Valuing American Options by Simulation: A Simple Least-Squares Approach

Abstract

This article presents a simple yet powerful new approach for approximating the value of America11 options by simulation. The kcy to this approach is the use of least squares to estimate the conditional expected payoff to the optionholder from continuation. This makes this approach readily applicable in path-dependent and multifactor situations where traditional finite difference techniques cannot be used. We illustrate this technique with several realistic exatnples including valuing an option when the underlying asset follows a jump-diffusion process and valuing an America11 swaption in a 20-factor string model of the term structure. One of the most important problems in option pricing theory is the valuation and optimal exercise of derivatives with American-style exercise features. These types of derivatives are found in all major financial markets including the equity, commodity, foreign exchange, insurance, energy, sovereign, agency, municipal, mortgage, credit, real estate, convertible, swap, and emerging markets. Despite recent advances, however, the valuation and optimal exercise of American options remains one of the most challenging problems in derivatives finance, particularly when more than one factor affects the value of the option. This is primarily because finite difference and binomial techniques become impractical in situations where there are multiple factors.' We are grateful for the cotnl~lelits of

Pricing Asset Scheduling Flexibility using Optimal Switching

“…This was the beautiful insight of Longstaff and Schwartz (2001). Let τ k (m∆t, X m∆t , i) ·∆t correspond to the smallest optimal switching time for J k (m∆t, X m∆t , i).…”

“…Therefore the conditional expectation can be viewed as simply a mapping x → E t 1 (x) = E[f (X t 2 )|X t 1 = x] and our strategy is to approximate this map. This is a well-known statistical problem; here we choose to concentrate on an approach first described by Longstaff and Schwartz (2001) and Tsitsiklis and Van Roy (2001). Many alternatives are available and we review and compare them in Section 5.2.…”

“…Several canonical choices have been proposed, including the Laguerre polynomials in the original paper of Longstaff and Schwartz (2001), the indicator functions of a partition of the domain of {X t } in Gobet et al (2005), and the logistic basis in Haugh and Kogan (2004). However, there is now widespread consensus that the numerical precision can be greatly improved by customizing the basis (for example, the customization advantage is documented by Andersen and Broadie (2004)).…”

Decision-making in incomplete markets with ambiguity—a case study of a gas field acquisition

“…Another complication is related to the various endogenous exercise moments; no analytical solution exists for such options either (in asset pricing jargon: we are dealing with what would be called American options in continuous time, or, more appropriately given our discrete time framework, Bermudatype options). We solve the valuation problem using SDP, and reduce the dimensionality problem using the Least Squares Monte Carlo approach proposed by Longstaff and Schwartz (2001). SDP sets the current value equal to current utility plus the value of continuing in the future.…”

Pricing Bermudan Interest Rate Swaptions via Parallel Simulation under the Extended Multi-factor LIBOR Market Model

“…Due to its large number of state variables, lattice-based methods are not feasible for this model class, and the pricing generally requires Monte Carlo simulations. To make Monte Carlo simulation be able to handle the built-in early exercise feature in Bermudan-style options Longstaff and Schwartz [10] proposed the least-squares Monte Carlo approach, which determines the early exercise boundary through linear regressions. Prices for Bermudan options computed using this regression-based method are biased low to the true values, because the exercise strategies generated by the regressions are inferior to the optimal ones.…”