Quasi-Monte Carlo and Variance Reduction for Exotic Options
Summary
The document compares Sobol quasi-random sequences with ordinary pseudo-random sampling for Monte Carlo pricing of path-dependent options. It recommends pairing Sobol points with Brownian bridge path construction, which reduces effective dimensionality by assigning early sequence dimensions to important parts of the simulated path. The benefit depends on the payoff: the cited discussion expects stronger gains for Asian options than for barrier or early-exercise options, and smaller gains for basket options as dimensions must also represent multiple assets.
A reported example compares pseudo-random paths with antithetics against Sobol sampling with a Brownian bridge for a two-year, 500-monitoring-point Bermudan barrier put. A visual comparison favored the latter, though the improvement was described as modest; the pricer lacked error estimates, limiting the evidence. The document also suggests control variates, antithetic paths, and importance sampling near a barrier. These methods can be combined, but their effectiveness should be checked for the specific model and payoff.
Key ideas
- Sobol sequences paired with Brownian bridge construction can improve Monte Carlo convergence by reducing effective dimensionality.
- The gains depend on payoff structure and may be smaller for barrier, early-exercise, and basket options.
- A reported Bermudan barrier example favored Sobol with Brownian bridge, but the improvement was modest and lacked error estimates.
- Control variates, antithetic paths, and barrier-focused importance sampling are additional variance reduction methods.
- Test convergence for the specific pricing problem before choosing a simulation method.
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# Quasi Monte Carlo # Quasi Monte Carlo I read several articles regarding quasi Monte Carlo algorithm with Sobol sequences but I still have questions. I implemented MC simulations with an ordinary random generator in matlab. I'd like to improve the speed of convergence but I'm not sure it's the correct way. Say I have to price a 2 year American barrier option with daily monitoring, from what I understood this is going to be a high dimensional problem (around 500 steps). I read that it doesn't work better than an ordinary MC. Extrapolating, I will also price American basket options, therefore with even higher dimensions. I would need help not to start implementing useless things. Is QMC ok or should I stick to implementing other variance reduction techniques such as antithetic variable ? ## Answer by Yian Pap (score 0, accepted) https://quant.stackexchange.com/a/41111 General question: Are Sobol sequences any better than pseudo-random (PR) numbers (and thus still worth the effort) when the dimension is high? In most cases I'd say yes, if they are used with the Brownian Bridge (BB) path construction (or other effective dimensionality-reduction technique). Then again it depends on the type of option you are trying to price. Have a look at these two pages that delve a little bit into this subject: https://www.acenumerics.com/option-pricer-sobol-sequences.html https://www.acenumerics.com/option-pricer-brownian-bridge.html You can see there that for Asian options for example, Sobol numbers with BB give a huge advantage over PR numbers, even for high dimensions. For European barrier options the gain is smaller and for Bermudan/American options even less. But I still think that in all these cases you'll get better convergence wih Sobol & BB. You can actually download the pricer from the site above and perform your particular test (American barrier with daily monitoring) so that you get the answer to your question, or at least a much better feel of what to expect. By the way, this will also point you to a different way of speeding up your MC pricing: Forgetting Matlab and going to C++ :) By the way, this pricer does the American exercise with the Longstaff-Schwartz algo. How are you planning to do it? So I did a sample test for you, for an up and out Bermudan put with T=2Y and 500 monitoring points. First convergence graph is using PR numbers (what you call ordinary random numbers) with antithetics and the second is using Sobol sequences with BB path construction. For these I chose to use 10 basis functions (overkill, for accuracy) 65K paths for the L-S phase and 262K paths for the pricing. A quick visual check gives the edge to Sobol & BB, though the gain is not dramatic. Unfortunately the pricer doesn't give you error estimates, so you have to go by the convergence graphs, which should give you an idea anyway. You can of course use the pricer yourself to do many more tests with different parameters and get a better idea of what to expect under different circumstances. For Basket options I would expect the gains to be smaller. This is because the initial (better) dimensions of the Sobol variates will have to be allocated not only across the different time-points but also the different stocks in the basket. And if you use a stochastic volatility model then you'd also have the variance stochastic driver to take into account when deciding where to allocate your "best" Sobol dimensions. ## Answer by stans (score 1) https://quant.stackexchange.com/a/41103 Certainly, if the barrier is way out of the money and does not get hit often, quasi Monte Carlo is a good idea. It helps in achieving the right sampling frequency... This is not to say that you should not implement other variance reduction techniques as well. In my experience, 1) using a vanilla American option (no barrier) as a control variate helps; you should have prices of those readily available; 2) using the mirror image of each trajectory of random shocks as an antithetic variate helps; 3) applying importance sampling by putting a bigger weight around the barrier is likely to improve the accuracy.
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