Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article

Degree

Doctor of Philosophy

Program

Statistics and Actuarial Sciences

Supervisor

Stentoft, Lars

2nd Supervisor

Reesor, Mark

Affiliation

Wilfrid Laurier University

Co-Supervisor

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 \citet{LS2001}.

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.

The second chapter introduces a new importance sampling approach that performs regressions on paths simulated under the importance probability measure directly. 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. 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. Results indicate that the corrected strategy of \citet{R2005} 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.

Summary for Lay Audience

Options are widely traded assets that find many applications in financial risk management. These derivative contracts stipulate that one party can trade an underlying asset, index or security with another party at a fixed price at a future date. If the option owner decides to exercise the option, the seller is then obligated to enter the agreed-upon trade. Alternatively, the owner may decide to forgo exercise if the trade is not profitable. Options thus offer limited-term insurance to the owner by guaranteeing that the underlying asset can be traded at a fixed price in the future.

Depending on contract modalities, the valuation of the options may be more or less difficult. In the simplest cases, options are exercisable solely when the contract expires. These options are termed ``European options'' and are relatively easy to price. Another class of options called ``American options'' allows exercise at any time before expiry, providing the owner with more flexible insurance. Contrary to European options, American options prices are difficult to estimate. This is because the European option exercise date is specified by the contract, whereas the American option exercise time is uncertain and must be approximated. In turn, the management and pricing of American options are complicated by the fact that one must continuously consider whether it is best to exercise the option or to keep it alive. Consequently, any actual numerical approach to this ``optimal control problem’’ involves the joint estimation of the option price and the exercise strategy in a ``dynamic programming algorithm’’.

This thesis focuses on the renowned Least-Squares Monte Carlo (LSM) algorithm of \citet{LS2001} for pricing American options. This simulation-based dynamic programming algorithm is flexible, easy to implement, and robust to various option pricing problems. However, implementing the LSM algorithm can be computationally costly, and its optimization remains an active area of research. Throughout the four chapters presented in this thesis, we focus on developing and optimizing techniques to reduce the bias and variance of LSM estimators. Extensive numerical studies test the robustness of our proposed methods across a wide range of option characteristics and underlying asset processes.

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