1. Chapter 19 Monte Carlo Valuation

2. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-2 Monte Carlo Valuation Simulation of future stock prices and using these simulated prices to compute the discounted expected payoff of an option –Draw random numbers from an appropriate distribution. –Use risk-neutral probabilities, and therefore risk- free discount rate. –Generate distribution of payoffs a byproduct.

3. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-3 Monte Carlo Valuation (cont’d) Simulation of future stock prices and using these simulated prices to compute the discounted expected payoff of an option (cont'd) –Pricing of asset claims and assessing the risks of the asset. –Control variate method increases conversion speed.

4. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-4 Computing the Option Price As a Discounted Expected Value Option valuation can be performed as if all assets earned the risk-free rate of return and investors performed all discounting at this rate. Specially, we compute the time price of a claim, V[S(0), 0], as

5. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-5 Computing the Option Price As a Discounted Expected Value (cont’d) Assume a distribution for the stock price 3 months from now. For each stock price drawn from the distribution compute the payoff of a call option (repeat many times). Discount the average payoff at the risk-free rate of return.

6. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-6 Computing the Option Price As a Discounted Expected Value (cont’d) In a binomial setting, if there are n binomial steps, and i down moves of the stock price, the European Call price is Note:

7. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-7 Computing the Option Price As a Discounted Expected Value (cont’d)

8. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-8 Computing the Option Price As a Discounted Expected Value (cont’d) The above figure can be used to illustrate Monte Carlo simulation. Imagine a gambling wheel divided into 4 unequal sections, where each section has a probability corresponding to one of the option payoffs in the figure. The option price can be estimated as follows: 1.Spin of the wheel to selects one of the final stock price nodes and option payoffs. 2.Repeat (1) numerous times. 3.Average the payoffs obtained in (2). 4.Discounting (3) at the risk-free rate provides an estimate of the option value.

9. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-9 Random Number Generation There are several ways for generating random numbers: –Use “RAND” function in Excel to generate random numbers between 0 and 1 from a uniform distribution U(0,1). –To generate random numbers from a normal distribution N (for which an inverse cumulative distribution N –1 can be computed), generate a random number x from U(0,1). find z such that N(z) = x, i.e., N –1 (x) = z. Repeat. This procedure of using the inverse cumulative probability distribution works for any distribution for which you can compute the inverse cummulative distribution.

10. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-10 Simulating Lognormal Stock Prices Recall that if Z ~ N(0,1), a lognormal stock price is Randomly draw a set of standard normal Z’s (Z(1), Z(2), … ) and substitute the results into the equation above. The resulting S t ’s will be lognormally distributed random variables at time t. To simulate the path taken by S (which is useful in valuing path-dependent options) split t into n intervals of length h

11. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-11 Monte Carlo Valuation If V(S t,t) is the option payoff at time t, then the time-0 Monte Carlo price V(S 0,0) is –where S T 1, …, S T n are n randomly drawn time-T stock prices. For the case of a call option V(S T i,T) = max(0, S T i –K).

12. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-12 Monte Carlo Valuation (cont’d) Example 19.1: Value a 3-month European call where the S 0 =$40, K=$40, r=8%, =0%, and =30% For each stock price, compute Option payoff = max(0, S 3 months – $40) Average the resulting option payoffs. Discount the average back 3 months at the risk- free rate $2.804 versus $2.78 Black-Scholes price 2500x

13. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-13 Monte Carlo Valuation (cont’d) Table 19.2 shows the results from running five Monte Carlo valuation. The result of 2500 simulations is close to the correct answer. However, there is considerable variation among the individual trials of 500 simulations.

14. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-14 Monte Carlo Valuation (cont’d)

15. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-15 Monte Carlo Valuation (cont’d) Let be the call price generated from the randomly drawn. If there are n trials, the Monte Carlo estimate is Let  C denote the standard deviation of one draw and  n the standard deviation of n draws. or Thus, the standard deviation of the Monte Carlo estimate is inversely proportional to the square root of the number of draws.

16. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-16 Monte Carlo Valuation (cont’d) In the Monte Carlo results reported in Table 19.2,  C = $4.05. –500 observations n =$0.180 6.5% of Black-Schole price –2500 observations  n =$0.080 2.9% of Black-Schole price –21,000 observations n =$0.028 1.0% of Black-Schole price Monte Carlo valuation of options is especially useful when –Number of random elements in the valuation problem is too great to permit direct numerical valuation. –Underlying variables are distributed in such a way that direct solutions are difficult. –The options are path-dependent, (the payoff depends on the path of underlying asset price).

17. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-17 Monte Carlo Valuation (cont’d) Monte Carlo valuation of Asian options –The payoff is based on the average price over the life of the option –The value of the Asian option is computed as

18. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-18 Monte Carlo Valuation (cont’d)

19. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-19 Efficient Monte Carlo Valuation Control variate method –Estimate the error on each trial by using the price of a related option that does have a pricing formula. The error estimate obtained from this control price can be used to improve the accuracy of the Monte Carlo price on each trial. –Example: use information on the error in the geometric price to adjust our estimate of the arithmetic price, for which there is no formula. –To be specific, we use the same simulation to estimate both the arithmetic price, and the geometric price,. Let G and A represents the true geometric and arithmetic prices.

20. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-20 Efficient Monte Carlo Valuation Consider Then,

21. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-21 Efficient Monte Carlo Valuation (cont’d) Therefore, A * is an unbiased estimate of A. On the other hand, Var(A * ) is minimized by setting  =, where is slope coefficient from regressing on.

22. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-22 Efficient Monte Carlo Valuation (cont’d) The minimum value of Var(A * ) is where  is the correlation between and. In general, the variance of the control variate estimate will be less than that of naive Monte Carlo.

23. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-23 Efficient Monte Carlo Valuation (cont’d) Antithetic variate method –For every draw also obtain the opposite and equally likely realizations to reduce variance of the estimate. Stratified sampling –Treat each number as a random draw from each percentile of the uniform distribution. Other methods –Importance sampling, low discrepancy sequences.

24. © 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.19-24