Ph.D. Public Lecture (DSAS) - François-Michel Boire

Efficiency Improvements in the Least-Squares Monte Carlo Algorithm

Abstract:

This thesis presents a collection of four essays dealing with the efficient pricing of American options with Monte Carlo simulation techniques. Specifically, we focus on developing and optimizing techniques to reduce pricing bias and variance in the Least-Squares Monte Carlo (LSM) algorithm of Longstaff & Schwartz (2001).

In the first chapter, we take advantage of the put-call symmetry property to improve the efficiency of variance reduction techniques applied to American call options. Indeed, it is always more efficient to apply variance reduction techniques to corresponding symmetric put options instead. The second chapter introduces a new importance sampling approach that performs regressions on paths simulated under the importance probability measure directly. This enhances continuation value estimates by performing regressions on samples that have more density near the optimal exercise boundary. Results show that this method successfully reduces the bias plaguing the standard method where regressions are instead performed under the nominal probability measure. The third chapter derives an approximation of local LSM estimator bias when exercise decisions and option values are mutually dependent. The derivations presented in this paper explicitly deal with foresight bias and hold true for general asset price processes and option payoff structures. We then introduce a new bias-corrected LSM estimator and demonstrate its robustness in the presence of multiple stochastic factors and price discontinuities. Finally, the fourth chapter studies the effects of stopping time optimality on the variance of the LSM estimator. We show that exercise errors stifle the variance reduction efficiency gains of control variates and preclude an efficient combined implementation of control variates and importance sampling. Results indicate that the corrected strategy of Rasmussen (2005) substantially improves stopping time optimality, virtually eliminating bias and improving the variance reduction efficiency gains by orders of magnitude.

Throughout the four chapters, we provide an in-depth discussion about the causes leading to the LSM estimator bias and variance. Using these insights, we develop extensions of the LSM, detail their implementation, and make numerous recommendations aimed at improving performance and computational efficiency. The numerical studies in this thesis test the robustness of our proposed methods across a wide range of option characteristics.