Leave‐one‐out least squares Monte Carlo algorithm for pricing Bermudan options

• Published date: 01 August 2024
• Author: Jeechul Woo,Chenru Liu,Jaehyuk Choi
• Date: 01 August 2024
• DOI: http://doi.org/10.1002/fut.22515

Received: 8 February 2023

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Accepted: 6 May 2024

DOI: 10.1002/fut.22515

RESEARCH ARTICLE

Leave‐one‐out least squares Monte Carlo algorithm

for pricing Bermudan options

Jeechul Woo

1

|Chenru Liu

2

|Jaehyuk Choi

3

1

Moduli Technologies LLC, Springfield,

Illinois, USA

2

Department of Management Science and

Engineering, Stanford University,

Stanford, California, USA

3

Peking University HSBC Business

School, Shenzhen, China

Correspondence

Jaehyuk Choi, Peking University HSBC

Business School, University Town,

Nanshan, Shenzhen 518055, China.

Email: jaehyuk@phbs.pku.edu.cn

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.

KEYWORDS

American option, Bermudan option, least squares Monte Carlo, leave‐one‐out‐cross‐

validation, look‐ahead bias

JEL CLASSIFICATION

C63, G10, G13

1|INTRODUCTION

Derivatives with early exercise features, such as American and Bermudan options, are popular in the market.

Nonetheless, pricing these options is a difficult problem in the absence of closed‐form solutions, even in the simplest

case of American options on a single asset. Researchers have thus developed various numerical methods for pricing

that largely fall into two categories: the lattice‐based and simulation‐based approaches.

In the lattice‐based approach, pricing is performed on a dense lattice in the state space by valuing the options at

each point of the lattice using suitable boundary conditions and the mathematical relations among neighboring points.

Examples include the finite difference scheme (Brennan & Schwartz, 1977), binomial tree (Cox et al., 1979), and its

multidimensional generalizations (Boyle, 1988; Boyle et al., 1989; He, 1990). These methods are known to work well in

low‐dimensional problems. However, they become impractical in higher‐dimensional settings, mainly because the

lattice size grows exponentially as the number of state variables increases. This phenomenon is commonly referred to

as the curse of dimensionality.

In the simulation‐based approach, the price is calculated as the average of the option values over simulated paths,

each representing a future realization of the state variables with respect to the risk‐neutral measure. While the methods

in this category are not challenged by dimensionality, they entail finding the optimal exercise rules. Several simulation‐

based methods propose various approaches for estimating the continuation values as conditional expectations.

Equipped with stopping time rules, they calculate the option price by solving a dynamic programming problem whose

J Futures Markets. 2024;44:1404–1428.wileyonlinelibrary.com/journal/fut1404

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© 2024 Wiley Periodicals LLC.

Bellman equation is essentially the comparison between the continuation values and exercise values. In comparison to

the randomized tree method (Broadie & Glasserman, 1997) and stochastic mesh method (Broadie & Glasserman, 2004),

regression‐based methods (Carriere, 1996; Longstaff & Schwartz, 2001; Tsitsiklis & Van Roy, 2001) are computationally

tractable. Because the methods use regression to estimate the continuation values from the simulated paths, their

computation time is linear not only in the number of simulated paths, but also in the number of exercise times. Among

the regression‐based methods, the least squares Monte Carlo (LSM) algorithm proposed by Longstaff and Schwartz

(2001) is the most popular for its simplicity and efficiency. Fu et al. (2001) and Glasserman (2003) provide

comprehensive reviews of the implementation and comparison of simulation‐based methods.

The LSM method is essential for pricing callable structured notes whose coupons have complicated dependency on

other underlying assets such as equity prices, foreign exchange rates, and benchmark interest swap rates. Because a

multi‐factor model is required for the underlying assets and the yield curve term structure, using Monte Carlo

simulation along with the LSM method is inevitable for pricing and risk‐managing such notes. Like many previous

studies on this topic (Beveridge et al., 2013; Kolodko & Schoenmakers, 2006), this study is motivated and developed in

the context of callable structured notes.

1.1 |Biases in the LSM method

In simulation‐based methods including the LSM method, there are two main sources of bias, which run in opposite

directions. Low‐side bias is related to suboptimal exercise decisions owing to various approximations adopted in the

method. In the LSM method, for example, finite basis functions cannot fully represent the conditional payoff function.

The resulting exercise policy deviates from the most optimal one, leading to a lower option price. For this reason, it is

also called suboptimal bias. High‐side bias comes from using one simulation set for both the exercise decision and the

payoff valuation. As explained by Broadie and Glasserman (1997), this practice creates a fictitious positive correlation

between exercise decisions and future payoffs; the algorithm is more likely to continue (exercise) precisely when the

future payoff in the simulation is higher (lower). For this reason, it is called look‐ahead or foresight bias. The LSM

estimator has both low‐and high‐side biases; hence, Glasserman (2003) calls it an interleaving estimator. Other

simulation estimators in the literature are typically either low‐biased or high‐biased. For example, Broadie and

Glasserman (1997) carefully construct both low‐and high‐biased estimators to form a confidence interval for the true

option price.

In callable structured note markets, look‐ahead bias is more dangerous than suboptimal bias, and look‐ahead bias

mixed with suboptimal bias is a significant drawback of the LSM estimator. This is closely related to the typical roles of

the market players. The financial institutions that issue notes are the effective buyers of the Bermudan option to

redeem notes early. On the contrary, investors, be they individual or institutional, are the effective sellers of the option.

Investors receive the option premium in the form of an enhanced note yield compared to noncallable notes with the

same structure.

1

As a result, option buyers (financial institutions) act as pricing agents. Because buyers have to risk‐

manage and optimally exercise the option, they typically use the LSM method. Option sellers (investors) usually hold

the note until maturity without hedging and, therefore, are less sensitive to accurate valuation. From option buyers'

perspective, look‐ahead bias is malicious because it wrongly inflates the option premium they pay. No matter how well

the option is delta‐hedged, the option value attributed from look‐ahead bias shrinks to zero when the position is near

the maturity or the early exercise because there is no more future to look into by then. Suboptimal bias, on the contrary,

is benign. Although it deflates the option value, the gain realized through delta‐hedging under the suboptimal exercise

policy is just as much as the deflated option value. In short, option buyers get what they pay for. The only downside of

suboptimal bias for buyers is to lose trades to competitors who bid a higher (more optimal) option premium. Therefore,

buyers prefer the low‐biased estimator to ensure that the premium they pay is lower than the true value. However,

look‐ahead bias mixed in the LSM method makes a conservative valuation difficult for buyers.

A standard technique for eliminating look‐ahead bias is calculating the exercise decision using an additional

independent set of Monte Carlo paths, thereby eliminating the correlation between the exercise decision and simulated

payoff. While this two‐pass approach removes look‐ahead bias, it comes at the cost of doubling the computational cost,

1

The buyers and sellers of the Bermudan option rarely change in the market. Financial institutions hardly issue puttable notes, where they sell the

option to investors. Puttable notes are not attractive to investors because the yield would be lower (i.e., investors pay the option premium).

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