGROUND OF THE BINOMIAL LATTICE AND BLACK-SCHOLES MODELS Call S S X rf T T Xe S X rf T T Call e rf T = ++ − +− = ++ − − − ΦΦ Φ ln( / ) ( / ) ln( / ) ( / ) $ ln $ $ () . $ () σ σ σ σ 22 2 0 22 100 100 100 005 1 2 025 1 025 1 100 () ln $ $ () . $ (.)$(.)(.)$(.)$.(.)$. 05 1 2 100 100 005 1 2 025 1 025 1 100 0 3250 100 0 9512 0 0750 100 0 6274 95 12 0 5299 12 3360 Φ ΦΦ +− =− =−=Call ccc_mun_ch07_77-82.qxd 8/20/04 9:23 AM Page 78 As an example, entering the function in Excel: “=BlackScholesCall(100,100,1,5%,25%)” results in 12.3360 and entering the function in Excel: “=BlackScholesPut(100,100,1,5%,25%)” results in 7.4589. Note that Log is a natural logarithm function in VBA and Sqr is square root, and make sure there is a space before the underscore in the code. MONTE CARLO SIMULATION MODEL In the simulation approach for estimating European options, a series of forecast stock prices are created using the Geometric Brownian Motion stochastic process, and the options maximization calculation is applied to the end point of the series, and discounted back to time zero, at the risk- free rate. That is, starting with an initial seed value of the underlying stock price, simulate out multiple future pathways using a Geometric Brownian Motion, where . That is, the change in asset value δS t at time t is the value of the asset in the previous period S t–1 multiplied by the Brownian Motion . The term rf is the risk-free rate, δt is the time-steps, σ is the volatility, and ε is the simulated value from a standard-normal distribution with mean of zero and a variance of one. Other variations of Brownian Motions exist but for illustration purposes, this simple version will be used. The first step in Monte Carlo simulation is to decide on the number of time-steps to simulate. In the example, 10 steps were chosen for simplicity. Starting with the initial stock price of $100 (S 0 ), the change in value from this initial value to the first period is seen as . Hence, the stock price at the first time-step is equivalent to . The stock price at the second time-step is hence , and so forth, all the way until the terminal tenth time-step. Notice that because ε changes on each simulation trial, each simulation trial will produce an entirely SS SSSrft t 21 211 =+ =+ + () δδσεδ () SS SSSrft t 10 100 =+ =+ + () δδσεδ () δδσεδ SSrft t 10 =+ () () rf t t() δσεδ + () δδσεδ SSrft t tt =+ () −1 () Brief Technical Background 79 ccc_mun_ch07_77-82.qxd 8/20/04 9:23 AM Page 79 different asset-evolution pathway. At the end of the tenth time-step, the maximization process is then applied. That is, for a simple European op- tion with a $100 implementation cost, the function is simply C 10,i = Max[S 10,i – X, 0]. This is the call value C 10,i at time 10 for the i th simulation trial. This value is then discounted at the risk-free rate to obtain the call value at time zero, that is, C 0,i = C 10,i e –rf(T) . This is a single-point estimate for a single simulated pathway. A forecast distribution of the thousands of simulated pathways is collected and the mean of the distribution is the ex- pected value of the option. On the one hand, it must be stressed that Monte Carlo simulation can be applied only to calculate European op- tions, and not American options, making it less suitable for use in valuing ESOs. On the other hand, Monte Carlo can be used to simulate the uncer- tain input variables that go into the customized binomial lattice model as seen in Chapter 5. BINOMIAL LATTICES Binomial lattices, in contrast to the other methods, are easy to implement and easy to explain. They are also highly flexible but require significant com- puting power and time-steps to obtain good approximations, as will be seen later. It is important to note, however, that at the limit, results obtained through the use of binomial lattices tend to approach those derived from closed-form solutions, and hence, it is always recommended that both ap- proaches be used to benchmark the results. The results from closed-form so- lutions may be used in conjunction with the binomial lattice approach when presenting a complete financial options valuation solution in the most basic European options analysis. However, when real-life cases are added into the analysis (forfeitures, vesting, suboptimal exercise behavior multiples, and blackout dates), the results diverge because the closed-form models such as the BSM or GBM cannot account for these added variables. Following is an example to illustrate the point of binomial lattices ap- proaching the results of a closed-form model. Let us look again at the Eu- ropean call option presented previously, but this time, calculated using the GBM: Using the previous example where both the stock price (S) and the strike price (X) are $100, the time to expiration (T) is one year with a 5 per- 80 BACKGROUND OF THE BINOMIAL LATTICE AND BLACK-SCHOLES MODELS Call Se SX rf q T T Xe SX rf q T T qT rT = +−+ − +−− − − ΦΦ ln( / ) ( / ) ln( / ) ( / ) σ σ σ σ 22 22 ccc_mun_ch07_77-82.qxd 8/20/04 9:23 AM Page 80 cent risk-free rate (rf) for the same duration, while the volatility (σ) of the underlying asset is 25 percent with no dividends (q). The GBM cal- culation yields $12.3360, similar to the BSM calculations, while using a binomial lattice we obtain the following results: N = 10 steps $12.0923 N = 20 steps $12.2132 N = 50 steps $12.2867 N = 100 steps $12.3113 N = 1,000 steps $12.3335 N = 10,000 steps $12.3358 N = 50,000 steps $12.3360 Notice that even in this simplified example, as the number of time-steps (N) gets larger, the value calculated using the binomial lattice approaches the GBM closed-form solution. Suffice it to say, many steps are required for a good estimate using binomial lattices. It has been shown in past re- search that 1,000 time-steps are usually sufficient for a good approxima- tion for up to 2 decimals. Chapter 10 provides a case example of how to find the optimal number of lattice steps and to test for results convergence in a binomial lattice. SUMMARY AND KEY POINTS ■ The three mainstream approaches used to solve simple options are the GBM and BSM closed-form models, path-dependent simulation, and binomial lattices. ■ BSM is highly inflexible and can be applied to solve only European options. ■ Path-dependent simulations are also applicable for solving only Euro- pean options. ■ Binomial lattices are more flexible and can be used to solve both Amer- ican and European options and are capable of handling other exotic input variables that exist in real-life ESOs. Brief Technical Background 81 ccc_mun_ch07_77-82.qxd 8/20/04 9:23 AM Page 81 . PART Two Technical Background of the Binomial Lattice and Black-Scholes Models ccc_mun_pt2 _75 -76 .qxd 8/20/04 9:22 AM Page 75 ccc_mun_pt2 _75 -76 .qxd 8/20/04 9:22 AM Page 76 CHAPTER 7 Brief. / ) ln( / ) ( / ) () () σ σ σ σ σ σ σ 22 22 22 22 σσ T 77 ccc_mun_ch 07_ 77- 82.qxd 8/20/04 9:23 AM Page 77 The cumulative standard-normal distribution function can be solved in. 1 100 0 3250 100 0 9512 0 075 0 100 0 6 274 95 12 0 5299 12 3360 Φ ΦΦ +− =− =−=Call ccc_mun_ch 07_ 77- 82.qxd 8/20/04 9:23 AM Page 78 As an example, entering