Longstaff-Schwartz Least Squares Monte Carlo (LSM) Lab

1. Optimal Stopping Time Formulation

American option valuation is governed by an optimal stopping problem over a complete probability space $(\Omega, \mathcal{F}, \mathbb{Q})$ with filtration $\{\mathcal{F}_t\}_{t=0}^T$:

where $\mathcal{T}_{[0, T]}$ is the set of all stopping times taking values in the discrete exercise grid $\{t_0, t_1, \dots, t_N\}$, and $h(S)$ represents the intrinsic exercise payoff $(K - S)^+$ for puts or $(S - K)^+$ for calls.

2. Cross-Sectional Conditional Expectation via OLS

At time step $t_k$, the continuation value $\Phi(S_{t_k})$ of holding the option until optimal stopping time $\tau > t_k$ satisfies the conditional expectation:

By the Markov property, conditional expectation depends solely on $S_{t_k}$. Longstaff and Schwartz approximate this expectation as a linear combination of $P$ orthogonal basis functions $\psi_p(S)$:

3. Restriction to In-the-Money (ITM) Trajectories

A key insight of Longstaff and Schwartz is restricting the cross-sectional regression at time $t_k$ strictly to paths that are in-the-money, i.e., $\{m \in \{1, \dots, M\} : h(S_{t_k}^{(m)}) > 0\}$.

Paths that are out-of-the-money have zero immediate exercise value, making early exercise strictly impossible at $t_k$. Omitting OTM paths eliminates noise and dramatically improves the numerical accuracy of $\boldsymbol{\beta}$ in the neighborhood of the critical exercise boundary $S^*(t)$.

4. Backward Induction & Stopping Rule

At expiration $t_N = T$, the cash flow for each path $m$ is initialized to $C_N^{(m)} = h(S_T^{(m)})$ with stopping time $\tau^{(m)} = T$. Moving backward from $t_{N-1}$ to $t_1$:

1. Identify in-the-money paths at step $k$: $h(S_{t_k}^{(m)}) > 0$.
2. Regress discounted realized future cash flows $Y_k^{(m)} = e^{-r(\tau^{(m)} - t_k)} C_{\tau^{(m)}}^{(m)}$ onto basis vectors $X_k^{(m)} = [\psi_0(S_{t_k}^{(m)}), \dots, \psi_{P-1}(S_{t_k}^{(m)})]$.
3. Evaluate estimated continuation value: $\hat{\Phi}(S_{t_k}^{(m)}) = X_k^{(m)} \hat{\boldsymbol{\beta}}$.
4. Optimal Exercise Decision: If $h(S_{t_k}^{(m)}) > \hat{\Phi}(S_{t_k}^{(m)})$, set $C_k^{(m)} = h(S_{t_k}^{(m)})$ and update $\tau^{(m)} = t_k$. Otherwise, leave $\tau^{(m)}$ unchanged.