[Paper Review] Leave-one-out least squares Monte Carlo algorithm for pricing Bermudan options

Jeechul Woo, Chenru Liu+1 more|arXiv (Cornell University)|Oct 4, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance46 references3 citations

TL;DR

This paper introduces the Leave-One-Out Least Squares Monte Carlo (LOOLSM) algorithm to eliminate look-ahead bias in American and Bermudan option pricing using the LSM method. By training regression models on leave-one-out samples, LOOLSM removes the fictitious correlation between exercise decisions and future payoffs, achieving bias reduction without doubling simulations, with asymptotic bias proportional to the regressors-to-simulations ratio.

ABSTRACT

The least squares Monte Carlo (LSM) algorithm proposed by Longstaff and Schwartz (2001) is widely used for pricing Bermudan options. The LSM estimator contains undesirable look-ahead bias, and the conventional technique of avoiding it requires additional simulation paths. We present the leave-one-out LSM (LOOLSM) algorithm to eliminate look-ahead bias without doubling simulations. We also show that look-ahead bias is asymptotically proportional to the regressors-to-paths ratio. Our findings are demonstrated with several option examples in which the LSM algorithm overvalues the options. The LOOLSM method can be extended to other regression-based algorithms that improve the LSM method.

Motivation & Objective

• To address the look-ahead bias inherent in the standard Least Squares Monte Carlo (LSM) algorithm for pricing American and Bermudan options.
• To develop a computationally efficient alternative to the conventional double-simulation method for bias removal.
• To establish a theoretical foundation showing that look-ahead bias is asymptotically proportional to the ratio of regressors to simulation paths.
• To demonstrate the method’s effectiveness on multi-asset and complex structured note options where LSM overvalues the option price.
• To extend the LOOLSM framework to other regression-based Monte Carlo algorithms for improved accuracy in derivative pricing.

Proposed method

• Proposes a leave-one-out cross-validation approach where the regression model for continuation value estimation is trained on all simulation paths except one, eliminating the look-ahead bias from using the same path for both decision and payoff evaluation.
• Applies leave-one-out regression at each exercise date, ensuring that the estimated continuation value is conditionally independent of the future payoff path used in the decision rule.
• The method maintains computational efficiency by avoiding a full second simulation set, unlike the conventional double-simulation approach.
• Theoretical analysis shows that the look-ahead bias in LSM is asymptotically proportional to the ratio of the number of basis functions (regressors) to the number of simulation paths.
• Uses inductive and probabilistic arguments to prove that the bias of the LOOLSM estimator converges to zero in probability and in expectation at rate O_p(M/N), where M is the number of regressors and N the number of paths.
• Employs concentration inequalities and moment bounds to establish convergence properties under regularity conditions on the basis functions and payoff structure.

Experimental results

Research questions

• RQ1 Can look-ahead bias in the LSM algorithm be eliminated without doubling the number of simulations?
• RQ2 What is the theoretical relationship between the number of regressors and the magnitude of look-ahead bias in LSM?
• RQ3 How does the LOOLSM method perform on multi-asset Bermudan options where LSM is known to overvalue the option?
• RQ4 Does the LOOLSM estimator converge in probability and in expectation to the true option price under standard assumptions?
• RQ5 Can the LOOLSM framework be generalized to other regression-based Monte Carlo methods for derivative pricing?

Key findings

• The LOOLSM algorithm successfully eliminates look-ahead bias in American and Bermudan option pricing without requiring a second simulation set, offering a computationally efficient alternative to the double-simulation method.
• Theoretical analysis confirms that look-ahead bias in LSM is asymptotically proportional to the ratio of the number of regressors (M) to the number of simulation paths (N), i.e., O_p(M/N).
• In multi-asset option examples, the standard LSM method significantly overvalues the option price due to look-ahead bias, while LOOLSM corrects this overvaluation effectively.
• The LOOLSM estimator converges in probability to the true option price, with bias decaying to zero as the number of simulation paths increases.
• The expected bias of the LOOLSM estimator is o_p(1), and for any δ > 0 and ε > 0, the probability that the bias exceeds δ can be made arbitrarily small by increasing N, confirming consistency.
• The method is extendable to other regression-based Monte Carlo algorithms, offering a general framework for bias reduction in simulation-based derivative pricing.