Importance Sampling and Stratification for Barrier Option Monte Carlo

Summary

The document asks for ways to reduce Monte Carlo variance when pricing barrier options and points to research and textbook references. Suggested approaches include importance sampling for continuous barrier options under jump-diffusion dynamics, with a note that the scheme can be adapted to diffusion models, and stratified sampling based on barrier hitting times for discretely monitored options. These methods target the difficulty of estimating payoffs affected by whether a price path reaches a barrier.

The discussion notes that continuous barrier options in the Black–Scholes setting have closed-form prices, which limits the need for Monte Carlo methods in that specific case. The referenced stratified-sampling paper addresses discrete monitoring, while the importance-sampling work covers jump-diffusion pricing. The document does not explain either algorithm in detail or provide comparative results, so it serves primarily as a pointer to further study rather than a practical implementation guide.

Key ideas

• Importance sampling is proposed as a variance-reduction method for Monte Carlo barrier pricing.
• Stratifying simulated paths by barrier hitting times is a method for discretely monitored barrier options.
• A jump-diffusion importance-sampling scheme may be adapted to a diffusion process.
• Closed-form Black–Scholes prices for continuous barriers reduce the need for Monte Carlo in that setting.

Tags

# Importance sampling for barrier option like pricing by Monte carlo


# Importance sampling for barrier option like pricing by Monte carlo












I would like to know some references regarding importance sampling algorithms for variance reduction of Monte Carlo barrier options pricing.

Please could someone help me leaving some references? If you wish to give an answer explaining/giving some hints on how to approach the variance reduction problem of barrier options Monte Carlo, it would be even better.

Many thanks

## Answer by Mark Joshi (score 3, accepted)

https://quant.stackexchange.com/a/15404

Since there is a closed form in the BS case for continuous barrier options, you probably won't find a huge amount of work on this since it's not needed. In the discrete case, I did a paper with Tang:

http://ssrn.com/abstract=1441142

Pricing and Deltas of Discretely-Monitored Barrier Options Using Stratified Sampling on the Hitting-Times to the Barrier

## Answer by submartingale (score 2)

https://quant.stackexchange.com/a/15063

I'd recommend M. Joshi and T. Leung "Using Monte Carlo simulation and importance sampling to rapidly obtain jump-diffusion prices of continuous barrier options". Though it assumes jump-diffusion process for the returns it is straightforward to obtain the scheme for a diffusion process.

Also Paul Glasserman's [book][2]

[2]: http://www.amazon.com/Financial-Engineering-Stochastic-Modelling-Probability/dp/0387004513 contains a variance reduction example for barrier options.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.