Valuation and applications of compound basket options

• Published date: 01 June 2019
• Date: 01 June 2019
• Author: Kwangil Bae
• DOI: http://doi.org/10.1002/fut.21996

Received: 20 March 2018

|

Revised: 30 December 2018

|

Accepted: 4 January 2019

DOI: 10.1002/fut.21996

RESEARCH ARTICLE

Valuation and applications of compound basket options

Kwangil Bae

Department of Business Administration,

Chonnam National University, Gwangju,

South Korea

Correspondence

Kwangil Bae, College of Business

Administration, Chonnam National

University, Gwangju 61186, South Korea.

Email: k.bae@chonnam.ac.kr

Abstract

This study investigates compound basket options, which are options on portfolios of

options. Although they may be new to financial markets, they are available as equity

basket options, equity spread options, stocks of holding companies, and collateralized

debt obligations. Using moments of portfolio values, we provide formulas for pricing

compound basket options. According to numerical analysis, a lower bound and a

weighted average of bounds yield relatively small errors. Additionally, ignoring the

compound feature increases the pricing error of equity basket options because the

feature captures the capital structure and leverage effect of stock prices.

KEYWORDS

analytic approximation, compound basket options, compound spread options, equity basket options,

equity spread options

JEL CLASSIFICATION

G13

1

|

INTRODUCTION

Compound options, which are options on options, are known since the work of Geske (1977, 1979). Although

commonly categorized as exotic options, compound options are linked to basic derivatives on equities because equities

are regarded as call options on firm value. Thus, from this perspective, when we regard firm values as underlying assets,

stock prices reflect the capital structure and leverage effects. Geske, Subrahmanyam, and Zhou (2016) support this view

empirically and show that a compound option pricing model provides a relatively small error than the Black and

Scholes (1973) model in pricing individual equity options. This evidence indicates that by adopting the compound

option model, we can provide accurate formulas for other equity derivatives. In this study, we consider equity basket

options and equity spread options, as they are linked to options on portfolios of options. We call these options

compound basket options and investigate their pricing method.

To the best of our knowledge, compound basket options are not traded in financial markets. However, they can be

traded because both basket options and compound options are popular exotic options. In addition, the applicability of

our findings is not limited to derivatives strictly defined as compound basket options; our findings can also be applied to

equity basket options, including index options, spread options, and Asian options, as they have features similar to

compound basket options. Likewise, a holding company’s stock can be regarded as a compound basket option under the

interpretation of a call on a portfolio of calls rather than a stock on a portfolio of stocks. In addition, collateralized debt

obligations (CDOs) are also examples of compound basket options because corporate bonds are portfolios of risk‐free

assets and put options on firm values.

1

As mentioned above, when firm values are deemed underlying assets, stock

J Futures Markets. 2019;39:704–720.wileyonlinelibrary.com/journal/fut704

|

© 2019 Wiley Periodicals, Inc.

1

Note that a corporate bond is a digital call option on firm value when the recovery rate is zero. Accordingly, a CDO is related to an option on a portfolio of standard put options, digital call options, and

risk‐free assets.

prices reflect the leveraging effects. Therefore, by adopting the compound basket options model, we can estimate

accurate values of equity basket options, stocks of holding companies, and CDOs.

Despite the usefulness of compound basket options, they do not have an exact closed‐form solution and this could be

problematic. Accordingly, the literature on valuation of basket options suggests simulation, the finite different method,

or analytical approximations. Among these, simulation and the finite different method are too time‐consuming to be

accurate. Therefore, analytical approximations are developed as alternatives.

There are several analytical approximation methods. The first approach uses a well‐known distribution. It matches

the first two to four moments of the portfolio value to the moments of specific well‐known distributions, such as the

lognormal distribution, shifted lognormal distribution, reciprocal gamma distribution, shifted reciprocal gamma

distribution, or Johnson’s (1949) family function (Borovkova, Permana, & Weide, 2007; Chang, Chen, & Wu, 2012;

Levy, 1992; Lo, Palmer, & Yu, 2014; Milevsky & Posner, 1998b; Posner & Milevsky, 1998). The existing body of literature

shows that the option price can be obtained if the value of an underlying asset follows one of these distributions.

Therefore, by matching moments and approximating distributions, we can estimate the values of basket options. The

second approach adds terms to the first (Jarrow & Rudd, 1982; Ju, 2002; Turnbull & Wakeman, 1991). In this approach,

the Taylor expansion method of Ju (2002) seems to be accurate. However, it cannot be applied to spread options because

it does not allow for negative weights. The other approach uses upper and lower bounds, which are derived from the

decomposition of option prices or the conditional expectation of variables (Curran, 1994; Li, Deng, & Zhoc, 2008;

Nielsen & Sandmann, 2003; Rogers & Shi, 1995; Vanmaele, Deelstra, & Liinev, 2004). According to the literature, lower

or upper bounds work well, not only as bounds but also as approximations.

In this study, we derive the moments and conditional momentsofthevalueofoptionsinaportfolio.Thestudies

mentioned above address the fact that moment‐matching methods and application of conditional moments introduce small

errors in formulas. Therefore, we investigate the value of compound basket options by obtaining moments for portfolios of

options. In addition, we conduct numerical analysis based on these formulas. According to our analysis, considering the

capital structure of firms reduces errors in equity basket option pricing formulas, which is consistent with Geske et al.’s

(2016) empirical study on individual equity options. Moreover, our numerical analysis shows that the weighted average of

bounds is effective in compound basket options, while the lower bound approach is useful in compound spread options.

In the existing literature, our compound basket options are related to the compound spread options of Eberlein and

Madan (2012) and heating degree days (HDD) and cooling degree days (CDD) options in weather derivatives. First, Eberlein

and Madan (2012) note that bonds can be regarded as options on spread options and call them compound spread options.

These relate to the compound basket options discussed in this study because a spread option is a special case of a basket

option. However, the difference is that we refer to an option on a spread of options, whereas Eberlein and Madan (2012)

representanoptiononaspreadoption.Second,CDDisthe sum of daily nonnegative excessive temperature over a

benchmark (generally 65° F) for a specific period.

2

Therefore, a CDD option is a call option on a portfolio of cash‐settled call

options on temperatures. However, most studies, including Huang, Yang, and Chang (2018), focus on identifying the

dynamics of the temperatures rather than investigating analytical pricing formulas because of the unique characteristics of

the temperatures. Moreover, the compounding feature is usually ignored because CDDs measure cooling energy in summer

and most temperatures exceed the benchmark in summer, as Alaton, Djehiche, and Stillberger (2002) note.

The rest of the paper is organized as follows. Section 2 presents the assumptions and moments of the value of option

portfolios. Section 3 presents the approximation formulas for the pricing of compound basket options. Section 4

compares the accuracy of the formulas via numerical analysis. The final section concludes the study.

2

|

MODEL

2.1

|

Assumptions

Assume a risk‐free asset with a rate of return r. We consider nunderlying assets. Let

∈

S

ti n( ), {1, 2,…, }

ibe the price of

the ith security at time t, with the following dynamics under the risk‐neutral probability measure:

d

St r qStdt σStdWt()=( −) () + () ()

,

iiiiii

(1)

2

More precisely, CDD is definedas ∑tmax ( −65,0

)

i

Ii

=1 where

t

iis the average temperature in Fahrenheit at day

i

. Similarly, HDD is definedas ∑tmax (65 −,0

)

i

Ii

=1 . Due to similarity, we omit the

explanation for HDD options.

BAE

|

705