Sum of all Black–Scholes–Merton models: An efficient pricing method for spread, basket, and Asian options

• Published date: 01 June 2018
• Date: 01 June 2018
• DOI: http://doi.org/10.1002/fut.21909

Received: 7 August 2017

|

Accepted: 6 January 2018

DOI: 10.1002/fut.21909

RESEARCH ARTICLE

Sum of all Black–Scholes–Merton models: An efficient pricing

method for spread, basket, and Asian options

Jaehyuk Choi

Peking University HSBC Business School,

Shenzhen, China

Correspondence

Jaehyuk Choi, Peking University HSBC

Business School, University Town,

Nanshan District, 518055 Shenzhen, China.

Email: jaehyuk@phbs.pku.edu.cn

Funding information

Bridge Trust Asset Management Research

Fund

Contrary to the common view that exact pricing is prohibitive owing to the curse of

dimensionality, this study proposes an efficient and unified method for pricing

options under multivariate Black–Scholes–Merton (BSM) models, such as the

basket, spread, and Asian options. The option price is expressed as a quadrature

integration of analytic multi-asset BSM prices under a single Brownian motion. Then

the state space is rotated in such a way that the quadrature requires much coarser nodes

than it would otherwise or low varying dimensions are reduced. The accuracy and

efficiency of the method is illustrated through various numerical experiments.

KEYWORDS

Asian option, basket option, curse of dimensionality, multi-asset Black–Scholes–Merton, spread

option

1

|

INTRODUCTION

1.1

|

Background

Ever since the celebrated success of the Black–Scholes–Merton (BSM) model, the effort to extend its simple analytic solution to

the derivatives written on multiple underlying assets has been of great interest to researchers (Broadie & Detemple, 2004). One

important class of such derivatives is the options on a linear combination of assets following correlated geometric Brownian

motions (GBMs); these include the following three popular option types:

Spread option: European-style option on the difference between two asset prices;

Basket option: European-style option on the sum of multiple asset prices with positive weights;

Asian option: option on the average price of one underlying asset on a predetermined discrete time set or continuous time

range.

These three option types arguably represent the most actively traded non-vanilla options on exchanges or in over-the-counter

markets. This is because a linear combination is a common way of associating multiple prices—of different assets or various

times—and such options provide customized hedge or risk exposure. For examples and financial motivations, see the

introductions of Carmona and Durrleman (2003) and Linetsky (2004).

However, such option pricings under the BSM model, not to mention the models beyond the BSM, is not a trivial matter. This

is because, unlike normal random variables (RVs), a linear combination of correlated log-normal RVs neither falls back to the

same class of distribution nor has a distribution expressed in any analytic form in general. Therefore, the exact valuation of option

prices involves a multidimensional integral over positive payoffs under the risk-neutral measure. The numerical evaluation of

such an integral, however, suffers from the curse of dimensionality. For example, even the coarse discretization of a standard

J Futures Markets. 2018;38:627–644. wileyonlinelibrary.com/journal/fut © 2018 Wiley Periodicals, Inc.

|

627

normal distribution from −5 to 5 with a grid size of 0.25 (41 points per dimension) leads to three million points for four assets and

116 million points for five assets, substantially exceeding the size of a typical Monte Carlo simulation.

1.2

|

Literature review

A vast amount of the literature has examined each of the three option types. First, a review is conducted of the analytic methods of

approximating the option price or the lower and upper bounds of the option price. For the spread option, Kirk's formula (Kirk,

1995) that is widely used in practice is an approximate generalization of Margrabe's formula (Margrabe, 1978) for an exchange

option, that is, a spread option with zero-strike price; several improvements to the formula have followed (Bjerksund &

Stensland, 2014, Lo, 2015). Carmona and Durrleman (2003) computes the lower bound of a spread option price as the maximum

over the prices from all possible linear, and therefore sub-optimal, exercise boundaries. Li et al. (2008) proposes a closed-form

formula based on quadratic approximation of the exercise boundary.

Basket and Asian options share the underlying ideas for analytic approximation because the payoff of an Asian option

depends on a basket of correlated prices over the observation period. One of the most popular approaches is to approximate the

distribution of the GBM sum with other analytically known distributions, such as log-normal (Levy, 1992; Levy & Turnbull,

1992), reciprocal gamma (Milevsky & Posner, 1998a,b), shifted log-normal (Borovkova, Permana, & Weide, 2007), and

log-extended-skew-normal (Zhou & Wang, 2008) distributions, and with perturbation expansions from known distributions

(Ju, 2002; Turnbull & Wakeman, 1991). Another popular idea is to exploit the geometric mean of GBMs—as opposed to

arithmetic mean—whose distribution is log-normal and hence analytically solvable. The option price on the geometric mean is a

reasonable proxy for that on the arithmetic mean (Gentle, 1993). Thus, it can be used as a control variate, thereby reducing the

Monte Carlo variance for Asian (Kemna & Vorst, 1990) and basket (Krekel, de Kock, Korn, & Man, 2004) options. Curran

(1994) uses the geometric mean as a conditioning variable to analytically estimate option prices. The conditioning approach is

further refined by Beisser (1999) and Deelstra et al. (2004) and also applied to the continuously monitored Asian options (Rogers

& Shi, 1995). For other basket and Asian option pricing approaches as well as classifications, see Zhou and Wang (2008) and the

references therein.

Analytic approximation methods are appealing because of their simple computation, but has a major limitation in that the

results are not precise. While each method is accurate for certain parameter ranges where the underlying assumptions are valid,

the accuracy of any one method can hardly be validated for all ranges. For example, no single method performs well in basket

options under various parameter sets (Krekel et al., 2004), although the method of Ju (2002) is outstanding overall. Therefore,

practitioners must carefully identify the parameter range in which the method of interest works best. Given the multi-asset aspect

of such problems, the charting of parameter maps in advance is not a trivial matter. Moreover, since errors cannot be controlled in

approximation methods, the tendency is to eventually apply external methods, typically a Monte Carlo simulation, to obtain a

benchmark value.

Convergent pricing methods are fewer in number compared to approximation methods. Convergent implies that the method

can produce a deterministic price, as opposed to Monte-Carlo methods, and converge to the true value with a reasonable amount

of computation when the computational parameters (e.g., grid size) are tuned. Convergent methods are feasible for spread

options because the problem is two-dimensional. Ravindran (1993) and Pearson (1995) reduce the pricing problem to a

one-dimensional integration over the BSM prices with varying spot and strike prices. In addition, Dempster and Hong (2002) and

Hurd and Zhou (2010) apply a two-dimensional fast Fourier transform (FFT).

To the best of the author's knowledge, few studies on basket options use methods that can be defined as convergent.

Particularly, no attempt has been made to use direct integration, even for error measurement, thus indicating the challenge posed

by the approach (see section 4.1). Although the FFT approach proposed by Leentvaar and Oosterlee (2008) reduces the

computation time through parallel partitioning, it does not significantly reduce the computation amount.

Previous convergent methods used for Asian options are based on the idea that pricing involves a single price process over

time. The continuously averaged Asian option, although hardly traded in practice owing to contractual difficulty, has analytic

solutions comprising the triple integral (Yor, 1992), Laplace transform in maturity (Geman & Yor, 1993), and a series expansion

(Linetsky, 2004). For discrete averaging, a series of studies have exploited the recursive convolution, referred to as

the Carverhill–Clewlow–Hodges factorization, on the probability density function (PDF) or price (Benhamou, 2002, Carverhill

& Clewlow, 1990, Černý & Kyriakou, 2011, Fusai & Meucci, 2008, Fusai, Marazzina, & Marena, 2011, Zhang & Oosterlee,

2013). Cai et al. (2013) describes the price of a discretely monitored Asian option as an asymptotic expansion on a small

observation interval.

Despite structural similarity, only a few studies are applicable to all the three option types. Carmona and Durrleman (2005)

extend the lower bound approach (Carmona & Durrleman, 2003) to a multi-asset problem, while Deelstra et al. (2010) use the

628

|

CHOI