American options

\[V_0 = \sup_{\tau} \mathbb{E}(e^{-r \tau} \Phi(X_{\tau}))\]

The exercise policy is estimated with the Longstaff-Schwartz algorithm (Longstaff and Schwartz, 2001): at each date the continuation value is regressed on $1, S, S^2$ over the in the money paths, and the option is exercised where the payoff exceeds it. The regression uses values only, so the policy carries no derivative.

Why freezing the exercise is not enough

The usual approach (Pagès et al., 2018) freezes the stopping time of each path and differentiates the cash flows as for a European contract. Since the policy is optimal, the first order effect of the moving exercise boundary vanishes and the first order sensitivities are correct. The second order ones are not: the boundary contributes a term proportional to the jump of the slope of the value at the boundary, and a frozen stopping time removes it. For a put, the frozen gamma only counts the paths that reach maturity near the strike, so it is even below the European gamma (-23% to -61% on the Longstaff-Schwartz put with $K = 40$).

Vibrato over every exercise date

A policy frozen in price space, i.e. an exercise region $E_k$ of prices at each date, does not have this problem for the sensitivities with respect to $S_0$: the optimal exercise boundary does not depend on $S_0$, so the policy is optimal for every $S_0$, and the error of the regressed boundary enters every derivative with respect to $S_0$ only at second order. The discontinuity of the exercise decision is then handled as the payoff at maturity: by vibrato. The paths are simulated at $s_0 = 0$ and $s_k = (k - \frac{1}{2}) h$, so that every exercise date is the midpoint of $[s_k, s_{k+1}]$ and, given the ends, the log price at $t_k$ is a Brownian bridge:

\[X_{t_k} | X_{s_k}, X_{s_{k+1}} \sim \mathcal{N}(\frac{X_{s_k} + X_{s_{k+1}}}{2}, \frac{\sigma^2 h}{4})\]

The bridge values are conditionally independent, and the future of a path depends on $X_{t_k}$ only through the decision at $t_k$. The value of each path is computed backwards:

\[W_N = \mathbb{E}(e^{-r T} \Phi(X_T) | X_{s_N})\]

\[W_k = \mathbb{E}(1_{X_{t_k} \in E_k} e^{-r t_k} \Phi(X_{t_k}) + 1_{X_{t_k} \notin E_k} W_{k+1} | X_{s_k}, X_{s_{k+1}})\]

and $V_0 = \mathbb{E}(W_1)$. Every conditional expectation is computed with the LRM over the bridge density, so the exercise indicator and the payoff are never differentiated: the estimator is smooth in all the parameters and unbiased for the chosen policy. The price of this exactness is variance: at the exercise boundary the integrand jumps by the gap between the exercise value and the realized continuation, and the weights of a bridge of standard deviation $\sigma \sqrt{h} / 2$ amplify it, more and more as the number of exercise dates grows.

The bridge is Gaussian only for Black-Scholes dynamics. For Lévy and Heston models the exercise decision for $t_k$ is taken at $t_{k-1}$, the payoff is replaced by its vibrato expectation over the step, and the decisions are frozen: their first order sensitivities are accurate, the second order ones are biased.