# 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 ?

## 2 Answers

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.

• Thx for your answer, I am indeed interested in these "new" techniques. Commented Aug 11, 2020 at 7:32
• arxiv.org/pdf/1903.11275.pdf arxiv.org/pdf/1905.09474.pdf These two papers are really good if we talk about new techniques for bemudan/american option pricing.
– ltrd
Commented Aug 11, 2020 at 7:49

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.

Cheers