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Least-Squares Monte Carlo as Machine Learning for Derivative Valuation

Article Quant Q&A · Author: kim
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Summary

The document explains how least-squares Monte Carlo (LSM) estimates the value of callable derivatives and exposure measures without repeatedly running costly nested simulations. It simulates model states and their discounted future cash flows, then fits a regression function to estimate continuation value conditional on each state. That fitted function is applied in Monte Carlo valuation or exposure calculations. The method is framed as supervised learning, with state variables as inputs and cash-flow values as labels.

The original approach uses linear or polynomial regression over basis functions; neural networks and deep learning are presented as possible extensions for settings with many state variables. The discussion connects this application to risk calculations such as CVA and other regulated measures. It offers a conceptual account and references further material, but gives no comparative numerical results. It also notes that applying deep learning effectively in high dimensions remains an active research topic, so accuracy and efficiency are not guaranteed by the description alone.

Key ideas

  • LSM estimates continuation values from simulated states and corresponding future cash flows.
  • The fitted conditional value function can replace expensive nested simulations in Monte Carlo valuation and exposure calculations.
  • Classical LSM uses linear or polynomial regression over functions of the state variables.
  • Neural networks may extend the regression approach to high-dimensional problems, but effective application remains an active research area.
  • The document provides no numerical comparison of model accuracy or efficiency.

Tags

Machine learningDerivatives pricingStatisticsRisk management
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# Machine Learning usage in Q part of Quant Finance


# Machine Learning usage in Q part of Quant Finance












Machine Learning algorithms is broadly used in trading strategies and in general when it comes to working with financial time series. The webpage Quantopian is a platform to see some of the possibilities that ML provides in finance.

I was then wondering if ML is also being used in the Q-part of finance when dealing with pricing derivatives, constructing hedging etc. Since this SE doesn't allow opinion based questions I try to be more precise: Are there some research examples out there when ML are being used in Q part of quant finance?

## Answer by Antoine Savine (score 7)

https://quant.stackexchange.com/a/43077

There is at least one clear area of application of ML in Q quant finance, it is the LSM algorithm invented by Longstaff, Schwartz and Carriere in the late 1990s for the valuation of callable exotics in the context of Monte-Carlo simulations, and widely adopted for more recent bank-wide risk calculations like CVA.

In order to estimate the continuation value on an exercise date, or a portfolio value on an exposure date, in the context of Monte-Carlo simulations, one would normally need extremely costly nested simulations. The widely adopted LSM algorithm resolves the problem in a particularly elegant manner:

1) Simulate a training set consisting of the simulated state variables of the model on the exercise/exposure date as inputs x, and (discounted) values of future cash-flows in the corresponding scenarios as labels y.

2) Use the simulated training set to train a regression/ANN/deep ANN to estimate f(x) = E[y|x], aka the value on the exercise/exposure date of the remaining cash-flows as a function of the state variables of the model on that date in this scenario.

3) Run Monte-Carlo simulations, applying the trained function f(x) to estimate the future value of the transaction(s) on exercise/exposure dates.

Now I kind of reformulated LSM in modern ML lingo. The original algorithm recommended to find f by linear regression or polynomial regression or more generally by linear regression over basis functions of the state variables.

We are now acutely aware that deep learning, a powerful generalization of linear regression, produces more accurate results much more effectively (when applied correctly). The correct, efficient estimation of future values in Monte-Carlo simulations is a crucial problem in modern finance, because it is at the heart of most regulated risk calculations: CVA, XVA, CCR, FRTB, PRIIPS, ... Recent advances of ML and DL naturally apply to resolve difficulties with LSM, particularly with many regression variables (high dimensional x)

You will find more on LSM and its application for CVA on my SSRN paper: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2966155 although the paper does not discuss exactly why and how ML/DL helps resolve LSM problems in high dimension, which is an active topic of current research.

I also posted a (more gentle) presentation on similar topics (and with the same limitations) here: https://www.slideshare.net/AntoineSavine/financial-cashflow-scripting-beyond-valuation

Finally, you may want to look at my lecture notes on back-propagation, and particularly the first part, which introduces (very basic) deep learning as an extension of linear regression and shows how DL naturally resolves problems in high dimension: https://github.com/asavine/CompFinance/blob/master/Intro2AADinMachineLearningAndFinance.pdf

Shown 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.