Regression Methods for Bermudan Option Monte Carlo Pricing
Summary
The document asks whether regression approaches beyond Andersen’s method and Longstaff–Schwartz can estimate exercise decisions for Bermudan claims. It frames the price as an optimization over exercise times and explains that these methods estimate an exercise boundary, then use that boundary in forward Monte Carlo pricing.
The responses mention Gaussian process regression as a machine learning approach that may help with high-dimensional options, where tree methods can be impractical. They also identify the stochastic mesh method as an alternative numerical technique. The discussion is suggestive rather than a comparative study: it supplies no implementation details, benchmarks, or evidence about accuracy. One respondent says stochastic mesh is not widely used in practice and reports little advantage over Longstaff–Schwartz in their experience; the machine learning response offers no specific references or results.
Key ideas
- Bermudan option pricing can be expressed as choosing the best exercise time from a discrete set.
- Regression methods estimate an exercise boundary that can guide subsequent forward Monte Carlo pricing.
- Gaussian process regression is proposed as an option for high-dimensional problems.
- Stochastic mesh is another cited numerical approach, though the response reports limited practical use and benefit.
Tags
Full text
# Regression techniques for bermudan Monte-Carlo
# Regression techniques for bermudan Monte-Carlo
One knows that the price of a bermudan claim exercisable at times $T_1, T_2,\ldots, T_N$ is $$V_0 = \sup_{\tau\in\Gamma} \mathbf{E} \left[ e^{\int_0^{\tau} r_s ds} \varphi_{\tau}\left( x_{\tau} \right) \right]$$ where
- $x$ is the $d$-dimensional underlying
- $\varphi_t(x_t)$ is the payoff value if exercised at $t$
- $\Gamma$ is the set of all stopping times with values in $\{T_1, T_2,\ldots, T_N\}$, also called exercise strategies, $\Gamma$ being also called the exercise boundary.
In both methods that I know of (Andersen and Longstaff & Schwartz), an exercise boundary is computed and then the computed exercise boundary is used for forward pricing using classical Monte-Carlo, as once the bounday is known, it can be used to price the option like a trigger option.
Both methods are regressions somehow. Are there other regression methods than these two ?
## Answer by ltrd (score 1)
https://quant.stackexchange.com/a/57217
Nowadays there are a lot of methods related to the machine learning. Most of them are based on Gaussian Process Regression and they are particulary good if you would like to price high dimensional options which can not be even possible for Tree based methods. These are new methods, most of papers are from 2019 and 2020 but they are quite easy to understand and provider completely new perspective. I am not able to find any paper related to this topic now but if you are interested in this topic, please comment this answer and I will provide you resources.
## Answer by Ezy (score 0)
https://quant.stackexchange.com/a/42683
There is a method called stochastic mesh that has been proposed in the literature but it is not much used in practice
https://www0.gsb.columbia.edu/faculty/pglasserman/Other/bgh.pdf
There are numerical methods to make it faster (fast gauss transform for instance), but in the end not a lot of advantage compared to using good old LS in my experience.
CheersShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.