# 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?

• It would have been very easy for you to tell us in which paper and on which page that formula is. To the question: the authors have the habit to write $\omega$s where there should be none. Apr 19 at 20:38
• @KurtG. I would appreciate to hear your input. I don't know why this question was closed; I am knowledgeable about quantitative finance, and the answer to my question is unclear to me. I therefore assume it is also unclear to other people interested in the field.
– arni
Apr 21 at 1:32
• My reasons for voting to close this question were that it was not self-contained (lack of exact reference and lack of explaining notation). The correct LS notation should be \begin{align} F(\,.\,;t_k) = \mathbb{E}_Q\Big[\sum_{j=k+1}^K\exp\Big(-\int_{t_k}^{t_j}r(\,.\,,s)ds\Big)C(\,.\,;t_j;t_k,T)\Big|\mathcal{F}_{k}\Big]. \end{align} This is the PV seen at time $t_k$ of the sum of all future cashflows. We need the conditional expectation at $t_k$ since that represents what we know at time $t_k$ and that is on what we base our decision to exercise at $t_k$ or not. Apr 21 at 1:45
• On p123 in the paper it says "We approximate $F(\omega; t_{K-1})$ using $M$ basis functions, and denote this approximation $F_M(\omega; t_{K-1})$. Once the basis functions have been specified, $F(\omega; t_{K-1})$ is estimated by projecting or regressing the discounted values of $C(\omega, s; t_{K-1},T)$ onto the basis functions." To me this is abuse of notation, because in the formula I provided (Eq. 1 in the paper) they are averaging over future paths, but in the algorithm itself, the future cash flows are discounted pathwise. In BOTH cases they use the "fixed $\omega$"-notation.
– arni
Apr 21 at 3:46
• I think it's worthwhile to clear up needless confusion here. Apr 21 at 10:56