January 26, 2025
An American put option with strike K and expiration T on an underlying having price S_t at time t pays \max\{K - S_\tau,0\} at a time \tau\le T chosen by the option holder. The sample space for the Black-Scholes/Merton model is \Omega = C[0, \infty), the set of continuous functions on the non-negative real numbers. The risk-neutral stock price is S_t = S_0\exp(\rho t + \sigma B_t - \sigma^2 t/2), where \rho is the risk-free rate and \sigma is the B-S/M volatility.
The sample space for an American option is the cartesian product \Omega\times (0, T] where the second factor is the exercise time. We assume the option cannot be exercised at t = 0. If the holder exercises at the stopping time \tau\colon\Omega\to (0,T] then the option value is V_0 = (\sum_{0 < t \le T} \nu(S_t) 1(\tau=t) D_t)(\Omega\times (0,T]), where D_t = \exp(-\rho t)P is the deflator at time T.
The option value at time t\le T is determined by V_t D_t = (\sum_{t \le s \le T} \nu(S_s) 1(\tau=s) D_s)|_{{\mathcal{A}}_t}.
A common assumption is the option holder exercises optimally so the option value is v = \max_{0\le\tau\le T} E[\nu(S_\tau) \exp(-\rho\tau)], where \tau\colon\Omega\to[0,T] is a stopping time and \nu(s) = \max\{K - s,0\} is the put payoff.
While this may be plausible for American options on equities, it is untenable for, e.g., mortgage backed securities. Their value depends on the actual prepayments and those are never optimal in reality. It is important models have knobs for all possible outcomes and not make the mistake of implicit optimization.
It is common to underspecify the sample space for models. For example, the exercise time might depend on tax considerations or social media data.
A Bermudan option specifies a set of times T_0 < \cdots < T_n at which the option holder can receive the payoff. Assuming the option holder exercises optimally, the above formula for the value is restricted to stopping times \tau\colon\Omega\to\{T_0, \ldots, T_n\}. To compute this define v_j by v_j \exp(-\rho T_j) = \max_{T_j\le\tau\le T} E[\nu(S_\tau) \exp(-\rho \tau)\mid\tau > T_j], where \nu is the payoff and {\mathcal{A}}_t is the information available at time t.
Lemma. We have v_j\exp(-\rho T_j) = \max\{\nu(S_{t_j}), v_{j + 1}\exp(-\rho T_{j+1})\}.