Regress-Later LSMC for American Option Valuation

Summary

The document raises practical questions about regress-later least-squares Monte Carlo (LSMC), a method discussed for valuing American options. In the stated formulation, a continuation value at the next time step is projected onto basis functions of the next state. The conditional expectations of those basis functions given the current state are then computed analytically, allowing the continuation estimate to be evaluated at the current state. The post describes this as an implementation question, rather than providing a complete algorithm or derivation.

The author reports that constructing the projection can be computationally expensive, potentially exceeding the cost of ordinary regression, and that analytic conditional expectations restrict convenient use to processes with tractable formulas. In an in-sample comparison, regress-later and regress-now produced similar standard errors in the example tried. These are limited observations from one implementation, not general evidence that the methods perform alike. The document points to research reporting benefits in an energy real-option application, leaving open how outcomes depend on model, basis, sampling, and valuation setup.

Key ideas

• Regress-later LSMC projects future continuation values onto basis functions of the next state.
• The method uses analytic conditional expectations of basis functions given the current state.
• Computing the projection can be more expensive than the regression used by regress-now LSMC.
• Analytic conditional expectations limit straightforward application to models with tractable transitions.
• The reported example found similar standard errors in-sample, but does not establish general comparative performance.

Tags

# Regress later LSMC


# Regress later LSMC












I am looking at the regress-later LSMC introduced by Broadie, Glasserman Ha.

This can be found here:

Simulation for American Options: Regression Now or Regression Later? by Paul Glasserman and Bin Yu

Let $X_t$ and $V_{t}$ denote the underlying Markov process value function at $t$. My understanding of this method is that it project $\Gamma V_{t+1}$ onto basis functions $\phi_j(X_{t+1})$, where $\Gamma$ is chosen such subject to some the sample conditional expectation of the basis function matches the the conditional expectation of the basis function. The conditional $\mathbb{E}(\phi(X_{t+1})|X_t)$ is computed analytically.

I tried to implement this and came up with a few observations:

- The projection $\Gamma$ is very expensive and can be more expensive than the linear regression itself.

- The requirement of computation of the conditional expectation affectively limit the Markov process to things with analytical values (such as GBM or Exponential OU type of model with polynomials.)

- In the example I tried, I saw no improvement. I implemented in-sample LSMC. The values from the naive regress now and this regress later approach gave similar standard error.

I am wondering if others tried to use this method and has similar type of observation. I found papers claiming contrary results such as this one:

https://www.researchgate.net/publication/314420750_Regress-Later_Least_Squares_Monte_Carlo_Duality_Perspective_and_Energy_Real_Option_Application

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.