John Hull [Numerical method, Other Derivatives]

CHAPTEI{ 19 Basic Numerical Procedures for the Instructor Chapter 19 presents the standard numerical procedures used to value derivatives when analytic results are not avai1able. These involve binomialjtrinomial trees, Monte Carlo simulation, difference methods. Binomial trees are introduced in Chapter 11, and Section 19.1 and 19.2 can be regarded as a review and more in-depth treatment of that material. When covering Section 19.1, 1 usually go through in some detail the calculations for a number of nodes in an example such as the one in Figure 19.3. Once the basic tree building and roll back procedure has been covered it is fairly easy to explain how it can be extended to currencies, indices, futures, and stocks that pay dividends. Also the calculation of hedge statistics such as delta, gamma, and vega can be explained. The software DerivaGem is a convenient way of displaying trees in class as well as an important calculation tool for students. The binomial tree and Monte Carlo simulation approaches use risk-neutral valua-tion arguments. By contrast, the finite difference method solves the underlying differential equation directly. However , as explained in the book the explicit finite difference method is essentially the same as the trinomial tree method and the implicit finite difference method is essentially the same as a multinomial tree approach where the are branches ema-nating from each node. Binomial trees and finite difference methods are most appropriate for American options; Monte Carlo simulation is most appropriate for path-dependent options. Any of Problems 19.25 to 19.30 work well as assignment questions. QUESTIONS AND PROBLEMS Problem 19. 1. Which of the following can be estimated for an American option by constructing a single binomia1 tree: delta, gamma, theta, rho? Delta, gamma, and theta can be determined from a single binomial tree. Vega is determined by making a small change to the volatility and recomputing the option price using a new tree. Rho is calculated by making a small change to the interest rate and recomputing the option prce using a new tree. Problem 19.2. Calculate the price of a American put option on a non-dividend-paying stock wlwn the stock prce is $60, the strike price is $60, the risk-free interest rate is 10% 237 per annum, and the volatility is 45% per annum. Use a binomial tree with a time interval of one month. In this case, 80 = 60, K = 60, r = 0.45, T = 0.25, and 6.t = 0.0833. Also u = eaVD. t = 1. 1387 d21:0.8782 u eTD.t = eO.lXO.0833 = 1.0084 1- P = 0.5002 The output from DerivaGem for this example is shown in the Figure 819. 1. The calculated price of the option is $5.16. Problem 19.3. Node lim e: o 0000 o 0833 0.1667 0.2500 Figure 819.1 Tree for Problem 19.2 Explain how the control variate technique is implemented when a tree is to value American options. 238 The control variate technique is implemented by (a) valuing an American option using a binomial tree in the usual way (= f A)., (b) valuing the European option with the same parameters as the American option using the same tree (= fE). (c) valuing the European option using Black-8choles (= f BS). The price of the American option is Problem 19.4. prce of a nne--month Amercan call opton on corn futures when the current futures prce s 198 cents, the strke prce s 200 cents, the interest rate per annum, and s 30% per annum. Use a binomal tree with a tme nterval of three months. In this case K = 200, r = 0.3, T = 0.75, and f}. t = 0.25. Also u = eO 3y'o'25 = 1.1618 d =1=08607 u 1 - p = 0.5373 The output DerivaGem for this example is shown in the Figure 819.2. The calculated price of the option is 20.34 cents. Problem 19.5. Consder an option that pays off the amount by which stock prce exceeds the average stock prce acheved during the life of the option. Can ths be valued using the binomial tree approach? Explain your answer. A binomial tree cannot be used in the way described in this chapter. This is an example of what is known as a history-dependent option. The payoff depends on the path followed by the stock price as well as its final value. The option cannot be valued by starting at the end of the tree and working backward since the payoff at the final branches is not known unambiguously. Chapter 26 describes an extension of the binomial tree approach that can be used to handle options where the on the average value of the stock price. Problem 19.6. For a dvidend-paying stock, the tree for the stock price does not but the tree for the stock prce less the present value of future dvdends does recombne." Explan this statement 239 Growth factor per step , a = 1 0000 P robability of up m ove , p = 0..4 626 u p s te p s iz e, U = 1 1 61 8 00 W n s te p s iz e, d = 0..8607 Node Tim e-o 0000 o 2500 o 5000 o 7500 Figure 819.2 Thee for Problem 19.4 Suppose a dividend equal to D is paid during a certain time interval. If S is the stock price at the beginning of the time interval, it will be either Su - D or Sd -- D at the end of the time interval. At the end of the next time interval, it will be one of (Su - (Su - D)d, (Sd - D)u and (Sd - D)d. Since (Su - D)d does not equal (Sd ._- D)u the tree does not recombine. If S is equal to the stock price less the present value of future dividends, this problem is avoided. Problem 19.7. Show that the in a Cox, Ross, and Rubinstein binomial tree are negative when the condition in footnote 9 holds. With the usual notation p= u-d d or u , one of the two probabilities is negative. This happens when e(T-q)At