73323357-17-Basic-Numerics

73323357-17-Basic-Numerics - B asic utnerical Procedures...

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Unformatted text preview: B asic utnerical Procedures This chapter discusses three numerical procedures for valuing derivatives when exact formulas are not available. The first involves representing the asset price movements in the form of a tree and was introduced in Chapter 11. The second involves Monte Carlo simulatiop., which we encountered briefly in Chapter 12 when explaining stochastic processes. The third involves finite difference methods. . Monte Carlo simulation is usually used for derivatives where the payoff is dependent on the history of the underlying variable or where there are several underlying variables. Trees and finite difference methods are usually used for American options and other derivatives where the holder has early exercise decisions to make prior to maturity. In addition to valuing a derivative, all the procedures can be used to calculate Greek letters such as delta, gamma, and vega. The basic procedures we discuss in this chapter can be used to handle most of the derivatives valuations problems that are encountered in practice. However, sometimes they have to be adapted to cope with particular situations. We discuss this in Chapter 24. 17.1 BINOMIAL TREES We introduced binomial trees in Chapter 11. They can be used to value either European or American options. The Black-Scholes formulas and their extensions that we presented in Chapters 13 and 14 provide analytic valuations for European options.! There are no analytic valuations for American options. Binomial trees are therefore most useful for valuing these types of options. 2 As explained in Chapter 11, the binomial tree valuation approach involves dividing the life of the option into a large number of small time intervals of length ~t. It assumes that in each time interval the price of the underlying asset moves from its initial value of 1 The Black-Scholes formulas are based on the same set of assumptions as binomial trees. As one might expect, in the limit as the number of time steps is increased, the price given for a European option by the binomial method converges to the Black-Scholes price. 2 Some analytic approximations for valuing American options have been suggested. The most well-known one is the quadratic approximation approach. See Technical Note 8 on the author's website for a description of this approach. 391 392 CHAPTER 17 Figure 17.1 Asset price movements in time !:it under the binomial model. Sit S Sd S to one of two new values, Su and Sd. The approach is illustrated in Figure 17.1. In general, u > 1 and d < 1. The movement from S to Su, therefore, is an "up" movement and the movement from S to Sd is a "down" movement. The probability of an up movement will be denoted by p. The probability of a down movement is 1 - p. Risk-Neutral Valuation The risk-neutral valuation principle, explained in Chapters 11 and 13,states that an option (or other derivative) can be valued on the assumption that the world is risk neutral. This means that for valuation purposes we can use the following procedure: 1. Assume that the expected return from all traded assets is the risk-free interest rate. 2. Value payoffs from the derivative by calculating their expected values and discounting at the risk-free interest rate. This principle is a key element of the ways in which trees are used. Determination of P, u, and d The parameters p, u, and d must give correct values for the mean and variance of asset price changes during a time interval of length !:it. Because we are working in a riskneutral world, the expected return from the asset is the risk-free interest rate, r. Suppose that the asset provides a yield of q. The expected return in the form of capital gains must be r - q. This means that the expected value of the asset price at the end of a time interval of length !:it must be Se(r-q)D.t, where S is the stock price at the beginning of the time interval. It follows that seCr-q)D.t = pSu + (1 - p)Sd (17.1) or /r-q)D.t = pu + (1 - p)d (17.2) As explained in Section 13.4, the variance of the percentage change in the stock price in a small time interval of length !:it is a 2 1:it. The variance of a variable Q is defined as E(Q2) - [E(Q)f There is a probability p that the percentage change is u and 1 - p that it is d. The expected percentage change is e(r-q)Ac. It follows that pu 2 + (1 - p)d2 - i(r-q)AC = a 2!:it 393 B asic Numerical Procedures Substituting for p from equation (17-2) gives e(r-q)!J.\u + d) - e2(r-q)!J.l ud = c? tlt ".:; (17.3) Equations (17.2) and (17.3) impose two conditions on p, u, and d. A third condition used by Cox, Ross, and Rubinstein (1979) is 3 1 U=- d A solution to equations (17.2) and (17.3) when terms of higher order than tlt are ignored is 4 a-d p=-(17.4) u-d u = eu.JF:i (17.5) (17.6) where a = e<r-q)!J.t (17.7) The variable a is sometimes referred to as the growth factor. Equations (17.4) to (17.7) are the same as those in Section 11.9. Tree of Asset Prices Figure 17.2 illustrates the complete tree of asset prices that is considered when the binomial model is used. At time zero, the asset price, So, is known. At time tlt, there are two possible asset prices, Sou and Sod; at time 2tlt, there are three possible asset prices, Sou 2 , So, and SOd 2 ; and so on. In general, at time i tlt, we consider i + 1 asset prices. These are Note that the relationship u = lid is used in computing the asset price at each node of the tree in Figure 17.2. For example, Sou 2d = Sou. Note also that the tree recombines in the sense that an up movement followed by a down movement leads to the same asset price as a down movement followed by an up movement. Working Backward through the Tree Options are evaluated by starting at the end of the tree (time T) and working backward. The value of the option is known at time T. For example, a put option is worth max(K - ST, 0) and a call option is worth max(ST - K, 0), where ST is the asset price at time T and K is the strike price. Because a risk-neutral world is being assumed, the 3 See J. C. Cox, S. A. Ross, and M. Rubinstein, "Option Pricing: A Simplified Approach," JOt/mal of Financial Economics, 7 (October 1979), 229-63. 4 To see this, we note that equations (17.4) and (17.7) satisfy the condition in equation (17.2) exactly. The exponential function e' can be expanded as 1 + x +.r2 /2 + .. '. When terms of higher order than At are ignored, equation (17.5) implies that t/ = 1 + (T.JE"t + t:J.t and equation (17.6) implies that d = 1 - (T.JE"t + At. Also, e(r-q)t:.r = 1 + (r - q)t:J.t and e 2(r-q)t:.r = 1 + 2(r - q)t:J.t. By substitution we see that equation (17.3) is satisfied when terms of higher order than At are ignored. ta2 ta2 3 94 CHAPTER 17 Figure 17.2 Tree used to value an option. So value at each node at time T - /::;.t can be calculated as the expected value at time T discounted at rate r for a time period /::;.t. Similarly, the value at each node at time T - 2/::;.t can be calculated as the expected value at time T - /::;.t discounted for a time period /::;.t at rate r, and so on. If the option is American, it is necessary to check at each node to see whether early exercise is preferable to holding the option for a further time period /::;.t. Eventually, by working back through all the nodes, we are able to obtain the value of the option at time zero. Example 17.1 Consider a 5-month American put option on a non-dividend-payigg stock when the stock price is $50, the strike price is $50, the risk-free interest r~te is 10% per annum, and the volatility is 40% per annum. With our usual notation, this means that So = 50, K = 50, r = 0.10, a = 0.40, T = 0.4167, and q = O. Suppose that we divide the life of the option into five intervals of length 1 month (= 0.0833 year) for the purposes of constructing a binomial tree. Then /::;.t = 0.0833 and, using equations (17.4) to (17.7), we have II = eU./ill = 1.1224, p a-d u-d d = e-u./ill = 0.8909, = - - = 0.5073, 1- p a = erAC = 1.0084 = 0.4927 . Figure 17.3 shows the binomial tree produced by DerivaGem. At each node there are t~o numbers. The top one shows the stock price at the node; the lower one shows the value of the option at the node. The probability of an up movement is always 0.5073; the probability of a down movement is always 0.4927. 395 B asic Numerical Procedures Binomial tree from DerivaGem for American put on non-dividendpaying stock (Example 17.1). r~ Figure 17.3 At each node: Upper value = Underlying Asset Price Lower value =Option Price Shading indicates where option is exercised Strike price =50 Discount factor per step = 0.9917 Time step, dt = 0.0833 years, 30.42 days Growth factor per step, a = 1.0084 Probability of up move, p = 0.5073 Up step size, u = 1.1224 Down step size, d =0.8909 .--~:.-.-......., Node Time: 0.0000 0.0833 0.1667 0.2500 0.3333 0.4167 The stock price at the jth node (j = 0, 1, ... , i) at time i Ilt (i = 0,1, ... ,5) is calculated as SOlJ d i - j. For example, the stock price at node A (i = 4, j = 1) (i.e., the second node up at the end of the fourth time step) is 50 x 1.1224 x 0.8909 3 = $39.69. The option prices at the final nodes are calculated as max(K - Sr, 0). For example, the option price at node Gis 50.00 - 35.36 = 14.64. The option prices at the penultimate nodes are calculated from the option prices at the final nodes. First, we assume no exercise of the option at the nodes. This means that the option price is calculated as the present value of the expected option price one time step later. For example, at node E, the option price is calculated as (0.5073 x 0 + 0.4927 x 5.45)e-O.lOxO.0833 = 2.66 whereas at node A it is calculated as (0.5073 x 5.45 + 0.4927 x l4.64)e-O.lOxO.0833 = 9.90 396 CHAPTER 17 We then check to see if early exercise is preferable to waiting. At node E, early exercise would give a value for the option of zero because both the stock price and strike price are $50. Clearly it is best to wait. The correct value for the option at node E, therefore, is $2.66. At node A, it is a different story. If the option is exercised, it is worth $50.00 - $39.69, or $10.31. This is more than $9.90. If node A is reached, then the option should be exercised and the correct value for the option at node A is $10.31. Option prices at earlier nodes are calculated in a similar way. Note that it is not always best to exercise an option early when it is in the money. Consider node B. If the option is exercised, it is worth $50.00 - $39.69, or $10.31. However, if it is held, it is worth (0.5073 x 6.38 + 0.4927 ~ 14.64)e-O.lOxO.0833 = 10.36 The option should, therefore, not be exercised at this node, and the correct option value at the node is $10.36. Working back through the tree, the value of the option at the initial node is $4.49. This is our numerical estimate for the option's current value. In practice, a smaller value of 1St, and many more nodes, would be used. DerivaGem shows that with 30, 50, 100, and 500 time steps we get values for the option of 4.263, 4.272, 4.278, and 4.283. Expressing the Approach Algebraically Suppose that the life of an American put option on a non-dividend-paying stock is divided into N subintervals oflength 1St. We will refer to the jth node at time i 1St as the (i, j) node, where 0 ~ i ~ Nand 0 ~ j ~ i). Define Ji,j as the value of the option at the j (i, j) node. The stock price at the (i, j) node is Sou d H . Since the value of an American put at its expiration date is max(K - ST, 0), we know that fN.j = max(K - Soujd N- j , 0), j = 0,1, ... , N There is a probability p of moving from the (i, j) node at time i 1St to the (i + 1, j + 1) node at time (i + 1) 1St, and a probability 1 - p of moving from the\(i, j) node at time i 1St to the (i + 1, j) node at time (i + 1) 1St. Assuming no early exercise, risk-neutral valuation gives for 0 ~ i ~ N - 1 and 0 ~ j ~ i. When early exercise is taken into account, this value for fL must be compared with the option's intrinsic value, and we obtain Note that, because the calculations start at time T and work backward, the value at time i 1St captures not only the effect of early exercise possibilities at time i 1St, but also the effect of early exercise at subsequent times. In the limit as 1St tends to zero, an exact value for the American put is obtained. In practice, N = 30 usually gives reasonable results. Figure 17.4 shows the convergence of the' option price in the example we have been considering. This figure was calculated 397 B asic Numerical Procedures Convergence of the price of the option in Example 17.1 calculated from ,~ the DerivaGem Application Builder functions. Figure 17.4 Option value 5.00 4.80 4.60 4.40 4.20 4.00 3.80 No. of steps 3.60 0 5 10 15 20 25 30 35 40 45 50 using the Application Builder functions provided with the DerivaGem software (see Sample Application A). Estimating Delta and Other Greek letters It will be recalled that the delta (.6.) of an option is the rate of change of its price with respect to the underlying stock price. It can be calculated as .6.f .6.S where .6.S is a small change in the stock price and .6.f is the corresponding small change in the option price. At time .6.t, we have an estimate fll for the option price when the stock price is Sou and an estimate flO for the option price when the stock price is Sod. In other words, when .6.S = Sou - Sod, .6.f = fll - flO' Therefore an estimate of delta at time .6.t is .6. = fll - flO (17.8) Sou-Sod To determine gamma (f), note that we have two estimates of .6. at time 2.6.t. When S = (Sou 2 + So)/2 (halfway between the second and third node), delta is 2 2 U22 - 121)/(SOU - So); when S = (So + Sod )/2 (halfway between the first and second node), delta is U21 - 12o)/(So - Sod\ The difference between the two values of Sis h, where h = O.S(Sou 2 - 2 Sod ) Gamma is the change in delta divided by h: f = fU22 - 121)/(SOU 2 - So)] h fU21 - 12o)/(So - Sod 2 )] (17.9) 398 CHAPTER 17 These procedures provide estimates of delta at time J::.t and of gamma at time 2J::.t. In practice, they are usually used as estimates of delta and gamma at time zero as well. 5 A further hedge parameter that can be obtained directly from the tree is theta (8). This is the rate of change of the option price with time when all else is kept constant. If the tree starts at time zero, an estimate of theta is 8= hi - foo (17.10) 2M Vega can be calculated by making a small change, J::.a, in the volatility and constructing a new tree to obtain a new value of the option. (The time step J::.t should be kept the same.) The estimate of vega is 1*:...- f v=-J::.a where f and f* are the estimates of the option price from the original and the new tree, respectively. Rho can be calculated similarly. Example 17.2 . Consider again Example 17.1. From Figure 17.3, we have fl,l = 2.16. Equation (17.8) gives an estimate for delta of 2.16 - 6.96 56.12 - 44.55 ho = 6.96 and = -0.41 From equation (17.9), an estimate of the gamma of the option can be obtained from the values at nodes B, C, and F as [(0.64 - 3.77)/(62.99 - 50.00)] - [(3.77 - 10.36)/(50.00 - 39.69)] = 0.03 11.65 From equation (17.10), an estimate of the theta of the option can be obtained from the values at nodes D and C as 3.77 - 4.49 0.1667 = -4.3 per year or -0.012 per calendar day. These are only rough estimates. They become progressively better as the number of time steps on the tree is increased. Using 50 time steps, DerivaGem provides estimates of -0.415, 0.034, and -0.0117 for delta, gamma, and theta, respectively. By making small changes to parameters and recomputing values, vega and rho are estimated as 0.123 and -0.072, respectively. 17.2 USING THE BINOMIAL TREE FOR OPTIONS ON INDICES, CURRENCIES, AND FUTURES CONTRACTS As explained in Chapters 11 and 14, stock indices, currencies, and futures contracts can, for the purposes of option valuation, be considered as assets providing known yields. In 5 If slightly more accuracy is required for delta and gamma, we can start the binomial tree at time -2.6.t and assume that the stock price is So at this time. This leads to the option price being calculated for three different stock prices at time zero. 399 B asic Numerical Procedures the case of a stock index, the releva~t yield is the dividend yield on the stock portfolio underlying the index; in the case of a currency, it is the foreign risk:free interest rate; in the case of a futures contract, it is the domestic risk-free interest f"ate. The binomial tree approach can therefore be used to value options on stock indices, currencies, and futures contracts provided that q in equation (17.7) is interpreted appropriately. Example 17.3 Consider a 4-month American call option on index futures where the current futures price is 300, the exercise price is 300, the risk-free interest rate is 8% per annum, and the volatility of the index is 30% per annum. We divide the life of the option into four I-month periods for the purposes of constructing the tree. In this case, Fa = 300, K = 300, r = 0.08, (J = 0.3, T = 0.3333, and !:It = 0.0833. Because a futures contract is analogous to a stock paying dividends at a rate r, q should be set equal to r in equation (17.7). This gives a = 1. The other parameters Figure 17.5 Binomial tree produced by DerivaGem for American call option on an index futures contract (Example 17.3). At each node: Upper value = Underlying Asset Price Lower value = Option Price Shading indicates where option is exercised Strike price = 300 Discount factor per step = 0.9934 Time step, dt = 0.0833 years, 30.42 days Growth factor per step, a = 1.0000 Probability of up move, p = 0.4784 Up step size, L! = 1.0905 Down step size, d = 0.9170r-_ _---r 327.14 33.64 275.11 6.13 Node Time: 0.0000 0.0833 0.1667 0.2500 0.3333 400 CHAPTER 17 necessary to construct the tree are II = eU.;t;i = 1.0905, 1 d =- = 0.9170 II a-d p = - - = 0.4784, ll-d 1- p = 0.5216 The tree, as produced by DerivaGem, is shown in Figure 17.5. (The upper number is the futures price; the lower number is the option price.) The estimated value of the option is 19.16. More accuracy is obtained using more steps. With 50 time steps, DerivaGem gives a value of 20.18; with 100 time steps it gives 20.22. Example 17.4 Consider a I-year American put option on the British pound. The current exchange rate is 1.6100, the strike price is 1.6000, the US risk-free interest rate is 8% Figure 17.6 Binomial tree produced by DerivaGem for AI:ner'ican put option on a currency (Example 17.4). , At each node: Upper value::: Underlying Asset Price Lower value::: Option Price Shading indicates where option is exercised Strike price::: 1.6 Discount factor per step::: 0.9802 Time step, dt ::: 0.2500 years, 91.25 days Growth factor per step, a ::: 0.9975 Probability of up move, p ::: 0.4642 Up step size, u ::: 1.0618 Down step size, d ::: 0.9418,..-_ _-1 Node Time: 0.0000 0.2500 0.5000 0.7500 1.0000 401 Basic Numerical Procedures per annum, the sterling risk-free interest rate is'9% per annum, and the volatility of the sterling exchange rate is 12% per annum. In this case, So = 1.61, K = 1.60, r = 0.08, '"1 = 0.09, a = 0.12, and T = 1.0. We divide the !jfe of the option into four 3-month periods for the purposes of constructing the tree, so that /:it = 0.25. In this case, q = rf and equation (17.7) gives a = e(O.08-0.09)xO.25 = 0.9975 The other parameters necessary to construct the tree are u = eU..jj;i 1 d=-=0.9418 u = 1.0618, a-d P = --d = 0.4642, u- 1- p = 0.5358 The tree, as produced by DerivaGem, is shown in Figure 17.6. (The upper number is the exchange rate; the lower number is the option price.) The estimated value of the option is $0.0710. (Using 50 time steps, DerivaGem gives the value of the option as 0.0738; with 100 time steps it also gives 0.0738.) 17.3 BINOMIAL MODEL fOR A DIVIDEND-PAYING STOCK We now move on to the more tricky issue of how the binomial model can be used for a dividend-paying stock. As in Chapter 13, the word dividend will, for the purposes of our discussion, be used to refer to the reduction in the stock price on the ex-dividend date as a result of the dividend. Known Dividend Yield If it is assumed that there is a single dividend, and the dividend yield (i.e., the dividend as a percentage of the stock price) is known, the tree takes the form shown in Figure 17.7 and can be analyzed in similar manner to that just described. If the time i /:it is prior to the stock going ex-dividend, the nodes on the tree correspond to stock prices j Soujd i - , j = 0,1, ... , i where u and d are defined as in equations (17.5) and (17.6). Ifthe time i /:it is after the stock goes ex-dividend, the nodes correspond to stock prices So(1- o)ujdi-j, j = 0,1, ... , i where 0 is the dividend yield. Several known dividend yields during the life of an option can be dealt with similarly. If OJ is the total dividend yield associated with all exdividend dates between time zero and time i /:it, the nodes at time i /:it correspond to stock prices Known Dollar Dividend In some situations, the most realistic assumption is that the dollar amount of the dividend rather than the dividend yield is known in advance. If the volatility of the 402 CHAPTER 17 stock, 0-, is assumed constant, the tree then takes the fonn shown in Figure 17.8. It does not recombine, which means that the number of nodes that have to be evaluated, particularly if there are several dividends, is liable to become very large. Suppose that there is only one dividend, that the ex-dividend date, T, is between k M and (k + 1) M, and that the dollar amount of the dividend is D. When i ~ k, the nodes on the tree at time i /),.t correspond to stock prices as before. When i = k + 1, the nodes on the tree correspond to stock prices Soujd i - j When i = k + 2, - D, j =_ 0,1,2, ... , i the nodes on the tree correspond to stock prices (Soujd i- 1- j - D)u and (Soujdi-I-j - D)d for j = 0,1,2, ... , i-I, so that there are 2i rather than i + 1 nodes. When i = k + 11l, there are m(k + 2) rather than k + 11l + 1 nodes. The problem can be simplified by assuming, as in the analysis of European options in Section 13.12, that the stock price has two components: a part that is uncertain and a part that is the present value of all future dividends during the life of the option. SuPP?se, as before, that there is only one e_x-dividend date, T, during the life of the Figure 17.7 Tree when stock pays a known dividend yield at one particular time. So1l4(1 -0) So So(1-0) t Ex.-dividend date So(1-0) B asic NUlnerical Procedures 403 Tree when dollar amount of dividend is assumed known and volatility is assumed constant. Figure 17.8 ~~ So Ex-dividend date option and that k flt :;:;; r:;:;; (k + 1) flt. The value of the uncertain component, S*, at time i M is given by S* = S when i flt > r and S* = S - De-r(r-iMl when i flt :;:;; r where D is the dividend. Define a* as the volatility of S* and assume that a* is constant. 6 The parameters p, u, and d can be calculated from equations (17.4), (17.5), (17.6), and (17.7) with a replaced by a* and a tree can be constructed in the usual way to model S*. By adding to the stock price at each node, the present value of future dividends (if any), the tree can be converted into another tree that models S. Suppose that So is the value of S* at time zero. At time i flt, the nodes on this tree correspond to the stock prices *zzid i - i So + De-r(r-i !!.tl , J. = 0 , 1,... ,z. when i flt < rand * SOU idi-j , j=O,l, ... ,i when i flt > 1:. This approach, which has the advantage of being consistent with the approach for European options in Section 13.12, succeeds in achieving a situation where 6 As mentioned in Section 12.13, a* is in theory slightly greater than (1, the volatility of S. In practice, the use of implied volatilities avoids the need for analysts to distinguish between (1 and (1*. . CHAPTER 17 404 the tree recombines so that there are i + 1 nodes at time i Llt. It can be generalized in a straightforward way to deal with the situation where there are several dividends. Example 17.5 Consider a 5-month American put option on a stock that is expected to pay a single dividend of $2.06 during the life of the option. The initial stock price is $52, the strike price is $50, the risk-free interest rate is 10% per annum, the volatility is 40% per annum, and the ex-dividend date is in 3~ months. We first construct a tree to model S*, the stock-price less the present value of future dividends during the life of the option. At time zero, the present value of the dividend is 2.06e-O.2917XO.J = 2.00 Figu re 17.9 Tree produced by DerivaGem for Example 17.5. At each node: Upper value Underlying Asset Price Lower value Option Price Shading indicates where option is exercised = = Strike price = 50 Discount factor per step 0.9917 Time step, dt = 0.0833 years, 30.42 days Growth factor per step, a = 1.0084 Probability of up move, p 0.5073 Up step size, u = 1.1224 Down step size, d 0.8909,--_ _--( = = Node Time: 0.0000 0.0833 0.1667 0.2500 0.3333 0.4167 B astc Nzimerical Procedures 405 The initial value of S* is therefore 50.00. Assuming that the 40% per annum volatility refers to S*, we find that Figure 17.3 provides a binomial tree for S*. (This is because S* has the same initial value and volatility ;:is the stock price that Figure 17.3 was based upon.) Adding the present value of the dividend at each node leads to Figure 17.9, which is a binomial model for S. The probabilities at each node are, as in Figure 17.3, 0.5073 for an up movement and 0.4927 for a down movement. Working back through the tree in the ust,Ial way gives the option price as $4.44. (Using 50 time steps, DerivaGem gives a value for the option of 4.202; using 100 steps it gives 4.212.) When the option lasts a long time (say, 3 or more years) it is usually more appropriate to assume a known dividend yield rather than a known cash dividend because the latter cannot reasonably be assumed to be the same for all the stock prices that might be encountered in the future. 7 Often for convenience the dividend yield is assumed to be paid continuously. Valuing an option on a dividend paying stock is then similar to valuing an option on a stock index. Control, Variate Technique A technique known as the control variate technique can improve the accuracy of the pricing of an American option. 8 This involves using the same tree to calculate both the value of the American option, fA, and the value of the corresponding European option, !E. We also calculate the Black-Scholes price of the European option, fBs. The error given by the tree in the pricing of the European option is assumed equal to that given by the tree in the pricing of the American option. This gives the estimate of the price of the American option as fA + fBs - fE To illustrate this approach, Figure 17.10 values the option in Figure 17.3 on the assumption that it is·European. The price obtained is $4.32. From the Black-Scholes formula, the true European price Of the option is $4.08. The estimate of the American price in Figure 17.3 is $4.49. The control variate estimate of the American price, therefore, is 4.49 + 4.08 - 4.32 = 4.25 A good estimate of the American price, calculated using 100 steps, is 4.278. The control variate approach does, therefore, produce a considerable improvement over the basic tree estimate of 4.49 in this case. The control variate technique in effect involves using the tree to calculate the difference between the European and the American price rather than the American price itself. We give a further application of the control variate technique when we discuss Monte Carlo simulation later in the chapter. Another problem is that, for long-dated options, S* is significantly less than So and volatility estimates can be very high. 7 S See J. Hull and A. White, "The Use of the Control Variate Technique in Option Pricing," Joul7lal oj Financial and Quantitative Analysis, 23 (September 1988): 237-51. 406 CHAPTER 17 Tree, as produced by DerivaGem, for European version of option in Figure 17.3. At each node, the upper number is the stock price, and the lower number is the option price. Figure 17.10 At each node: Upper value = Underlying Asset Price Lower value = Option Price Shading indicates where option is exercised = Strike price 50 Discount factor per step = 0.9917 Time step, dt = 0.0833 years, 30.42 days Growth factor per step, a = 1.0084 Probability of up move, p= 0.5073 Up step size, u = 1.1224 Down step size, d = 0.8909.--_ _-1 62.99 0.64 50.00 3.67 39.69 9.86 Node Time: 0.0000 0.0833 0.1667 0.2500 0.3333 0.4167 17 .4 ALTERNATIVE PROCEDURES FOR CONSTRUCTING TREES The Cox, Ross, and Rubinstein approach is not the only way of building a binomial tree. Instead of imposing the assumption u = 1/d on equations (17.2) and (17.3), we can set p = 0.5. A solution to the equations when terms of higher order than .6..t are ignored is then d = i r - q- cl /2)b.r-<f-Jj;i B asic Numerical PTOcedures 407 This allows trees with p = 0.5 to be built for options on indices, foreign exchange, and futures. . This alternative tree-building procedure has the advantage ove: the Cox, Ross, and Rubinstein approach that the probabilities are always 0.5 regardless of the value of (J or the number of time steps.9 Its disadvantage is that it is not as straightforward to calculate delta, gamma, and rho from the tree because the tre.e is no longer centered at the initial stock price. . Example 17.6 Consider a 9-month American call option on the Canadian dollar. The current exchange rate is 0.7900, the strike price is 0.7950, the US risk-free interest rate is 6% per annum, the Canadian risk-free interest rate is 10% per annum, and the Binomial tree for American call option on the Canadian dollar. At each node, upper number is spot exchange rate and lower number is option price. All probabilities are 0.5. Figure 17.11 At each node: Upper value = Underlying Asset Price Lower value = Option Price Shading indicates where option is exercised Strike price = 0.795 Discount factor per step = 0.9851 Time step, dt = 0.2500 years, 91.25 days Probability of up move, p = 0.5000 .0.8056 0.7900 0.0026 Node Time: 0.0000 0.2500 0.5000 0.7500 When time steps are so large that a < ICr - q).JI;i I, the Cox, Ross, and Rubinstein tree gives negative probabilities. The alternative procedure described here does not have that drawback. 9 408 CHAPTER 17 volatility of the exchange rate is 4% per annum. In this case, So = 0.79, K = 0.795, r = 0.06, r f = 0.10, (T = 0.04, and T = 0.75. We divide the life of the option into 3-month periods for the purposes of constructing the tree, so that M = 0.25. We set the probabilities on each branch to 0.5 and u = e(O.06-0.1O-0.0016/2)O.25+0.04J[25 = 1.0098 d = /O.06-0.10-0.0016/2)O.25-0.04J[25 = 0.9703 The tree for the exchange rate is shown in Figure 17.11. The tree gives the value of the option as $0.0026. Trinomial Trees Trinomial trees can be used as an alternative to binomial trees. The general form of the - tree is as shown in Figure 17.12. Suppose that P,,, Pm, and Pd are the probabilities of up, middle, and down movements at each node and I1t is the length of the time step. For a non-dividend-paying stock, parameter values that match the mean and standard deviation of price changes when terms of higher order than I1t are ignored are u Pd = -J (rM 120-2 2 q- 0- ) 2 d='£ u = eu..l3"i1i, +~ 6' 2 P;1l =3" 409 B asic Numerical Procedures Calculations for a trinomial tree are analogous to those for a binomial tree. We work from the end of the tree to the beginning. At each node we .calculate the value of exercising and the value of continuing. The value of continuing is where 1,,, 1,,1' and fd are the values of the option at the subsequent up, middle, and down nodes, respectively. The trinomial tree approach proves to be equivalent to the explicit finite difference method, which will be described in Section 17.8. Figlewski and Gao have proposed an enhancement of the trinomial tree method, which they call the adaptive mesh model. In this, a high-resolution (small-.6.t) tree is grafted onto a low-resolution (large-.6.t) tree. lO When valuing a regular American option, high resolution is most useful for the parts of the tree close to the strike price at the end of the life of the option. 17.5 TIME-DEPENDENT PARAMETERS Up to now we have assumed that r, q, r j, and a are constants. In practice, they are usually assumed to be time dependent. The values of these variables between times t and t + .6.t are assumed to be equal to their forward valuesY We can make rand q (or rj) a function of time in a Cox-Ross-Rubinstein binomial tree. We set a = e(f(I)-g(t)]ll.t (17.11) for nodes at time t, where f(t) is the forward interest rate between times t and t + .6.t and get) is the forward value of q between these times. This does not change the geometry of the tree because II and d do not depend on a. The probabilities on the branches emanating from nodes at time tare: 12 e(f(t)-g(t)]ll.t _ p= d ll-d (17.12) II _ e(f(t)-g(t)]M l-p=----ll-d The rest of the way that we use the tree is the same as before, except that when discounting between times t and t + .6.t we use f(t). Making a a function of time in a binomial tree is more challenging. One approach is to make the lengths of time steps inversely proportional to the variance rate. The values of II and d are then always the same and the tree recombines. Suppose that aCt) is the volatility for a maturity t so that a(t)2 t is the cumulative variance by time t. Define V = a(T)2T, where T is the life of the tree, and let t; be the end of the ith time step. If there is a total 10 See s. Figlewski and B. Gao, "The Adaptive Mesh Model: A New Approach to Efficient Option Pricing," Journal of Financial Economics, 53 (1999): 313-51. II The forward dividend yield and fonvard variance rate are calculated in the same way as the fonvard interest rate. (The variance rate is the square of the volatility.) 12 For a sufficiently large number of time steps, these probabilities are always positive. CHAPTER 17 410 Business Snapshot 17.1 Suppose the sides of the square in Figur you fire darts randomly at the square circle. What should you find? The square has an of 0.5 The area of the circle is Jr times the radiu percentage of darts· that lie in the circle shou multiplying the percentag 'e in the circle by We can use an Excel s heet to simulate th Table 17..1. We define both cell Al a ell BIas numbers between 0 and 1 and defin lands in the square in Figure 17.13. =IF((AI-O.Sf2+(BI-O.S)A2<OS2,4, , This has the effect of setting C1 equal to 4 i Define the next 99 rows of the· sprea "select and drag" operation in Excel.) C103 as =STDEV(C1:ClOO); C102(which . calculated from 100 random trials. C103 is t ta as we will see in Example 17.7 can be used to Increasing the number of trials improves accura value of 3.14162 is slow. of N time steps, we choose ti to satisfy a(ti)\ = iV / N. The variance between times ti-I and ti is then V / N for all i. With a trinomial tree, a generalized tree-building procedure can be used to match timedependent interest rates and volatilities (see Technical Note 9 on the author's website). 17.6 MONTE CARLO SIMULATION We now explain Monte Carlo simulation, a quite different approach for valuing derivatives from binomial trees. Business Snapshot 17.1 illustrates the random sampling idea underlying Monte Carlo simulation by showing how a simple Excel program can be constructed to estimate If. Figure 17.13 Calculation of If by throwing darts. Basic lVumerical Procedures 411 Sample spreadsheet calculations in Business Snapshot 17.1. Table 17.1 A B C 1 2 3 0.207 0.271 0.007 0.690 0.520 0.221 4 4 0 100 101 102 103 0.198 0.403 4 Mean: SD: 3.04 1.69 When used to value an option, Monte Carlo simulation uses the risk-neutral valuation result. We sample paths to obtain the expected payoff in a risk-neutral world and then discount this payoff at the risk-free rate. Consider a derivative dependent on a single market variable S that provides a payoff at time T. Assuming that interest rates are constant, we can value the derivative as follows: 13 . 1. Sample a random path for S in a risk-neutral world. 2. Calculate the payoff from the derivative. 3. Repeat steps 1 and 2 to get many sample values of the payoff from the derivative in a risk-neutral world. 4. Calculate the mean of the sample payoffs to get an estimate of the expected payoff in a risk-neutral world. 5. Discount the expected payoff at the risk-free rate to get an estimate of the value of the derivative. Suppose that the process followed by the underlying market variable in a risk-neutral world is dS = (lS dt + (J"S dz (17.13) where dz is a Wiener process, (l is the expected return in a risk-neutral world, and (J" is the volatility.14 To simulate the path followed by S, we can divide the life of the derivative into N short intervals of length D.t and approximate equation (17.13) as Set + D.t) - Set) = (lS(t) D.t + (J"S(t)E,J!;j (17.14) where Set) denotes the value of S at time t, E is a random sample. from a normal distribution with mean zero and standard deviation of 1.0. This enables the value of S at time D.t to be calculated from the initial value of S, the value at time 2 D.t to be calculated from the value at time D.t, and so on. An illustration of the procedure is in Section 12.3. One simulation trial involves constructing a complete path for S using N random samples from a normal distribution. 13 We discuss how Monte Carlo simulation can be used with stochastic interest rates in Section 25.4. If S is the price of a non-dividend-paying stock then fl = r, if it is an exchange rate then fl = r - rf, and so on. Note that the volatility is the same in a risk-neutral world as in the real world, as shown in Section 11.7. 14 412 CHAPTER 17 In practice, it is usually more accurate to simulate In S rather than S. From Ito's lemma the process followed by In S is dIn S = (tl - ~2) dt + adz (17.15) so that In Set + t1t) -In Set) = (tl- ~) t1t + aE-liS:i or equivalently (17.16) This equation is used to construct a path for S. - The advantage of working with In S is that it follows a generalized Wiener process. This means that the equation In SeT) - In S(Q) = (tl - ~2) T + aE-IT is true for all T. 15 It follows that SeT) = S(Q) exp [ (i1- ~2) T + aE-IT ] (17.17) This equation can be used to value derivatives that provide a nonstandard payoff at time T. As indicated in Business Snapshot 17.2, it can also be used to check the Black-Scholes formulas. The key advantage of Monte Carlo simulation is that it can be used when the payoff depends on the path followed by the underlying variable S as well a& when it depends only on the final value of S. (For example, is can be used when payoffs depend on the average value of S.) Payoffs can occur at several times during the life of the derivative rather than all at the end. Any stochastic process for S can be accommodated. As will be shown shortly, the procedure can also be extended to accommodate situations where the payoff from the derivative depends on several underlying market variables. The drawbacks of Monte Carlo simulation are that it is computationally very time consuming and cannot easily handle situations where there are early exercise opportunities. 16 Derivatives Dependent on More than One Market Variable Consider the situation where the payoff from a derivative depends on n variables e i (1 :::;; i :::;; n). Define Si as the volatility of e , Ini as the expected growth rate of e in a riski i neutral world, and Pik as the instantaneous correlation between ei and e • 17 As in the k 15 By contrast, equation (17.14) is true only in the limit as llt tends to zero. 16 As we wi1(discuss in Chapter 24, a number of researchers have suggested ways Monte Carlo simulation can ,be extended 10 value American options. 17 Note that Sj, lii j , and Pik are not necessarily constant; they may depend on the Bi • 413 B asic Numerical Procedures Business Snapshot 17.2 This is the present value of the payoff from a call 0 of the ~preadsheet similarly to the first one. (This i Excel.) Define B1002 a AVERAGE(B1:B1000) a Bl002 (which is 4.98 i ample spreadsheet option. This should be not too far from the c Example 17.8, B1003 can be used to assess the accurac single-variable case, the life of the derivative must be divided into N subintervals of length b.t. The discrete version of the process for 8i is then (17.18) where Ei is a random sample from a standard normal distribution. The coefficient of correlation between €i and Ek is Pik (1 :::;;; i; k :::;;; n). One simulation trial involves obtaining N samples of the Ei (1 :::;;; i :::;;; n) from a multivariate standardized normal distribution. These are substituted into equation (17.18) to produce simulated paths for each 8i , thereby enabling a sample value for the derivative to be calculated. Table 17.2 Monte Carlo simulation to check Black-Scholes A B 1 2 3 4 5 45.95 54.49 50.09 47.46 44.93 0 4.38 0.09 0 0 1000 1001 1002 1003 68.27 17.82 Mean: SD: 4.98 7.68 t::=:':.---y"" C D E F So K 50 d] 0.2239 r 0.05 d2 0.0118 a 0.3 BS price 4.817 50 G T 0.5 414 CHAPTER 17 Generating the Random Samples from Normal Distributions An approximate sample from a univariate standardized normal distribution can be obtained from the formula 12 E= L (17.19) Ri -6 i=I ° where the Ri (1 :s;; i :s;; 12) are independent random numbers between and 1, and E is the required sample from ¢(O, 1). This approximation is satisfactory for most purposes. An alternative approach in Excel is to use =NORMSINV(RANDO) as in Business Snapshot 17.2. When two correlated samples EI and E2 - from standard normal distributions are required, an appropriate procedure is as follows. Independent samples ~I and X2 from a univariate standardized normal distribution are obtained as just described. The -required samples EI and E2 are then calculated as follows: EI =XI E2 = PXI + X2"\!l - p 2 where p is the coefficient of correlation. More generally, consider the situation where we require Il correlated samples from normal distributions with the correlation qetween sample i and sample j being Pij' We first sample n independent variables Xi (1:S;; i :s;; n), from univariate standardized normal distributions. The required samples, Ei (1 :s;; i :s;; n), are then defined as follows: El = Q'IlXl E2 = Q'21 X l + Q'n X 2 E3 = Q'3I X I + Q'32 X 2 + Q'33 X 3 and so on. We choose the coefficients Q'ij so that the correlations and variances are correct. This can be done step by step as follows. Set Q'll = 1; choose Q'21 so that ? ? Q'21Q'II = P2I; choose Q'n so that Q'21 +Q'22 = 1; choose Q'31 so that Q'31Q'II = P3I; choose IS Q'32 so that Q'31 Q'21 + Q'32Q'22 = P32; choose Q'33 so that Q'~I + Q'~2 + Q'~3 = 1; and so on. This procedure is known as the Cholesky decomposition. Number of Trials The accuracy of the result given by Monte Carlo simulation depends on the number of trials. It is usual to calculate the standard deviation as well as the mean of the discounted payoffs given by the simulation trials. Denote the mean by f.J., and the standard deviation by (v. The variable f.J., is the simulation's estimate of the value of the derivative. The standard error of the estimate is where M is the number of trials. A 95% confidence interval for the price f of the 18 IT the equations for the a's do not have real solutions, the assumed correlation structure is internally inconsistent This will be discussed further in Chapter 19. B asic Numerical Procedures 415 derivative is therefore given by This shows that our uncertainty about the value of the derivative is inversely proportional to the square root of the number of trials. To double the accuracy of a simulation, we must qlfadruple the number of trials; to increase' the accuracy by a factor of 10, the number of trials must increase by a factor of 100; and so on. Example 17.7 In Table 17.1, n: is calculated as the average of 100 numbers. The standard deviation of the numbers is 1.69. In this case, {J) = 1.69 and M = 100, so that the standard error of the estimate is 1.69/.JI50 = 0.169. The spreadsheet therefore gives a 95% confidence interval for n: as (3.04 - 1.96 x 0.169) to (3.04+ 1.96 x 0.169) or 2.71 to 3.37. Example 17.8 In Table 17.2, the value of the option is calculated as the average of 1000 numbers. The standard deviation of the numbers is 7.68. In this case, {J) = 7.68 and M = 1000. The standard error of the estimate is 7.68/-J1000 = 0.24. The spreadsheet therefore gives a 95% confidence interval for the option value as (4.98 - 1.96 x 0.24) to (4.98 + 1.96 x 0.24), or 4.51 to 5.45. Applications Monte Carlo simulation tends to be numerically more efficient than other procedures when there are three or more stochastic variables. This is because the time taken to carry out a Monte Carlo simulation increases approximately linearly with the number of variables, whereas the time taken for most other procedures increases exponentially with the number of variables. One advantage of Monte Carlo simulation is that it can provide a standard error for the estimates that it makes. Another is that it is an approach that can accommodate complex payoffs and complex stochastic processes. Also, it can be used when the payoff depends on some function of the whole path followed by a variable, not just its terminal value. Calculating the Greek Letters The Greek letters discussed in Chapter 15 can be calculated using Monte Carlo simulation. Suppose that we are interested in the partial derivative of f with respect to x, where f is the value of the derivative and x is the value of an underlying variable or a parameter. First, Monte Carlo simulation is used in the usual way to calculate an estimate j for the value of the derivative. A small increase Llx is then made in the value of x, and a new value for the derivative, j*, is calculated in the same way as j. An estimate for the hedge parameter is given by j*-j Llx In order to minimize the standard error of the estimate, the number of time intervals, N, 416 CHAPTER 17 the random number streams, and the number of trials, M, should be the same for calculating both j and j*. Sampling through a Tree Instead of implementing Monte Carlo simulation by randomly sampling from the stochastic process for an underlying variable, we can use an N-step binomial tree and sample from the 2N paths that are possible. Suppose we have a binomial tree where the probability of an "up" movement is 0.6. The procedure for sampling a random path through the tree is as follows. At each node, we sample a random number between 0 and 1. If the number is less than 0.4, we take the down path. If it is greater than 0.4, we take the up path. Once we have a complete path from the initial node to the end of the tree, we can calculate a payoff. This completes the first trial. A similar procedure is used to complete more trials. The mean of the payoffs is discounted at the risk-free rate to - get an estimate of the value of the derivative. 19 Example 17.9 Suppose that the tree in Figure 17.3 is used to value an option that pays off max(Save - 50, 0), where Save is the average stock price during the 5 months (with the first and last stock price being included in the average). This is known as an Asian option. When ten simulation trials are used one possible result is shown in Table 17.3. Table 17.3 Monte Carlo simulation to value Asian option from the tree in Figure 17.3. Payoff is amount by which average stock price exceeds $50. U = up movement; D = down movement. Trial Path A verage stock price Option paYoff 1 UUUUD UUUDD DDDUU UUUUU UUDDU UDUUD DDUDD UUDDU UUUDU DDUUD 64.98 59.82 42.31 68.04 55.22 55.22 42.31 55.22 62.25 45.56 14.98 9.82 0.00 18.04 5.22 5.22 0.00 5.22 12.25 0.00 2 3 4 5 6 7 8 9 10 Average The value of the option is calculated as the average payoff discounted at the risk-free rate. In this case, the average payoff is $7.08 and the risk-free rate is 10% and so the calculated value is 7.08e-O.lx5/l2 = 6.79. (This illustrates the methodology. In practice, we would have to use more time steps on the tree and many more simulation trials to get an accurate answer.) 19 See D. Mintz, "Less is More," Risk, July 1997: 42-45, for a discussion of how sampling through a tree can be made efficient. B asic Numerical Procedures 417 17.7 VARIANCE REDUCTION PROCEDURES ? If the simulation is carried out as described so far, a very large number of trials is usually necessary to estimate I with reasonable accuracy. This is very expensive in terms of computation time. In this section, we examine a number of variance reduction procedures that can lead to dramatic savings in computation ·~ime. Antithetic Variable Technique In the antithetic variable technique, a simulation trial involves calculating two values of the derivative. The first value 11 is calculated in the usual way; the second value h is calculated by changing the sign of all the random samples from standard normal distributions. (If E is a sample used to calculate 11, then -E is the corresponding sample used to calculate h.) The sample value of the derivative calculated from a simulation trial is the average of 11 and h. This works well because when one value is above the true value, the other tends to be below, and vice versa. Denote ] as the average of II and h: =/l+h 2 The final estimate of the value of the derivative is the average of the ]'s. If w is the standard deviation of the ]'s, and M is the number of simulation trials (i.e., the number of pairs of values calculated), then the standard error of the estimate is This is usually much less than the standard error calculated using 2M random trials. Control Variate !echnique We have already given one example of the control variate technique in connection with the use of trees to value American options (see Section 17.3). The control variate technique is applicable when there are two similar derivatives, A and B. Derivative A is the security being valued; derivative B is similar to derivative A and has an analytic solution available. Two simulations using the same random number streams and the same I::i.t are carried out in parallel. The first is used to obtain an estimatel:l of the value of A; the second is used to obtain an estimate I;, of the value of B. A better estimate IA of the value of A is then obtained using the formula IA = l:l - I; + Is (17.20) where Is is the known true value of B calculated analytically. Hull and White provide an example of the use of the control variate technique when evaluating the effect of stochastic volatility on the price of a European call option. 2o In this case, IA is the estimated value of the option assuming stochastic volatility and Is is its Black-Scholes value assuming constant volatility. 20 See J. Hull and A. White, "The Pricing of Options on Assets with Stochastic Volatilities," Journal oJ Finance, 42 (June 1987): 281-300. 418 CHAPTER 17 Importance Sampling Importance sampling is best explained with an example. Suppose that we wish to calculate the price of a deep-out-of-the-money European call option with strike price K and maturity T. If we sample values for the underlying asset price at time T in the usual way, most of the paths will lead to zero payoff. This is a waste of computation time because the zero-payoff paths contribute very little to the determination of the value of the option. We therefore try to choose only important paths, that is, paths where the stock price is above K at maturity. Suppose F is the unconditional probability distribution function for the stock price at time T and q, the probability of the stock price being greater than K at maturity, is known analytically. Then G = F jq is the probability distribution of the stock price conditional on the stock price being greater than K. To implement: importance sampling, we sample from G rather than F. The estimate of the value of the option - is the average discounted payoff multiplied by q. Stratified Sampling Sampling representative values rather than random values from a probability distribution usually gives more accuracy. Stratified sampling is a way of doing this. Suppose we wish to take 1000 samples from a probability distribution we would divide the distribution into 1000 equally likely intervals and choose a representative value Ctypically the mean or median) for each interval. In the case of a standard normal distribution when there are n intervals, we can calculate the representative value for the ith interval as where N- 1 is the inverse cumulative normal distribution. For example, wheh n = 4 the representative values corresponding to the four intervals are N- 1CO.125), N- 1C0.375), N- 1CO.625), N- 1CO.875). The function N- 1 can be calculated using the NORMSINV function in Excel. Moment Matching Moment matching involves adjusting the samples taken from a standardized normal distribution so that the first, second, and possibly higher moments are matched. Suppose that we sample from a normal distribution with mean 0 and standard deviation 1 to calculate the change in the value of a particular variable over a particular time period. Suppose that the samples are Ei (1 ::s; i ::s; n). To match the first two moments, we calculate the mean of the samples, m, and the standard deviation of the samples, s. We then define adjusted samples E7 (l ::s; i ::s; n) as E·-m E'!'=-'-I s These adjusted samples have the correct mean of 0 and the correct standard deviation of 1.0. We use the adjusted samples for all calculations. 419 B asic Numerical Procedures Moment matching saves compl,ltation time, but can lead to memory problems because every number sampled must be stored until the end of the simulation. Moment matching is sometimes termed quadratic resampling. It is often USed in conjunction with the antithetic variable technique. Because the latter automatically matches all odd moments, the goal of moment matching then becomes that of matching the second moment and, possibly, the fourth moment. Using Quasi-Random Sequences A quasi-random sequence (also called a [ow-discrepancy sequence) is a sequence of representative samples from a probability distribution. 21 Descriptions of the use of quasi-random sequences appear in Brotherton-Ratcliffe, and Press et al. 22 Quasi-random sequences can have the desirable property that they lead to the standard error of an estimate being proportional to 1/ M rather than l/.JM, where M is the sample size. Quasi-random sampling is similar to stratified sampling. The objective is to sample representative values for the underlying variables. In stratified sampling, it is assumed that we know in advance how many samples will be taken. A quasi-random sampling scheme is more flexible. The samples are taken in such a way that we are always "filling in" gaps between existing samples. At each stage of the simulation, the sampled points . are roughly evenly spaced throughout the probability space. Figure 17.14 shows points generated in two dimensions using a procedure suggested by Sobol,.23 It can be seen that successive points do tend to fill in the gaps left by previous points. 17.8 FINITE DIFFERENCE METHODS Finite difference methods value a derivative by solving the differential equation that the derivative satisfies. The differential equation is converted into a set of difference equations, and the difference equations are solved iteratively. To illustrate the approach, we consider how it might be used to value an American put option on a stock paying a dividend yield of q. The differential equation that the option must satisfy is, from equation (14.6), (17.21) Suppose that the life of the option is T. We divide this into N equally spaced intervals of length I::.t = T / N. A total of N + 1 times are therefore considered 0, !::.t, 21::.t, ... , T 21 The term quasi-random is a misnomer. A quasi-random sequence is totally deterministic. See R. Brotherton-Ratcliffe, "Monte Carlo Motoring," Risk, December 1994: 53-58; W. H. Press, S. A. Teukolsky, W. T. Vetteriing, and B. P. Flannery, Numerical Recipes in C: The Art oj Scielllific Computing, 2nd edn. Cambridge University Press, 1992. 22 See 1. M. Sobol', USSR Computational Mathematics and Mathematical Physics, 7, 4 (1967): 86-112. A description of Sobol's procedure is in W. H. Press, S. A. Teukolsky, W. T. Vetteriing, and B. P. Flannery, Numerical Recipes in C: The Art oj Scientific Computing, 2nd edn. Cambridge University Press, 1992. 23 420 CHAPTER 17 Figure 17.14 First 1024 points of a Sobol' sequence. 1.0 r-r"-ro-..--r-,-r-r"-r-r...,.-,-,-,,,.-, : .: °0 •• : ° 0.8 '. .°0·· • .. •° .. ".. "0 0·. . .' . ." 0 " ••• 0.4 • • •0 0.2 eO· . • 0 ••• 0 00': :. . 0: .0° .0 ° 0.6 0.4 • ~.;:: >~. ~ . / :::. • 0.6 0° 0 • 0.8 :-.... . • 'I. " : .' :. '.. . ": ":. '. ..:' "0 . . .: •• '0: ':.. :'.0 ., :.: ."0 .. 0:- . •• 0.2 . ": •• ' .. ".- • I 00 00 '. • Points 129 to 512 Points 1 to 128 I •• 0.8 . • "0" ". S;J::~i::i;l.::~';~;;::t:~}: 1.0 Points 1 to 1024 Points 513 to 1024 Suppose that Smax is a stock price sufficiently high that, when it is reached, the put has virtually no value. We define /::,.S = Sma:,j M and consider a total of M + I equally spaced stock prices: 0, /::,.S, 2/::,.S, ... , Smax The level Smax is chosen so that one of these is the current stock price. The time points and stock price points define a grid consisting of a total of (M + 1)(N + 1) points, as shown in Figure 17.15. The (i, j) point on the grid is the point that corresponds to time i /::,.t and stock price j /::"S. We will use the variable fi.j to denote the value of the option at the (i, j) point. Implicit Finite Difference Method For an interior point (i, j) on the grid, afjaS can be approximated as af fi.j+! - fi.j as= /::,.s (17.22) B asic Numerical Procedures 421 or as af h,j - fi,j-l = as !:lS (17.23) Equation (17,22) is known as the forward difference approximation; equation (17,23) is known as the backward difference approximation. We use a more sYIIllll,etrical approximation by averaging the two: af as = h,j+l - fi,j-l 2!:lS (17.24) For af/at, we will use a forward difference approximation so that the value at time i!:lt is related to the value at time (i + 1) !:It: Ji+l,j - h,j !:It af at (17.25) The backward difference approximation for af/as at the (i, j) point is given by equation (17.23). The backward difference at the (i, j + 1) point is h,j+l - h,j !:lS Figure 17.15 Grid for finite difference approach. Stock price, S ••• • • • • • ••• • •• ••• •• ••• ••••• • • •• •• .. • • • • Ii) III Ii) III III III Ii) iii III III It It iii It CD Ii) CD It Ii) CD It Ii) •• III III III • • iii • III Ii) III • • •• •• It • •• ••• ••• • • .. • • • III CD It It It It Time,! ill T 422 CHAPTER 17 Hence a finite difference approximation for a2 flaS 2 at the (i, j) point is 2 af .as 2 (AHI - = fi,j _ fi,j - Aj-I)//::,.s /::,.S /::,.S or (17.26) Substitu,ting equations (17.24), (17.25), and (17.26) into the differential equation (17.21) and noting that S = j /::,.S gives - for j = 1,2, ... , M - 1 and i = 0,1 ... , N - 1. Rearranging terms, we obtain (17.27) where aj = 1(r- q)j b..t -la2 /M hj = 1 +a2/M+rM Cj I I = -2 ( r-q ) }. t i t - 2a2·2 tit } A A The value of the put at time T is max(K - ST, 0), where ST is the stock price at time T. Hence, (17.28) fN,j = max(K - j /::"S, 0), j = 0, 1, ... , M The value of the put option when the stock price is zero is K. Hence, Ao = K, i = 0, 1, ... , N (17.29) We assume that the put option is worth zero when S = Sma:", so that AM = 0, i = 0, 1, ... , N (17.30) Equations (17.28), (17.29), and (17.30) define the value of the put option along the three edges of the grid in Figure 17.15, where S = 0, S = Smax' and t = T. It remains to use equation (17.27) to arrive at the value of f at all other points. First the points corresponding to time T - /::"t are tackled. Equation (17.27) with i = N - 1 gives (17.31) for j = 1,2, ... , M - 1. The right-hand sides of these equations are known from equation (17.28). Furthermore, from equations (17.29) and (17.30), fN-I,O =K (17.32) fN-I,M =0 (17.33) 423 B asic Numerical Procedures Equations (17.31) are therefore M - 1 simultaneous equations that can be solved for the M - 1 unknowns: fN-I 1> fN-I J, .... , fN-1 M_I· 24 After this has been done, each value of fN-I,j is compared' with i - j I:lS. If fN-I,j < K - j I:lS; early exercise at time ·T - b..t is optimal and fN-I,j is set equal to K - j I:lS. The nodes corresponding to time T - 2 I:lt are handled in a similar way, and so on. Eventually, fO,I' fO,2' fO,3,"" fO,M-I are obtained. One of these is the option price of interest. The control variate technique can be used in conjuncti'on with finite difference methods. The same grid is used to value an option similar to the one under consideration but for which an analytic valuation is available. Equation (17.20) is then used. Example 17.10 Table 17.4 shows the result of using the implicit finite difference method as just described for pricing the American put option in Example 17.1. Values of 20, 10, and 5 were chosen for M, N, and I:lS, respectively. Thus, the option price is evaluated at $5 stock price intervals between $0 and $100 and at half-month time intervals throughout the life of the option. The option price given by the grid is $4.07. The same grid gives the price of the corresponding European option as $3.91. The true European price given by the Black-Scholes formula is $4.08. The control variate estimate of the American price is therefore 4.07 + 4.08 - 3.91 = $4.24 Explicit Finite Difference Method The implicit finite difference method has the advantage of being very robust. It always converges to the solution of the differential equation as I:lS and I:lt approach zero. 25 One of the disadvantages of the implicit finite difference method is that M - 1 simultaneous equations have to be solved in order to calculate the fi,j from the fi+l,j' The method can 2 be simplified if the values of aflaS and a f/as 2 at point (i, j) on the grid are assumed to be the same as at point (i + 1, j). Equations (17.24) and (17.26) then become af _ fi+l,j+1 - as - fi+l,j-1 21:lS J a- f _ fi+l,j+1 as 2 - + fi+l,j-I - 2fi+l,j I:lS 2 The difference equation is fi+l,j - /;,j + (r _ ). I:lS fi+l,j+l - fi+l,j-I b..t qJ. 21:lS I +2 2·2 AS2 -(J ] II fi+l,j+1 + fi+l,j-1 I:ls 2 - 2fi+l,j - rJ... l,j This does not involve inverting a matrix. The j= 1 equation in (17.31) can be used to express iN-I.1 in terms of iN-I.I; the j = 2 equation, when combined with the j = I equation, can be used to express iN-I.3 in terms of iN-I.I; and so on. The j = M - 2 equation, together with earlier equations, enables iN-I.M-1 to be expressed in terms of iN-I.I. The final j = M - I equation can then be solved for iN.,-I,I, which can then be used to determine the other iN-l.j. 14 A general rule in finite difference methods is that t1S should be kept proportional to zero. 15 .JKi as they approach 424 CHAPTER 17 Table 17.4 Grid to value American option in Example 17.1 using implicit finite difference methods. Stock price (dollars) 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 100 95 90 85 80 75 70 - 65 60 55 50 45 40 35 30 25 20 15 10 5 0 0.00 0.02 0.05 0.09 0.16 0.27 0.47 0.82 1.42 2.43 4.07 6.58 10.15 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 0.00 0.02 0.04 0.07 0.12 0.22 0.39 0.71 1.27 2.24 3.88 6.44 10.10 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 0.00 0.01 0.03 0.05 0.09 0.17 0.32 0.60 1.11 2.05 3.67 6.29 10.05 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 0.00 0.01 0.02 0.03 0.07 0.13 0.25 0.49 0.95 1.83 3.45 6.13 10.01 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 0.00 0.00 0.01 0.02 0.04 0.09 0.18 0.38 0.78 1.61 3.19 5.96 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 0.00 0.00 0.01 0.01 0.03 0.06 0.13 0.28 0.62 1.36 2.91 5.77 10.00 15.00 - 20.00 25.00 30.00 35.00 40.00 45.00 50.00 0.00 0.00 0.00 0.01 0.02 0.03 0.08 0.19 0.45 1.09 2.57 5.57 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 0.00 0.00 0.00 0.00 0.01 0.02 0.04 0.11 0.30 0.81 2.17 5.36 0.00 0.00 0.00 0.00 0.00 0.01 0.02 0.05 0.16 0.51 1.66 5.17 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.02 0.05 0.22 0.99 5.02 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 Time to maturity (months) 1O.QO 15.00 20.00 25.00 30.00 35.00 40.00 45.00 50.00 or Aj = aj ii+l,j-1 + bj ii+l.j + cj ii+l,j+1 (17.34) where aj 1 = 1 +rM (-t(r - bj = 1 +r/::;.t (1 1 cj 1 = 1 +rM et(r - q)j M+ t(j2/ M ) - (j2 j 2 M) q)j /::;.t + t(j2 / - /::;.t) This creates what is known as the explicit finite difference method. 26 Figure 17.16 shows the difference between the implicit and explicit methods. The implicit method leads to equation (17.27), which gives a relationship between three different values of the option at time i M (i.e., ii,j-lo ii,j, and ii,HI) and one value of the option at time (i + 1) M 26 We !llso obtain the explicit finite difference method if we use the backward difference approximation instead of the forward difference approximation for aflat. B asic Numerical Procedures Figure 17.16 425 Difference between implicit and explicit finite difference methods. Ji+l,j.f.l fi,j+l fij .-------41 Ji+l,j fij ._------0 fi,j-l fi+l,j fi+l,j-J Implicit finite difference method Explicit finite difference method (i.e., li+l). The explicit method leads to equation (17.34), which gives a relationship between one value of the option at time i !::l.t (i.e., h) and three different values of the option at time (i + l)!::l.t (i.e., h+I,j-b h+I,j' h+I,i+z)' Example 17.11 Table 17.5 shows the result of using the explicit version of the finite difference method for pricing the American put option in Example 17.1. As in Example 17.10, values of 20, 10, and 5 were chosen for M, N, and !::l.S, respectively. The option price given by the grid is $4.26. 27 Change of Variable It is computationally more efficient to use finite difference methods with In S rather than S as the underlying variable. Define Z = In S. Equation (17.21) becomes 2 2 a 1 a- all a 1 -+ ( r - q -2 )- az - a2 -=1'1 + at 2 az 2 The grid then evaluates the derivative for equally spaced values of Z rather than for equally spaced values of S. The difference equation for the implicit method becomes h+l,j - li,j + (. - q - a 2/7) Aj+! - li,j-I 1 !::l.t - 2!::l.Z + -a2 Aj+1 + li,j-I 1 2 !::l.Z2 21i,j - 'Ii., 1 l,j or (17.35) 27 The negative numbers and other inconsistencies in the top left-hand part of the grid will be explained later. CHAPTER 17 4 26 Table 17.5 Grid to value American option in Example 17.1 using explicit finite difference method. Time to maturity (months) Stock price (dollars) 100 95 90 85 80 75 70 65 60 55 50 45 40 .35 30 25 20 15 10 5 0 5 , 4.5 3.5 4 2.5 3 2 1 1.5 0.5 0 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.11 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.28 -0.05 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -0.13 0.20 0.00 0.05 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.46 0.06 0.20 0.04 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.32 0.46 0.23 0.25 0.10 0.09 0.00 0.00 0.00 0.00 0.00 0.91 0.68 0.63 0.44 0.37 0.21 0.14 0.00 0.00 0.00 0.00 1.48 1.37 1.17 1.02 0.81 0.65 0.42 0.27 0.00 0.00 0.00 2.59 2.39 2.21 1.99 1.77 1.50 1.24 0.90 0.59 0.00 0.00 4.26 4.08 3.89 3.68 3.44 3.18 2.87 2.53 2.07 1.56 0.00 6.76 6.61 6.47 6.31 6.15 5.96 5.75 5.50 5.24 5.00 5.00 10.28 10.20 10.13 10.06 10.01 10.00 10.00 W.OO 10.00 10.00 10.00 15.00 15.00 15.00 15.00 15.00 15.00 15.00 15.00 15.00 15.00 15.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00 25.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 30.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.00 35.0@ 35.00 35.00 40.00 40.00 40.00 40.00 40.00 40.00 40.00 40.00 40.00 40.00 40.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 45.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 50.00 where llt 7 llt 7 q - (r /2) - 2llZ 2 (r aj = 2llZ (I" - f3j = 1 +--7 a- +rllt llZ- Yj =- llt 7 llt 7 llt 7 2llZ (I" - q - a- /2) - 2llZ 2 a- The difference equation for the explicit method becomes fi+l. j - Ji,j llt + (1. - q - a 2 j?) fi+I,j+I - fi+I,j-l 2llZ + -a2 I fi+I,j+I + fi+I,j-l 2 llZ2 - 2fi+I,j - 'J... 1 ',J or aj fi+l,j-l + f3j fi+l,j + yj fi+l,j+l = Ji,j (17.36) B asic Numerical Procedures 427 where (17.37) (17.38) (17.39) The change of variable approach has the property that aj, fJj, and Yj as well as aj, fJj, and yj are independent of j. It can be shown that it is numerically most efficient to set I:!.Z = aJ3I:!. t. Relation to Trinomial Tree Approaches The explicit finite difference method is equivalent to the trinomial tree approach.2 8 In the expressions for aj, bj, and cj in equation (17.34), we can interpret terms as follows: -~(r -:- q)j M - + ~a2 i I:!.t: - Probability of stock price decreasing from j I:!.S to (j - l)I:!.S in time M. Probability of stock price remaining unchanged at j I:!.S in time I:!.t. Probability of stock price increasing from j I:!.S to (j + 1)I:!.S in time I:!.t. This interpretation is illustrated in Figure 17.17. The three probabilities sum to unity. They give the expected increase in the stock price in time I:!.t as (r - q)j I:!.S I:!.t = (r - q)S I:!.t. This is the expected increase' in a risk-neutral world. For small values Figure 17.17 Interpretation of ,explicit finite difference method as a trinomial tree. ~(r - q)j /:it + !p2j2 /:it , - - - - - - - - - - fi+ l.j+ 1 -~(r- q)j /:it + ~cr2j2/:it ' - - - - - - - - - - - fi+ l,j-l 28 It can also be shown that the implicit finite difference method is equivalent to a multinomial tree approach where there are M + I branches emanating from each node. 428 CHAPTER 17 of tlt, they also give the variance of the change in the stock price in time tlt as if- / tlS 2 M = (i S2 M. This corresponds to the stochastic process followed by S. The value of f at time i tlt is calculated as the expected value of f at time (i + 1) tlt in a risk-neutral world discounted at the risk-free rate. For the explicit version of the finite difference method to work well, the three "probabilities" l( A -2: r - q) ]. ut + 2: a- r ilt, 1 1.1 A 2 l-a /M -!(r - q)j tlt + -!a2j2 tlt should all be positive. In Example 17.11, 1 - a 2 / tlt is negative when j ~ 13 (i.e., when S ~ 65). This explains the negative option prices and other inconsistencies in the top -left-hand part of Table 17.5. This example illustrates the main problem associated with the explicit finite difference method. Because the probabilities in the associated tree may be negative, it does not necessarily produce results that converge to the solution of the differential equation. 29 When the change-of-variable approach is used (see equations (17.36) to (17.39)), the probability that Z = In S will decrease by tlZ, stay the same, and increase by tlZ are tlt 1 tlt 1 - 2tlZ (r - q -:: a- /2) + 2tlZ 2 atlt 1 1--atlz 2 tlt 1 tlt 1 2tlZ (r - q - a- /2) + 2tlZ 2 a- respectively. These movements in Z correspond to the stock price changing from S to Se- AZ , S, and Se AZ , respectively. If we set tlZ = a.J3tlt, then the tree and the probabilities are identical to those for the trinomial tree approach discussed in Section 17.4. Other Finite Difference Methods Many of the other finite difference methods that have been proposed have some of the features of the explicit finite difference method and some features of the implicit finite difference method. In what is known as the hopscotch method, we alternate between the explicit and implicit calculations as we move from node to node. This is illustrated in Figure 17.18. At each time, we first do all the calculations at the "explicit nodes" in the usual way. We can then deal with the "implicit nodes" without solving a set of simultaneous equations because the values at the adjacent nodes have already been calculated. 29 J. Hull a~d A. White, "Valuing Derivative Securities Using the Explicit Finite Difference Method," Journal of Financial and Qualltitatil'e Analysis, 25 (March 1990): 87-100, show how this problem can be overcome. In the situation considered here it is sufficient to construct the grid in In S rather than S to ensure convergence. B asic lVumerical Procedures 429 The hopscotch method. I indicates node at which implicit calculations are done; E indicates node at which explicit calculations are; done. Figure 17.18 Asset price Boundary 8E 81 8E oI eI eE 81 oE Q d E "0 c ::> 0 a:l eE 01 eI eE Boundary Time The Crank-Nicolson scheme is an average of the explicit and implicit methods. For the implicit method, equation (17.27) gives ii,j = aj ii-I,j-I + b j ii-I,j+ Cj ii-I,HI For the explicit method, equation (17.34) gives fi-I,j = aj fi,j-I + bj ii,j + cj ii,HI The Crank-Nicolson method averages these two equations to obtain ii,j + ii-I,j = aj ii-I,j-I + b/ii-I,j + Cj fi-I,HI + aj ii,j-I + bj fi,j + cj ii,HI Putting gi,j = ii,j - aj ii,j-I - bj ii,j - cj fi.HI we obtain gi,j = aj ii-I,j-I + bj ii-I,j + Cj ii-I,j+1 - ii-I,j This shows that implementing the Crank-Nicolson method is similar to implementing the implicit finite difference method. The advantage of the Crank-Nicolson method is that it has faster convergence than either the explicit or implicit method. Applications of Finite Difference Methods Finite difference methods can be used for the same types of derivative pricing problems as tree approaches. They can handle American-style as well as European-style derivatives but cannot easily be used in situations where the payoff from a derivative depends on the past history of the underlying variable. Finite difference methods can, at the expense of a considerable increase in computer time, be used when there are several state variables. The grid in Figure 17.15 then becomes multidimensional. 430 CHAPTER 17 The method for calculating Greek letters is similar to that used for trees. Delta, gamma, and theta can be calculated directly from the fi,j values on the grid. For vega, it is necessary to make a small change to volatility and recalculate the value of the derivative using the same grid. SUMMARY We ha'{e presented three different numerical procedures for valuing derivatives when no analytic solution is available. These involve the use of trees, Monte Carlo simulation, . and finite difference methods. Binomial trees assume that, in each short mterval oftime t:.t, a stock price either moves up by a multiplicative amount II or down by a multiplicative amount d. The sizes of II and d and their associated probabilities are chosen so that the change in the stock price has the correct mean and standard deviation in a risk-neutral world. Derivative prices are calculated by starting at the end of the tree and working backWards. For an American option, the value at a node is the greater of (a) the value if it is exercised immediately and (b) the discounted expected value if it is held for a further period of time t:.t. Monte Carlo simulation involves using random numbers to sample many different paths that the variables underlying the derivative could follow in a risk-neutral world. For each path, the payoff is calculated and discounted at the risk-free interest rate. The arithmetic average of the discounted payoffs is the estimated value of the derivative. Finite difference methods solve the underlying differential equation by converting it to a difference equation. They are similar to tree approaches in that the computations work back from the end of the life of the derivative to the beginning. The explicit method is functionally the same as using a trinomial tree. The imPl1fit finite difference method is more complicated but has the advantage that the user does not have to take any special precautions to ensure convergence. In practice, the method that is chosen is likely to depend on the characteristics of the derivative being evaluated and the accuracy required. Monte Carlo simulation works forward from the beginning to the end of the life of a derivative. It can be used for European-style derivatives and can cope with a great deal of complexity as far as the payoffs are concerned. It becomes relatively more efficient as the number of underlying variables increases. Tree approaches and finite difference methods work from the end of the life of a security to the beginning and can accommodate American-style as well as European-style derivatives. However, they are difficult to apply when the payoffs depend on the past history of the state variables as well as on their current values._ Also, they are liable to become computationally very time consuming when three or more variables are involved. FURTHER READING General Clew1ow, 'L., and C. Strickland, Implementing Derivatives Models. Chichester: Wiley, 1998. P.ress, W. H., S. A. Teuko1sky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. Cambridge University Press, 1992. B asic Numerical Procedures 431 On Tree Approaches Cox, J. C, S. A. Ross, and M. Rubinstein. "Option Pricing: A Simplifi~d Approach," Journal of Financial Economics, 7 (October 1979): 229-64. c~ Figlewski, S., and B. Gao. "The Adaptive Mesh Model: A New Approach to Efficient Option Pricing," Journal oj Financial Economics, 53 (1999): 313-5l. Hull, J. C., and A. White, "The Use of the Control Variate Technique in Option Pricing," Journal oj Financial and Quantitative Analysis, 23 (September 1988): 237-5l. Rendleman, R., and B. Bartter, "Two State Option Pricing," Journal of Finance, 34 (1979): 1092-1110. On Monte Carlo Simulation Boyle, P. P., "Options: A Monte Carlo Approach," Journal of Financial Economics, 4 (1977): 323-38. Boyle, P. P., M. Broadie, and P. G1asserman. "Monte Carlo Methods for Security Pricing," Journal oj Economic Dynamics and Control, 21 (1997): 1267-1322. Broadie, M., P. G1asserman, and G. Jain. "Enhanced Monte Carlo Estimates for American Option Prices," Journal of Derivatives, 5 (Fall 1997): 25-44. On Finite Difference Methods Hull, J. C., and A. White, "Valuing Derivative Securities Using the Explicit Finite Difference Method," Journal of Financial and Quantitative Analysis, 25 (March 1990): 87-100. Wilmott, P., Derivatives: The TheOlT and Practice of Financial Engineering. Chichester: Wiley, 1998. Questions and Problems (Answers in Solutions Manual) 17.1. Which of the following can be estimated for an American option by constructing a single binomial tree: delta, gamma, vega, theta, rho? 17.2. Calculate the price of a 3-month American put option on a non-dividend-paying stock when the stock price is $60, the strike price is $60, the risk-free interest rate is 10% per annum, and the volat~lity is 45% per annum. Use a binomial tree with a time interval of 1 month. 17.3. Explain how the control variate technique is implemented when a tree is used to value American options. 17.4. Calculate the price of a 9-month American call option on corn futures when the current futures price is 198 cents, the strike price is 200 cents, the risk-free interest rate is 8% per annum, and the volatility is 30% per annum. Use a binomial tree with a time interval of 3 months. 17.5. Consider an option that pays off the amount by which the final stock price exceeds the average stock price achieved during the life of the option. Can this be valued using the binomial tree approach? Explain your answer. 17.6. "For a dividend-paying stock, the tree for the stock price does not recombine; but the tree for the stock price less the present value of future dividends does recombine." Explain this statement. 17.7. Show that the probabilities in a Cox, Ross, and Rubinstein binomial tree are negative when the condition in footnote 9 holds. 17.8. Use stratified sampling with 100 trials to improve the estimate of TC in Business Snapshot 17.1 and Table 17.1. 430 CHAPTER 17 The method for calculating Greek letters is similar to that used for trees. Delta, gamma, and theta can be calculated directly from the Ji. j values on the grid. For vega, it is necessary to make a small change to volatility and recalculate the value of the derivative using the same grid. SUMMARY We have presented three different numerical procedures for valuing derivatives when no analytic'solution is available. These involve the use of trees, Monte Carlo simulation, and finite difference methods. Binomial trees assume that, in each short iriterval of time b..t, a stock price either moves up by a multiplicative amount II or down by a multiplicative amount d. The sizes of II and d and their associated probabilities are chosen so that the change in the stock price has - the correct mean and standard deviation in a risk-neutral world. Derivative prices are calculated by starting at the end of the tree and working backwards. For an American option, the value at a node is the greater of (a) the value if it is exercised immediately and (b) the discounted expected value if it is held for a further period of time b..t. Monte Carlo simulation involves using random numbers to sample many different paths that the variables underlying the derivative could follow in a risk-neutral world. For each path, the payoff is calculated and discounted at the'risk-free interest rate. The arithmetic average of the discounted payoffs is the estimated value of the derivative. Finite difference methods solve the underlying differential equation by converting it to a difference equation. They are similar to tree approaches in that the computations work back from the end of the life of the derivative to the beginning. The explicit method is functionally the same as using a trinomial tree. The implicit finite difference method is more complicated but has the advantage that the user does not have to take any special precautions to ensure convergence. In practice, the method that is chosen is likely to depend on the characteristics of the derivative being evaluated and the accuracy required. Monte Carlo simulation works forward from the beginning to the end of the life of a derivative. It can be used for European-style derivatives and can cope with a great deal of complexity as far as the payoffs are concerned. It becomes relatively more efficient as the number of underlying variables increases. Tree approaches and finite difference methods work from the end of the life of a security to the beginning and can accommodate American-style as well as European-style derivatives. However, they are difficult to apply when the payoffs depend on the past history of the state variables as well as on their current values. Also, they are liable to become computationally very time consuming when three or more variables are involved. FURTHER READING General Clewlow, L, and C. Strickland, Implementing Derivatives Models. Chichester: Wiley, 1998. Pr~ss, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. Cambridge University Press, 1992. Basic Numerical Procedures 431 On Tree Approaches . Cox, J. C, S. A. Ross, and M. Rubinstein. "Option Pricing: A Simplifi.ed Approach," Joul"llal of Financial Economics, 7 (October 1979): 229-64. .~ .Figlewski, S., and B. Gao. "The Adaptive Mesh Model: A New Approach to Efficient Option Pricing," Joul"Ilal of Financial Economics, 53 (1999): 313-5l. Hull, J. c., and A. White, "The Use of the Control Variate Technique in Option Pricing," Joul"Ilal of Financial and Quantitative Analysis, 23 (September 1988): 237-5l. Rendleman, R., and B. Bartter, "Two State Option Pricing," Joul"Ilal of Finance, 34 (1979): 1092-1110. On Monte Carlo Simulation Boyle, P.P., "Options: A Monte Carlo Approach," Journal of Financial Economics, 4 (1977): 323-38. Boyle, P. P., M. Broadie, and P. Glasserman. "Monte Carlo Methods for Security Pricing," Journal of Economic Dynamics and Control, 21 (1997): 1267-1322. Broadie, M., P. Glasserman, and G. Jain. "Enhanced Monte Carlo Estimates for American Option Prices," Journal of Derivatives, 5 (Fall 1997): 25--44. On Finite Difference Methods Hull, J. c., and A. White, "Valuing Derivative Securities Using the Explicit Finite Difference Method," Joul"Ilal of Financial and Quantitative Analysis, 25 (March 1990): 87-100. Wilmott, P., Derivatives: The Theory and Practice of Financial Engineering. Chichester: Wiley, 1998. Questions and Problems (Answers in Solutions Manual) 17.1. Which of the following can be estimated for an American option by constructing a single binomial tree: delta, gamma, vega, theta, rho? 17.2. Calculate the price of a 3-month American put option on a non-dividend-paying stock when the stock price is $60, the strike price is $60, the risk-free interest rate is 10% per annum, and the volat~lity is 45% per annum. Use a binomial tree with a time interval of 1 month. 17.3. Explain how the control variate technique is implemented when a tree is used to value American options. 17.4. Calculate the price of a 9-month American call option on com futures when the current futures price is 198 cents, the strike price is 200 cents, the risk-free interest rate is 8% per annum, and the volatility is 30% per annum. Use a binomial tree with a time interval of 3 months. 17.5. Consider an option that pays off the amount by which the final stock price exceeds the average stock price achieved dunng the life of the option. Can this be valued using the binomial tree approach? Explain your answer. 17.6. "For a dividend-paying stock, the tree for the stock price does not recombine; but the tree for the stock price less the present value of future dividends does recombine." Explain this statement. 17.7. Show that the probabilities in a Cox, Ross, and Rubinstein binomial tree are negative when the condition in footnote 9 holds. 17.8. Use stratified sampling with 100 trials to improve the estimate of TC in Business Snapshot 17.1 and Table 17.1. 432 CHAPTER 17 17.9. Explain why the Monte Carlo simulation approach cannot easily be used for Americanstyle derivatives. 17.10. A 9-month American put option on a non-dividend-paying stock has a strike price of $49. The stock price is $50, the risk-free rate is 5% per annum, and the volatility is 30% per annum. Use a three-step binomial tree to calculate the option price. 17.11. Use a three-time-step tree to value a 9-month American call option on wheat futures. The current futures price is 400 cents, the strike price is 420 cents, the risk-free rate is 6%, and the volatility is 35% per annum. Estimate the delta of the option from your tree. 17.12. A 3-month American call option on a stock has a strike price of $20. The stock price is $20, the risk-free rate is 3% per annum, and the volatility is 25% per annum. A dividelld of$2 is expected in 1.5 months. Use a three-step binomial tree to calculate the option price. 17.13. A I-year American put option on a non-dividend-paying stock has an exercise price of $18. The current stock price is $20, the risk-free interest rate is 15% per annum, and the volatility of the stock price is 40% per annum. Use the DerivaGem software with four 3-month time steps to estimate the value of the option. Display the tree and verify that the option prices at the final and penultimate nodes are correct. Use DerivaGem to value the European version of the option. Use the control variate technique to improve your estimate of the price of the American option. 17.14. A 2-month American put option on a stock index has an exercise price of 480. The current level of the index is 484, the risk-free interest rate is 10% per annum, the dividend yield on the index is 3% per annu.m, and the volatility of the index is 25% per annum. Divide the life of the option into four half-month periods and use the tree approach to estimate the value of the option. 17.15. How can the control variate approach improve the estimate of the delta of an American option when the tree approach is used? 17.16. Suppose that Monte Carlo simulation is being used to evaluate a European call option on a non-dividend-paying stock when the volatility is stochastic. How could the control variate and antithetic variable technique be used to improve numerical efficiency? Explain why it is necessary to calculate six values of the option in each simulation trial when both the control variate and the antithetic variable technique are used. 17.17. Explain how equations (17.27) to (17.30) change when the implicit finite difference method is being used to evaluate an American call option on a currency. 17.18. An American put option on a non-dividend-paying stock has 4 months to maturity. The exercise price is $21, the stock price is $20, the risk-free rate of interest is 10% per annum, and the volatility is 30% per annum. Use the explicit version of the finite difference approach to value the option. Use stock price intervals of $4 and time intervals of 1 month. 17.19. The spot price of copper is $0.60 per pound. Suppose that the futures prices (dollars per pound) are as follows: 3 months 6 months 9 months 12 months 0.59 0.57 0.54 0.50 The volatility of the price of copper is 40% per annum and the risk-free rate is 6% per B asic lVumerical Procedures 433 annum. Use a binomial tree to value an American, call option on copper with an exercise price of $0.60 and a time to qlaturity of 1 year. Divide the life of the option into four 3-month periods for the purposes of constructing the tree. (Hint: As explained in Section 14.7, the futures price of a variable is its expected fulure price in a riskneutral world.) . 17.20. Use the binomial tree in Problem 17.19 to value a security that pays off x 2 in 1 year where x is the price of copper. 17.21. When do the boundary conditions for S = 0 and S -+ derivative prices in the explicit finite difference method? 00 affect the estimates of 17.22. How would you use the antithetic variable method to improve the estimate of the European option in Business Snapshot 17.2 and Table 17.2? 17.23. A company has issued a 3-year convertible bond that has a face value of $25 and can be exchanged for two of the company's shares at any time. The company can call the issue when the share price is greater than or equal to $18. Assuming that the company will force conversion at the earliest opportunity, what are the boundary conditions for the price of the convertible? Describe how you would use finite difference methods to value the convertible assuming constant interest rates. Assume there is no risk of the company defaulting .. 17.24. Provide formulas that can be used for obtaining three random samples from standard normal distributions when the correlation between sample i and sample j is Pi,j' Assignment Questions 17.25. An American put option to sell a Swiss franc for dollars has a strike price of $0.80 and a time to maturity of 1 year. The volatility of the Swiss franc is 10%, the dollar interest rate is 6%, the Swiss franc interest rate is 3%, and the current exchange rate is 0.81. Use a three-time-step tree to value the option. Estimate the delta of the option from your tree. 17.26. A I-year American call option on silver futures has an exercise price of $9.00. The current futures price is $8.50, the risk-free rate of interest is 12% per annum, and the volatility of the futures price is 25% per annum. Use the DerivaGem software with four 3-month time steps to estimate the value of the option. Display the tree and verify that the option prices at the final and penultimate nodes are correct. Use DerivaGem to value the European version of the option. Use the control variate technique to improve your estimate of the price of the American option. 17.27. A 6-month American call option on a stock is expected to pay dividends of $1 per share at the end of the second month and the fifth month. The current stock price is $30, the exercise price is $34, the risk-free interest rate is 10% per annum, and the volatility of the part of the stock price that will not be used to pay the dividends is 30% per annum. Use the DerivaGem software with the life of the option divided into six time steps to estimate the value of the option. Compare your answer with that given by Black's approximation (see Section 13.12). 17.28. The current value of the British pound is $1.60 and the volatility of the pound/dollar exchange rate is 15% per annum. An American call option has an exercise price of $1.62 and a time to maturity of 1 year. The risk-free rates of interest in the United States and 434 CHAPTER 17 the United Kingdom are 6% per annum and 9% per annum, respectively. Use the explicit finite difference method to value the option. Consider exchange rates at intervals of 0.20 between 0.80 and 2.40 and time intervals of 3 months. 17.29. Answer the following questions concerned with the alternative procedures for constructing trees in Section 17.4: (a) Show that the binomial model in Section 17.4 is exactly consistent with the mean and variance of the change in the logarithm of the stock price in time Ilt. (b) Show that the trinomial model in Section 17.4 is consistent with the mean and variance of the change in the logarithm of the stock price in time Ilt when terms of order (!lti and higher are ignored. (c) Construct an alternative to the trinomial model in Section 17.4 so that the probabilities are 1/6, 2/3, and 1/6 on the upper, middle, and lower branches emanating from each node. Assume that the branching is from S to Su, Sm, or Sd with m2 = ud. Match the mean and variance of the change in the logarithm of the stock price exactly. 17.30. The DerivaGem Application Buider functions enable you to investigate how the prices of options calculated from a binomial tree converge to the correct value as the number of time steps increases. (See Figure 17.4 and Sample Application A in DerivaGem.) Consider a put option on a stock index where the index level is 900, the strike price is 900, the risk-free rate is 5%, the dividend yield is 2%, and the time to maturity is 2 years. (a) Produce results similar to Sample Application A on convergence for the situation where the option is European and the. volatility of the index is 20%. (b)· Produce results similar to Sample Application A on convergence for the situation where the option is American and the volatility of the index is 20%. (c) Produce a chart showing the pricing of the American option when the volatility is 20% as a function of the number of time steps when the control varIate technique is used. (d) Suppose that the price of the American option in the market is 85.0. Produce a chart showing the implied volatility estimate as a function of the number of time steps. ...
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This note was uploaded on 01/30/2012 for the course MATH 174 taught by Professor Donblasius during the Spring '11 term at UCLA.

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