Why Longstaff–Schwartz Continuation Value Uses Conditional Expectation

Summary

The document raises a conceptual question about the continuation value in Longstaff–Schwartz American option pricing. The displayed expression conditions on information available at the current exercise date and averages future discounted cash flows under the pricing measure. The questioner is unsure why an expectation is needed when a simulated sample path appears to fix the future cash flows already.

The key distinction is between conditioning on information known at the current time and conditioning on the entire realized future path. At the exercise date, future cash flows and rates remain uncertain; the conditional expectation averages over possible continuations consistent with the information then available. In a simulation, one estimates this conditional value using paths or regression, rather than treating one path’s realized future payoff as information available at the decision point. The document itself contains only the question, with no answer, derivation, or numerical example, so it does not explain implementation details or regression choices.

Key ideas

• Continuation value is a conditional expectation of future discounted cash flows under the pricing measure.
• Conditioning on current information does not reveal the future portion of a simulated path.
• The expectation averages over possible future continuations available from the current state.
• The document poses the question but does not provide a derivation or implementation method.

Tags

# Continuation value in Longstaff-Schwartz: Why the expected value?


# Continuation value in Longstaff-Schwartz: Why the expected value?












In the paper by Longstaff and Schwartz on American option pricing, the continuation value at time $t_k$ is given by: \begin{align} F(\omega;t_k) = \mathbb{E}_Q\Big[\sum_{j=k+1}^Kexp\Big(-\int_{t_k}^{t_j}r(\omega,s)ds\Big)C(\omega;t_j;t_k,T)\Big|\mathcal{F}_{k}\Big]. \end{align} Why do we need the expected value in the above equation? Note that the formula is pathwise ($\omega$ is fixed). In other words, this is the future discounted cash flow for a fixed path that has already been simulated. What is the expectation averaging over?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.