Monte Carlo Option Pricing with Variance Reduction
Summary
The discussion addresses Monte Carlo pricing for geometric Asian and discrete barrier options, focusing on reducing simulation noise and computational work. It recommends antithetic variates and control variates to improve estimates with fewer simulated paths. A plain vanilla option with a known Black–Scholes value can serve as a benchmark for checking how quickly a simulation converges; a geometric Asian option with a tractable value can also act as a control variate for an arithmetic Asian option.
The follow-up clarifies that using log prices does not simply mean replacing stock prices with log returns. For a geometric average, taking logarithms turns the product of observed prices into an average of their logarithms, which can yield a more efficient calculation. The answer also notes that log transformations are common in finite-difference methods for solving the Black–Scholes equation. These suggestions are conceptual rather than a complete implementation guide: variance reduction depends on the instrument and the relationship between the target and control, and the source treats combining techniques as potentially nontrivial.
Key ideas
- Antithetic and control variates can reduce Monte Carlo estimation variance.
- A vanilla option with a known analytical price can benchmark simulation convergence.
- A geometric Asian option can serve as a control variate for an arithmetic Asian option.
- Logarithms simplify calculations involving geometric averages of simulated prices.
- The effectiveness of variance reduction depends on the chosen instruments and setup.
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Full text
# How to price exotic options using Monte-Carlo? # 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. ## Answer by FreshF (score 3) https://quant.stackexchange.com/a/16448 You can use: - Antithetic variates and; - Control variates. 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. ## Answer by user2448864 (score 0) https://quant.stackexchange.com/a/16892 First of all, thank for your answer and your time. 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.
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