American options — Longstaff-Schwartz

American options can be exercised at any time up to expiry. Pricing them requires finding the optimal exercise strategy — at each time and state, decide whether to exercise immediately or continue holding. The Longstaff-Schwartz (2001) Least-Squares Monte Carlo (LSM) algorithm democratised American option pricing: simulate paths, regress to estimate continuation values, and exercise when the immediate payoff exceeds the estimated continuation value.

The American-option dilemma

Backward dynamic programming says: at expiry, hold or exercise based on the payoff. One step back: hold or exercise based on max(immediate payoff, present value of continuing). But the continuation value depends on the optimal strategy in the future, which itself depends on continuation values further out — circular. PDE methods solve this by backward induction in time and space. Monte Carlo is harder because we simulate forward but value backward.

Longstaff-Schwartz idea

Simulate N paths forward. At each exercise date (going backward in time), regress the realised cash flows on basis functions of the current state to estimate the continuation value. Exercise on path p if immediate exercise > regression-estimated continuation. Iterate backward.

Algorithm in detail

1. Simulate N paths of the underlying from t = 0 to T.
2. Initialise: at t = T, optimal action is exercise; record cash flow = payoff(S(T)).
3. Loop backward over exercise dates t_M, t_{M-1}, ..., t_1:
4. At each t_k: for in-the-money paths, regress future cash flows (discounted to t_k) on basis functions of S(t_k).
5. Compute continuation value ĉ(S) from the regression.
6. On each in-the-money path: if payoff(S(t_k)) > ĉ(S(t_k)), exercise at t_k. Update path's cash flow.
7. After the backward loop: average discounted optimal cash flows = LSM American price.

Choice of basis functions

Standard choices: polynomials of S (1, S, S², S³), Laguerre or Hermite polynomials, or moneyness-based features. Three or four basis functions usually suffice. Over-fitting can hurt; under-specification more rarely.

Why LSM works

The regression is an approximation to the true conditional expectation E[future discounted cash flows | S(t_k)]. Under mild regularity conditions (Clément-Lamberton-Protter 2002), LSM converges to the true American price as N → ∞ and the basis function set grows. The convergence rate is typically slower than European MC because of the regression error.

Bias structure

LSM gives a low-biased estimate: the recovered exercise policy is sub-optimal because of regression noise, so the realised value undervalues the true price. Andersen-Broadie (2004) high-biased estimator (based on dual martingale arguments) provides a complementary upper bound; averaging gives a tight interval.

American options in production

Equity index puts, equity options, callable bonds, mortgage-prepayment risk, Bermudan swaptions, real options in capital budgeting — all are American-style early-exercise problems. LSM is the workhorse Monte Carlo method; binomial/trinomial trees and PDE methods are alternatives for low-dimensional problems.

Variants

• Tsitsiklis-Van Roy: estimate value function (regression to past payoffs); related but less popular.
• Stochastic Mesh (Broadie-Glasserman): pseudo-mesh of nodes; high variance but unbiased.
• Andersen-Broadie dual method: provides an upper bound; complementary to LSM lower bound.
• Deep-LSM: use neural networks instead of polynomial regression. Modern frontier.