The Longstaff-Schwartz Algorithm in Simple Terms

In quantitative finance, the Longstaff-Schwartz algorithm plays a crucial role in options pricing, particularly for American options, which allow for early exercise. The algorithm uses stochastic processes, such as Monte Carlo simulations, to determine the optimal exercise strategy by comparing the immediate exercise payoff to the continuation value.

The algorithm estimates the continuation value of an option at various time steps using simulated paths. Let \( C(t) \) denote the continuation value at time \( t \), and let \( P(t) \) represent the immediate exercise payoff. The decision rule for early exercise is given by:

Monte Carlo simulations generate multiple paths of the underlying asset price, \( S_t \), which evolves according to a stochastic process:

Here, \( \mu \) is the drift rate, \( \sigma \) is the volatility, and \( W_t \) is a Wiener process representing the random component.

To estimate the continuation value \( C(t) \), the algorithm applies regression techniques. Using simulated asset price paths, it fits a regression model to approximate the expected value of the payoff:

This approach enables the calculation of the continuation value based on the observed states \( S_t \) at each time step.

The Longstaff-Schwartz algorithm offers several advantages:

• It handles the complexity of early exercise decisions in American options efficiently.
• It is flexible and can be applied to a wide range of options and derivatives.
• It eliminates the need for a closed-form solution, which is often unavailable for American options.