Chapter 19 Monte Carlo Valuation 1 Monte Carlo
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Chapter 19 Monte Carlo Valuation 1
Monte Carlo Valuation n Simulation of future stock prices and using these simulated prices to compute the discounted expected payoff of an option n n n Draw random numbers from an appropriate distribution Uses risk-neutral probabilities, and therefore risk-free discount rate Distribution of payoffs a byproduct Pricing of asset claims and assessing the risks of the asset Control variate method increases conversion speed Incorporate jumps by mixing Poisson and lognormal variables Simulated correlated random variables can be created using Cholesky distribution 2
Computing the option price as a discounted expected value n Assume a stock price distribution 3 months from now n For each stock price drawn from the distribution compute the payoff of a call option (repeat many times) n Take the expectation of the resulting option payoff distribution using the risk-neutral probability p* n Discount the average payoff at the risk-free rate of return 3
Computing the option price as a discounted expected value n In a binomial setting, if there are n binomial steps, and i down moves of the stock price, the European Call price is: 4
Computing the option price as a discounted expected value (cont. ) 5
Computing the option price as a discounted expected value (cont. ) 6
Computing random numbers n There are several ways for generating random numbers n n n Use “=RAND()” function in Excel to generate random numbers between 0 and 1 from a uniform distribution U(0, 1) To generate random numbers (approximately) from a standard normal distribution N(0, 1), sum 12 uniform (0, 1) random variables and subtract 6 To generate random numbers from any distribution D (for which an inverse cumulative distribution D– 1 can be computed), n generate a random number x from U(0, 1) – 1 n find z such that D(z) = x, i. e. , D (x) = z n repeat 7
Simulating lognormal stock prices n Recall that if Z ~ N(0, 1), a lognormal stock price is 2 St = S 0 e(a – 0. 5 s )t + s t. Z n Randomly draw a set of standard normal Z’s and substitute the results into the equation above. The resulting St’s will be lognormally distributed random variables at time t. n To simulate the path taken by S (which is useful in valuing path-dependent options) split t into n intervals of length h 2 Sh = S 0 e(a – 0. 5 s )h + s h. Z(1) 2 S 2 h = She(a –… 0. 5 s )h + s h. Z(2) 2 Snh = S(n-1)he(a – 0. 5 s )h + s h. Z(n) 8
Examples of Monte Carlo Valuation n If V(St, t) is the option payoff at time t, then the time-0 Monte Carlo price V(S 0, 0) is 1 n where ST , … , ST are n randomly drawn time-T stock prices n For the case of a call option, i i V(ST , T) = max (0, ST –K) 9
Examples of Monte Carlo Valuation (cont. ) n Example 19. 1: Value a 3 -month European call where the S 0=$40, K=$40, r=8%, and s=30% S 3 months = S 0 e(0. 08 – 0. 3^2/2)x 0. 25 + 0. 3 0. 25 Z n For each stock price, compute 2500 x Option payoff = max(0, S 3 months – $40) n Average the resulting payoffs n Discount the average back 3 months at the risk-free rate $2. 804 versus $2. 78 Black-Scholes price 10
Monte Carlo Valuation Pros and Cons n Monte Carlo valuation of American options is not as easy, although it can be done (section 19. 6 of textbook) n Monte Carlo valuation can require many simulated values for the option price to converge (somewhat inefficient) n n n 500 simulated values 21, 000 simulated values s=$0. 180 s=$0. 028 6. 5% 2. 9% 1. 0% n Monte Carlo valuation of options is especially useful when n Number of random elements in the valuation problem is two great to permit direct numerical valuation n Underlying variables are distributed in such a way that direct solutions are difficult n The payoff depends on the path of underlying asset price 11
Examples of Monte Carlo Valuation (cont. ) n Monte Carlo valuation of Asian options: n The payoff is based on the average price over the life of the option 2)t/3 + s t/3 Z(1) (r – 0. 5 s 40 e S 1 = S 2 = S 1 e(r – 0. 5 s 2) t/3 + s t/3 Z(2) S 3 = S 2 e(r – 0. 5 s 2) t/3 + s t/3 Z(3) n The value of the Asian option is computed as Casian = e–rt. E(max[(S 1+S 2+S 3)/3 – K, 0]) 12
Efficient Monte Carlo Valuation (cont. ) n Control variate method n n Estimate the error on each trial by using the price of a related option that does have a pricing formula Example: use errors from geometric Asian option to correct the estimate for the arithmetic Asian option price n Antithetic variate method n For every draw also obtain the opposite and equally likely realizations to reduce variance of the estimate n Stratified sampling n Treat each number as a random draw from each percentile of the uniform distribution 13
The Poisson Distribution n A discrete probability distribution that counts the number of events that occur over a period of time n n n lh is the probability that one event occurs over the short interval h Over the time period t the probability that the event occurs exactly m times is given by The cumulative Poisson distribution (the probability that there are m or fewer events from 0 to t) is 14
The Poisson Distribution (cont. ) n The mean of the Poisson distribution is lt n Given an expected number of events, the Poisson distribution gives the probability of seeing a particular number of events over a given time 15
Simulating jumps with the Poisson distribution n Stock prices sometimes move (“jump”) more than what one would expect to see under lognormal distribution n The expression for lognormal stock price with m jumps is where a. J and s. J are the mean and standard deviation of the jump and Z and Wi are random standard normal variables n To simulate this stock price at time t+h select n A standard normal Z n Number of jumps m from the Poisson distribution n m draws, W(i), i= 1, … , m, from the standard normal distribution 16
Simulating jumps with the Poisson distribution (cont. ) 17
Simulating correlated stock returns 18
Why it works (i. e. , why Z is a standard N(0, 1) and why Corr(W, Z)=r: 19
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