I've never used antithetics when writing monte carlo simulations and have a question, which I'll try to illustrate with a simple example.Assume I'm valuing a stock option that vests only if the stock price for 10 consecutive days is $110 (or more). Also assume that the underlying and strike price are $100 and there is a 1 year time to vest (so if it doesn't hit the hurdle within 1 year, the options forfeit). My general Monte Carlo would have me generate daily stock prices and when I get to the earlier of 1 year or 10 consecutive $110 prices. Assume my Monte Carlo's stock price path for the first 3 days is $100 (t=0), $101 (t=1 day), $102 (t=2), $101 (t=3). 1. (a) How do I generate the antithetic? For day 1, assume the antithetic price is $99. But what happens on day 2? I would assume that the antithetic is based off the $99 price antithetic price and not the $101 day 1 base path. (b) If so, I don't see any significant savings in computation time, or should I expect very little computational savings as this just gives me lower variance? 2. Also, let's assume that my base path vests (with a resulting option value of $10) and the antithetic path doesn't (so option value of $0). Is the value from this trial $5 (the average of the base and antithetic)? Or should I just treat this as 2 values.3. Assume also that I need to provide the average time to vesting. Does that change the answer to question 2?Thanks.

I'm not sure I fully understand this. I think you are suggesting that I develop two separate stock price paths (both base and antithetic) whereby the base path moves based on the 364 random numbers and the antithetic path moves based on the negative of the randon numbers, right?I understand about running more days than when I get 10 consecutive days on the base path as I need to check for stopping condition for the antithetic.But am I averaging the two results, or just treat it as 2 trials?

Antithetic sampling is not much help for binary simulations. If you have equal up and down probabilities, you can reduce variance slightly by starting the randomization a few steps into the tree. For example, you could start five steps in and do 1 simulation starting from 105, 5 starting from 103, 10 starting from 101, 10 starting from 99, 5 starting from 97 and 1 starting from 95. Your answer will have slightly lower variance than 32 simulations starting from 100.Antithetic sampling is more useful when each update is a continuous variable and there are fewer steps. If each step is a draw from an normal distribution with mean M, you can use X and 2*M - X. You do this at each node. Suppose the draws on your first simulation run are X1, X2, . . ., Xn. You create n new simulation paths, replacing Xi with 2*M - Xi. n has to be reasonably small for this to be sensible. Instead of doing 10,000 random 9-step simulations, you might do 1,000 simulations, each one with 10 antithetic variants.In your example problem, you should think about using control variates and importance sampling rather than antithetic sampling.

Aaron and outrun,Thanks for your suggestions. I'm not well versed with anything but basic Monte Carlo techniques and this is my first attempt to try to introduce variance reduction methods. To use control variates, what is my control? Would it be a knock-in barrier? For importance sampling, I guess I would start where the price stock price has already achieved the hurdle. However, I also need to be able to tell the average time for the option to vest. Is there a way to determine even the first time where the stock price achieves the hurdle?