# How to price exotic options using Monte-Carlo?

I am actually trying to solve some exercise problem using Monte-Carlo and C++ for exotic options. Namely, the exotic options are geometric Asian options and discrete barrier option.

It is claimed that using log values would enable to get accurate pricing using "fewer approximations” and though results in a gain of time required for computing.

I have tried to look all over the place to see where I could get some hint but failed to do so.

• It is claimed that... where did you see that? because I'm not sure what you mean by using log values. Plus, what are you looking for exactly? Implementation of Monte-Carlo methods for geometric Asian and discrete barrier options? Or some example relating to your claim?
– SRKX
Mar 6, 2015 at 9:28

You can use:

Both are variance reduction techniques which will allow you to use fewer paths/simulations. Usually antithetic variates are very efficient on their own. Combining both can be a bit tricky.

You could start by simulating the value of a plain vanilla call. Then include antithetic variates and/or control variates. The "right value" can be obtained via BS's closed form solution. You'll see which model converges faster towards the BS-value.

Same could be done if you have a closed form solution for your more complex derivative. Rubinstein/Reiner i think offer closed form for barrier options.

Using log-values is (i think) more common in finite-difference-methods where you try to find the value of a derivative by approximating the Black Scholes PDE.

Having looked all over the place, I come to realize that stock price cannot be rewritten using log return. That is

St = S0 * r1,0 * r2,1 * ..., with rt+1,t = log(St+1/St)

For the 1st case, that is the Asian geometric option. If you use the fact that the definition of geometric mean can be rewritten such as

x_geo_mean = exp(1/n*sum(log(xi))

and do the maths, you can come up with a much leaner expression that would require fewer exponentiations to compute under a monte carlo framework.

The price for the geometric option can then be used as control variate to compute the price of an arithmetic option for example.

Similar fashion goes for the discrete barrier option.

• I'm not exactly sure what you meant... Mar 8, 2015 at 23:54