16.3 Deriving the parameters 153 Our task is to find V 0 0 , the option value at time zero. We may do this by working backwards through the tree. Suppose {V i+1 n } i+1 n=0 are known; that is, we have the option values corresponding to time t = t i+1 and all possible asset prices. Then consider the option value V i n corresponding to asset price S i n at time t = t i .Because of our up/down assumption about the asset price movement, working from right to left, the asset price S i n comes either from S i+1 n+1 , with probability p,orfrom S i+1 n , with probability 1 − p.Now, recall the definition (3.1) for the expected value of a discrete random variable. The big idea in the binomial method is to multiply the two possible values V i+1 n+1 and V i+1 n by their associated probabilities to get an expected value. In this way the option value V i n corresponding to asset price S i n is taken to be pV i+1 n+1 + (1 − p)V i+1 n ,scaled by the appropriate factor that allows for the interest rate, r. This gives the fundamental relation V i n = e −rδt  pV i+1 n+1 + (1 − p)V i+1 n  , 0 ≤ n ≤ i, 0 ≤ i ≤ M − 1. (16.3) Once the parameters u, d, p and M have been chosen, the formulas (16.1)– (16.3) completely specify the binomial method. The recurrence (16.1) shows how to insert the asset prices in the binomial tree. Having obtained the asset prices at time t = t M = T , (16.2) gives the corresponding option values at that time. The relation (16.3) may then be used to step backwards through the tree until V 0 0 , the option value at time t = t 0 = 0, is computed. 16.3 Deriving the parameters Since the discrete asset price model in the binomial method fits into the framework of (6.2), by appealing to Exercise 6.2 we could tune the parameters by asking for the corresponding Y i to have zero mean and unit variance. This would lead to two constraints. However, to give more insight into the workings of the method, we will derive those constraints from first principles. Exercise 16.5 asks you to confirm that the two approaches lead to the same conclusion. As a means to write down an expression for the up/down asset price model used in the binomial method, we define a random variable R i such that R i = 1ifthe asset price goes up from time (i − 1)δt to iδt and R i = 0ifthe asset price goes down. Hence, R i = 1 with probability p and R i = 0 with probability 1 − p. This means that R i is a Bernoulli random variable with parameter p,sofrom (3.2) and (3.14) we see that E(R i ) = p and var(R i ) = p(1 − p). After n time increments the asset has undergone  n i=1 R i upward movements and n −  n i=1 R i downward movements. So the asset price S(nδt) at time t = nδt is given by S(nδt) = S 0 u  n i=1 R i d n−  n i=1 R i . 154 Binomial method We may re-arrange this to S(nδt) S 0 = d n  u d   n i=1 R i . Taking logs gives log  S(nδt) S 0  = n log d + log  u d  n  i=1 R i . (16.4) Now, by the Central Limit Theorem, for large n the sum  n i=1 R i behaves like a normal random variable. Hence, for large n, log(S(nδt)/S 0 ) will be close to normal. To match the continuous asset price model (6.8) used in the Black–Scholes analysis, we thus require the mean of log(S(nδt)/S 0 ) to be (µ − 1 2 σ 2 )nδt and the variance to be σ 2 nδt. Further, as the binomial method works with expected values, we impose the risk neutrality assumption µ = r. This leads to the conditions p log u + (1 − p) log d = (r − 1 2 σ 2 )δt, (16.5) log  u d  = σ  δt p(1 − p) , (16.6) see Exercise 16.2. Regarding δt = T/M as pre-specified, we now have two equa- tions in the three unknowns, p, u and d.Ingeneral, we can fix one of the three and solve for the other two. To pick out a particular solution this way, we may set p = 1 2 and solve to find that u = e σ √ δt+(r− 1 2 σ 2 )δt , d = e −σ √ δt+(r− 1 2 σ 2 )δt , (16.7) see Exercise 16.3. 16.4 Binomial method in practice The arguments in the previous section suggest that the binomial method asset model matches that used in the Black–Scholes analysis for small δt, that is, large M.Wemay thus hope that the option values computed from the binomial method agree well with those from the Black–Scholes formulas, and that the agreement improves if M is increased. Computational example We use the binomial method to value a European put with S 0 = 9, E = 10, T = 3, r = 0.06 and σ = 0.3. Table 16.1 shows the results for M = 100, M = 200 and M = 400, along with the Black–Scholes value 1.4728. Our first observation is that with all three choices of M the bi- nomial method approximation is correct to at least two decimal places. The 16.4 Binomial method in practice 155 Table 16.1. European put value approximations from binomial method Option value M = 100 1.4716 M = 200 1.4762 M = 400 1.4726 Black–Scholes 1.4728 0 50 100 150 200 250 1.46 1.48 1.5 1.52 M European put 200 220 240 260 280 300 320 340 360 380 400 1.472 1.473 1.474 1.475 1.476 1.477 M European put Fig. 16.2. Convergence of the binomial method for a European put as the num- ber of time points, M, increases. Upper picture: M goes from 20 to 250 in steps of 5. Dashed line is ‘exact’ solution. Lower picture: M goes from 200 to 400 in steps of 1. most accurate approximation of the three comes from the largest value of M, which is intuitively reasonable. However, it is perhaps surprising that M = 200 gives less accuracy that M = 100. To check whether this is simply a quirk, the upper picture in Figure 16.2 shows the computed option value for M = 20, 25, 30, ,250, with the Black–Scholes value superimposed as a dashed line. We see that although the binomial method approximations do appear to converge as M increases, the convergence is by no means monotonic – tak- ing a slightly bigger M may worsen the error – and there is a general ‘saw- tooth’ pattern to the sequence of approximations as M increases. The lower plot 156 Binomial method emphasizes the waviness. Here we have plotted the computed solution for all M between 200 and 400. The result appears to oscillate between two smooth curves, neither of which approaches the correct answer monotonically. ♦ Two features stand out in Figure 16.2. (i) The binomial method approximation converges to the Black–Scholes value as M →∞. (ii) The convergence is not monotonic. These may be shown to be generic. Moreover, it is possible to describe the rate at which convergence takes place. Letting e M =|V 0 0 − P(S 0 , 0)| denote the error in the binomial method approximation with δt = T/M,itcan be shown that there is a constant K such that e M ≤ K M . (16.8) In the upper picture of Figure 16.3 we display the errors in the example above for M between 100 and 400. The points have been joined by straight lines for clarity. The curve 1/M is added as a solid line, and we see that (16.8) appears to hold with K = 1. Taking logs in (16.8) gives log e M ≤ log K − log M,showing that the log of the error as a function of log M should lie below a straight line of slope −1. The lower picture of Figure 16.3 re-scales the axes logarithmically to confirm this behaviour. 16.5 Notes and references Cox, Ross and Rubinstein (Cox et al., 1979) wrote the original binomial method paper. Since then numerous authors have analysed and extended the ideas. It is possible to derive the parameters u, d and p from a number of different viewpoints. For example, with p = 1 2 the choice u = e rδt  1 +  e σ 2 δt − 1  , d = e rδt  1 −  e σ 2 δt − 1  (16.9) is common; see (Kwok, 1998; Wilmott et al., 1995). Exercise 16.4 shows that this is very close to the choice (16.7) for small δt. Although much literature has been devoted to establishing that the error in vari- ous classes of binomial methods tends to zero as M →∞, surprisingly little atten- tion was initially paid to the rate of convergence. Leisen and Reimer (Leisen and Reimer, 1996) developed a general convergence rate theory, and the bound (16.8) follows from their results. A more detailed analysis, with explicit error constants, appears in (Walsh, 2003). 16.5 Notes and references 157 100 150 200 250 300 350 400 0 0.002 0.004 0.006 0.008 0.01 M Error in binomial method 100 200 400 10 −6 10 −4 10 −2 M Error in binomial method Fig. 16.3. Upper picture: Error in the binomial method for a European put as the number of time points, M, increases from 100 to 400. Solid line is 1/M.Lower picture: same data on a log–log scale. The odd–even ripples in the error, as depicted in Figures 16.2 and 16.3, have been widely reported. The references (Leisen and Reimer, 1996; Rogers and Sta- pleton, 1998) give explanations for the effect and propose fixes. Applying the binomial method may be shown to be equivalent to using a finite difference method to approximate the Black–Scholes PDE, a point that we pursue in Section 24.4. This is one means of proving that the binomial method solution converges to the Black–Scholes value as M →∞, see (Kwok, 1998), for example, and numerical analysis insights can also be used to explain the odd-even ripples. The book (Clewlow and Strickland, 1998) covers a number of practical issues in the implementation of the binomial method, and provides pseudo-code listings. A case study with the aim of making the binomial method run as quickly as possible in MATLAB is given in (Higham, 2002), along with downloadable codes. It is possible to compute Greeks via the binomial method. For partial derivatives with respect to S or t, approximations can be obtained using information from the tree. Exercise 16.8 illustrates the idea. Other partial derivatives can be treated by re-running the method with perturbed data, in the manner outlined in Section 15.4. Further details can be found in (Hull, 2000), for example, and (Walsh, 2003) shows that delta can be approximated to the same order of accuracy as the option value. 158 Binomial method EXERCISES 16.1.  Consider the discrete asset price model used in the binomial method. Show that it may be written in the form (6.2) if we let Y i be defined as Y i =    u−1−µδt σ √ δt , with probability p, d−1−µδt σ √ δt , with probability 1 − p. (16.10) 16.2.  Starting from (16.4) show that E  log  S(nδt) S 0  = n log d + log  u d  np and var  log  S(nδt) S 0  =  log  u d  2 np(1 − p). Hence, obtain (16.5)–(16.6). 16.3.  Show that setting p = 1 2 in (16.5)–(16.6) produces (16.7). 16.4. For the parameters u and d in (16.7) show that u = 1 + σ √ δt + rδt + O(δt 3/2 ), d = 1 − σ √ δt + rδt + O(δt 3/2 ), as δt → 0. Show also that the corresponding u and d parameters in (16.9) have the same expansions up to O(δt 3/2 ). [Hint: recall that √ 1 + x = 1 + 1 2 x + O(x 2 ) and e x = 1 + x + 1 2 x 2 + O(x 3 ) as x → 0.] 16.5.  We know from Exercise 6.2 that if Y i in (16.10) has zero mean and unit variance, we recover the continuous asset price model in the limit δt → 0. Set µ = r and p = 1 2 and show that requiring E(Y i ) = 0 and var(Y i ) = 1 in (16.10) leads to u = 1 + σ √ δt + rδt, d = 1 − σ √ δt + rδt. Note that these values agree with those in Exercise 16.4 up to O(δt 3/2 ). 16.6. Returning to the recurrence (16.3) we see that for M = 1 V 0 0 = e −rδt  pV 1 1 + (1 − p)V 1 0  , and for M = 2 V 0 0 = e −rδt  pV 1 1 + (1 − p)V 1 0  = e −rδt  pe −rδt ( pV 2 2 + (1 − p)V 2 1 ) + (1 − p)e −rδt ( pV 2 1 +(1 − p)V 2 0 )  = e −2rδt  p 2 V 2 2 + 2p(1 − p)V 2 1 + (1 − p) 2 V 2 0  . 16.6 Program of Chapter 16 and walkthrough 159 Similarly for M = 3wefind that V 0 0 = e −3rδt  p 3 V 3 3 + 3p 2 (1 − p)V 3 2 + 3p(1 − p) 2 V 3 1 + (1 − p) 3 V 3 0  . The coefficients {1, 1}, {1, 2, 1}, {1, 3, 3, 1} are familiar from Pascal’s tri- angle. Having spotted this connection, prove by induction that V 0 0 = e −rT M  k=0  M k  p k (1 − p) M−k V M k , (16.11) where  M k  denotes the binomial coefficient,  M k  := M! k! (M − k)! . 16.7.  Letting W i :=         V i 0 V i 1 . . . . . . V i i         , write down the form of the i by (i + 1) matrix B i such that W i = B i W i+1 . 16.8.  Explain why the ratio (V 1 1 − V 1 0 )/(S 1 1 − S 1 0 ) can be regarded as an ap- proximation to the time-zero delta. 16.6 Program of Chapter 16 and walkthrough The program ch16 implements the binomial method for a European call. It is listed in Figure 16.4. First, parameters are initialized, using (16.7) for u and d. The quantity S*d.^([M:-1:0]’).*u.^([0:M]’) is an M+1 by 1 array whose components cover the values S M 0 , S M 1 , ,S M M in the expiry-time level of the asset price tree in Figure 16.1. Hence, the line W=max(S*d.^([M:-1:0]’).*u.^([0:M]’)-E,0); contains the expiry time option values, as in (16.2). We then work through the iteration (16.3) by exploiting MATLAB’s colon notation to extract subarrays. The syntax exp(-r*dt)*(p*W(2:i+1) + (1-p)*W(1:i)); 160 Binomial method %CH16 Program for Chapter 16 % % Implements binomial method for European call %%%%%%%% Problem and method parameters %%%%%%%%%%% S=3;E=2;T=1;r=0.05; sigma = 0.3; M=400; dt = T/M; p =0.5; u=exp(sigma*sqrt(dt) + (r-0.5*sigmaˆ2)*dt); d=exp(-sigma*sqrt(dt) + (r-0.5*sigmaˆ2)*dt); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %Time T option values W=max(S*d.ˆ([M:-1:0]’).*u.ˆ([0:M]’)-E,0); %Work back to option value at time zero for i = M:-1:1 W=exp(-r*dt)*(p*W(2:i+1) + (1-p)*W(1:i)); end disp(’Option value is’), disp(W) Fig. 16.4. Program of Chapter 16: ch16.m. represents e −rδt       p       W 2 W 3 . . . W i+1       + (1 − p)       W 1 W 2 . . . W i             . The line for i = M:-1:1 sets up a loop that is repeated M times; first with i=M, then with i= M-1,andso on, down to i=1.With this set-up, the dimension of W decreases by one each time around the loop. On exit, W is a scalar, whose value is V 0 0 . Running ch16.m produces the value 1.1175.Tocheck, we may call ch08. >> [C, Cdelta, P, Pdelta] = ch08(3,2,0.05,0.3,1) C=1.1175 Cdelta = 0.9524 P=0.0200 Pdelta = -0.0476 PROGRAMMING EXERCISES P16.1. Alter ch16 so that the choice (16.9) for u and d is used. P16.2. Implement the binomial method via the formula (16.11). 16.6 Program of Chapter 16 and walkthrough 161 Quotes ‘Would you tell me, please, which way I ought to go from here?’ ‘That depends a good deal on where you want to get to,’ said the Cat. ‘I don’t much care where ’said Alice. ‘Then it doesn’t matter which way you go,’ said the Cat. LEWIS CARROLL, Alice in Wonderland Sir, In your otherwise beautiful poem (The Vision of Sin) there is a verse which reads ‘Every moment dies a man, every moment one is born.’ Obviously, this cannot be true and I suggest that in the next edition you have it read ‘Every moment dies a man, every moment 1 1 16 is born.’ Even this value is slightly in error but should be sufficiently accurate for poetry. CHARLES BABBAGE (in a letter to Lord Tennyson), source (Fr ¨ oberg, 1985) In the literature, there are numerous contributions with limit proofs to European type options. Astonishingly, however, the convergence speed of binomially computed option prices has, so far, rarely been examined technically. Here, we present a theorem . . . DIETMAR LEISEN AND MATTHIAS REIMER (Leisen and Reimer, 1996) [...]... American call and put equivalence of European and American call Black–Scholes for American put binomial method for American options optimal exercise boundary Monte Carlo for American options 18.1 Motivation We now look at American options These are typically more common than Europeans The significant new feature here is the early-exercise facility For put options, this complicates the Black–Scholes analysis,... places analytic formulas out of reach, and puts a strain on computational methods 18.2 American call and put An American option is like a European option except that the holder may exercise at any time between the start date and the expiry date Definition An American call option gives its holder the right (but not the obligation) to purchase from the writer a prescribed asset for a prescribed price at any... date, an American call option must have the same value as a European call option Exercise 18.1 asks you to reach this conclusion by an alternative route As we will see shortly, the same is not true for put options 18.3 Black–Scholes for American options Our aim in this section is to show how the arguments in Chapter 8 that led to the Black–Scholes PDE can be adapted to cover an American put option. .. for this option, where d1 is defined in (8.20) How would the analogous asset-ornothing put option be defined, and what is its value? 17.8 Show that holding a European call option is equivalent to holding an asset-or-nothing call option (see Exercise 17.7 above) and writing a cashor-nothing call with A = E, for the same expiry date Use this to give another way to value the asset-or-nothing call option in...17 Cash-or-nothing options OUTLINE • • • • • cash-or-nothing call and put options Black–Scholes formulas Greeks behaviour of delta risk-neutral valuation 17.1 Motivation We now take our first step away from vanilla Europeans and look at cash-ornothing call and put options There are three good reasons to look at these options • They are widely traded, and hence of practical importance • The corresponding... date and a prescribed expiry date in the future ♦ Definition An American put option gives its holder the right (but not the obligation) to sell to the writer a prescribed asset for a prescribed price at any time between the start date and a prescribed expiry date in the future ♦ The holder of an American option is thus faced with the dilemma of deciding when, if at all, to exercise If, at time t, the option. .. time t and then purchase the asset at time t = T by doing the most favourable of (a) exercising the option at t = T , and (b) buying at the market price at time T With this strategy the holder has gained amount S(t) > E at time t and paid out an amount less than or equal to E at time T This is clearly better than gaining S(t) − E at time t Since it is never optimal to exercise an American call option. .. cash-or-nothing options C cash (S, t) We will let and P cash (S, t) denote the values of the cash-or-nothing call and put options, respectively, for asset price S and time t The hedging argument used in Chapter 8 is very general – it requires only that the option value is a smooth function of S and t Hence, we may ask for C cash (S, t) 17.3 Black–Scholes for cash-or-nothing options 165 and P cash (S, t) to satisfy... then it 173 174 American options is clearly best not to exercise However, if the option is in-the-money it may be beneficial to wait until a later time where the payoff might be even bigger American options are more widely traded than their European counterparts In many exchanges, the early-exercise feature is offered by default It is thus important to know how much extra value, if any, this flexibility... We write P Am (S, t) to denote the American put option value at asset price S and time t, and use (S(t)) = max(E − S(t), 0) for the corresponding payoff function Our first observation is that P Am (S, t) ≥ (S(t)), for all 0 ≤ t ≤ T, S ≥ 0 (18.1) This follows from a simple arbitrage argument If P Am (S, t) < (S(t)) then an instantaneous profit can be made by purchasing the option and immediately exercising . depicted in Figures 16.2 and 16.3, have been widely reported. The references (Leisen and Reimer, 199 6; Rogers and Sta- pleton, 199 8) give explanations for the effect and propose fixes. Applying. solution converges to the Black–Scholes value as M →∞, see (Kwok, 199 8), for example, and numerical analysis insights can also be used to explain the odd-even ripples. The book (Clewlow and Strickland, 199 8). zero mean and unit variance, we recover the continuous asset price model in the limit δt → 0. Set µ = r and p = 1 2 and show that requiring E(Y i ) = 0 and var(Y i ) = 1 in (16.10) leads to u =