1. Simulating the value of Asian Options Vladimir Kozak

2. Variance Reduction Techniques for Asian Options Some of the possible methods: Control Variates Stratified Sampling Importance Sampling

3. Definition Discretely sampled Asian Call Option has payoff at maturity: Where Our objective is to find the value of the Asian option where And the stock price dynamics under this measure is: It follows that the distribution of is where

4. The value of the the Asian Option is given by where We want to reduce the variance of the value of the Asian option calculated using Monte Carlo simulation From our discussion on control variates we know that we can estimate unbiasedly by estimator where The idea is to find a random variable such that it is close to so that the variance of is small Moreover, we must be able to evaluate analytically

5. Observe that if random variables X i i=1,2…n are lognormal, then geometric mean has also lognormal distribution Assuming that geometric mean is close to arithmetic mean a good control variate is the random variable where

6. The Closed form solution for E(V 2 ) is easily obtained since it can be written in the form An this is the same integral over lognormal density that leads to the Black-Scholes formula

7. Asian Options and Stratified Sampling For many options, the terminal value of the stock has a lot of influence on option price By stratifying the terminal value of a stock price, much of variability in the option`s payoff can be eliminated In the case of the asset price dynamics described by the geometric Brownian Motion, we can stratify the terminal value of the stock

8. The idea is to stratify the terminal value of a Brownian Motion W(T) and then randomly generate the rest of the path W(1), W(2) … W(T- 1) using the Brownian Bridge interpolation We know that under the risk-neutral measure, Where, To stratify into K strata of equal probability for S T, we generate Z T :

9. We can interpolate the rest of each stock price path using the fact that the distribution of Brownian Motion increments W(jk)-W((j-1)k), j=1,2…N, Nk=T conditionally on the values W(t(j-1)k) and W(T) is Normally distributed with Mean = (((N-j)/N )*W((j-1)k)+j*W(T)/N) and Variance=(N-j)/(N-j+1)

10. Importance Sampling and Pricing of Asian Options Suppose that an Asian call option is well out of money, so that most of the randomly generated values for are below the strike K, contributing 0 to option price. To decrease variance we can generate stock prices from a geometric Brownian Motion with a drift larger than the the original drift so that it is more likely that and multiply the result by the Radon Nikodym derivative of one process with respect to the other

11. To price an Asian option, we need to evaluate Under this measure we assume : (1) But since (2) We may simulate the stock prices assuming a different stock price dynamics: (3)

12. We simulate the path recursively through: Where If we change the Stock price dynamics from (1) to (3), then: Where is the pdf of under measure Q and is the pdf of under the new measure P