Why Longstaff–Schwartz Uses Regression for Continuation Values
Summary
The document explains why the Longstaff–Schwartz method uses regression when estimating the value of an American option. At each exercise date, the relevant continuation value is the expected future payoff conditional on information currently available, rather than the payoff realized on one simulated path in the next period. A single path’s next-step outcome is only one realization of that conditional random variable, so it cannot by itself supply the expectation needed to decide whether to exercise.
One direct way to estimate the conditional expectation would be nested Monte Carlo: simulate many future paths from each current state. The response notes that this is computationally costly. Longstaff–Schwartz instead uses outcomes across simulated paths and regression to estimate continuation value from current information, supporting the exercise decision without a separate inner simulation at every path and time. The explanation is conceptual and does not specify regression basis choices, convergence behavior, or implementation details; those choices can affect practical accuracy.
Key ideas
- The exercise decision compares immediate payoff with conditional expected continuation value.
- A next-period payoff on one simulated path is a realization, not the conditional expectation at the current time.
- Nested Monte Carlo can estimate that expectation but requires substantial computation.
- Longstaff–Schwartz uses regression across simulated paths to approximate continuation values from current state information.
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# What is the point of the regression in Longstaff Schwartz method?
# What is the point of the regression in Longstaff Schwartz method?
In the Longstaff and Schwartz method of pricing American options, what is the point of the regressions at each step?
The goal is to approximate an optimal stopping time for each path. However, why approximate, when you can get an exact stopping time for each path by just ... you know ... using the values you've simulated to figure out when the best stopping time is?
For example, say the maturity is $T$, and at time $T - 1$, we want to estimate the continuation value for each path. But I already know what the continuation value is for each path, I know it exactly, I just use what the stock value for that particular path is, and then calculate what my option payoff will be in the next period. Why do I need to use regression? What is the point?
## Answer by byouness (score 7)
https://quant.stackexchange.com/a/40360
You are mixing up the realization of a random variable with its expected value at a certain stage.
Let's say you are at path $i$ and time step $t_j$, what you want is not the realization of the stock at $t_{j+1}$ but rather its expected value at $t_{j+1}$ conditional on the info you have up to $t_j$.
The brute force approach here would be to do a (nested) Monte Carlo starting at $t_j$ to get this expectation, which is very costly in terms of computing power.
Longstaff and Schwarz' approach uses a regression to kind of extract this info from all realizations $t_{j+1}$ across all paths.
I won't go further into the details. As everybody said in the comments, the paper is a must read and is very well explained using a simple example. If anything is unclear for you in the paper then everybody here will be happy to help.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.