Pricing callable–puttable convertible bonds with an integral equation approach

• Published date: 01 October 2022
• Author: Sha Lin,Song‐Ping Zhu
• Date: 01 October 2022
• DOI: http://doi.org/10.1002/fut.22363

Received: 13 January 2022

|

Accepted: 15 June 2022

DOI: 10.1002/fut.22363

RESEARCH ARTICLE

Pricing callable–puttable convertible bonds with an

integral equation approach

Sha Lin

1

|Song‐Ping Zhu

2

1

School of Finance, Zhejiang Gongshang

University, Hangzhou, China

2

School of Mathematics and Applied

Statistics, University of Wollongong,

Wollongong, New South Wales, Australia

Correspondence

Sha Lin, School of Finance, Zhejiang

Gongshang University, Hangzhou

310018, China.

Email: linsha@mail.zjgsu.edu.cn

Abstract

In this paper, the pricing problem of callable–puttable convertible bonds

written on a single underlying asset is studied with an integral equation (IE)

approach. The complication of the pricing exercise results from the tangled

presence of callability, puttability,aswellasconversion,whichhave

led to possible coexistence of two moving boundaries at the same time,

depending on the call price, the put price, and the conversion ratio. If a

callable–puttable convertible bond needs to be priced at a time sufficiently

far away from the expiry, only the moving boundary associated with the

puttability needs to be dealt with. When the pricing time is closer to expiry

beyond a critical point, it is then possible to have two distinct cases. While

the two moving boundaries associated with conversion and puttability

coexist in one case, they may both disappear in another case; callability

remains to be the only issue that needs to be dealt with. Furthermore, there

exists another critical value, beyond which a callable–puttable convertible

bond can be treated as its vanilla counterpart. Mathematically, various

different scenarios demand different systems of IEs to be formulated and

solved numerically.

KEYWORDS

callable–puttable convertible bond, integral equation, three different scenarios

1|INTRODUCTION

Convertible bonds (CBs)

1

are widely used financial instruments, which are different from bonds and stocks. However,

CBs can be treated as a combination of bonds and options, since they possess the essential characteristics of these two.

A CB gives its holders a right to convert the bond into a predetermined number of underlying stocks either only at the

expiry (the so‐called European‐style) or during the entire life of the bond (the so‐called American‐style). Although such

a right enables the holders to benefit from both the security of a bond as well as a possible higher return through a

more risky underlying asset, such as stocks, it also results in a much more complex pricing problem, especially for

those of American‐style since they are allowed to be converted at any time.

Various models have been used to price CBs. A simple choice was the Black–Scholes model (Jensen et al., 1972).

J. E. Ingersoll (1977b) and Brennan and Schwartz (1977a) were the first to work on the problem under this model. In

J Futures Markets. 2022;42:1856–1911.wileyonlinelibrary.com/journal/fut1856

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© 2022 Wiley Periodicals LLC.

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We call the American‐style CB with only the conversion feature embedded as the vanilla CB.

their approach, the firm value was utilized as the underlying asset. However, firm values are not observable in real

markets and thus their approach has some drawbacks in practice as far as model calibration is concerned. Later on,

McConnell and Schwartz (1986) proposed to adopt the stock price instead of the firm value as the underlying variable

to price CBs.

Since then, research activities in the area of pricing CBs intensified. Among a large number of papers published

in the past 30years, numerical approaches, such as the finite element method (Barone‐Adesi et al., 2003), the

finite difference method (Tavella & Randall, 2000), and the finite volume method (Zvan et al., 2001), are

often adopted. However, two main drawbacks, that is, the accuracy problem and the time‐consuming feature that

exist in most of the numerical methods, prompted researchers to seek analytical solution approaches for their

simplicity and analytical elegancy, though they are quite often restricted to some relatively simple cases.

For example, a closed‐form solution for a simple CB, which can only be converted at maturity, was obtained by

Nyborg (1996), while Zhu (2006) presented an analytical solution in the form of a Taylor series expansion for

the simplest American–style CB without any other clauses being added, using the Homotopy Analysis Method

(Liao, 2003).

As one of the most popularly used financial derivatives in financial practice for firms to raise needed capital, CBs

today, stemming from the very basic original concept, have many variations with some quite involved terms, clauses,

and conditions. Among them, callable CBs and puttable CBs are two kinds of the most popular CBs (Ammann et al.,

2003). The former is a bond that allows the issuer to call (repurchase) the bond from the holder for a predetermined call

price, which is used to protect the issuer against the risk of the underlying running unreasonably much higher than

initially expected. When the underlying asset price increases beyond a preset critical value that is related to the

conversion ratio and the call price, the issuer can call back the bond at the call price. Therefore, the price of the callable

CB should be less than that of the vanilla counterpart, as a result of the holder's potential return being capped from

the above. On the other hand, puttability permits the holder to sell the bond back to the issuer at a predetermined put

price. Obviously, the put feature benefits the holder of the bond, and thus a puttable CB is traded at a higher price than

that of its vanilla counterpart.

Regarding solving the pricing problem of callable CBs, there are many reference materials. While Brennan and

Schwartz (1977a) explained in theory on how to price the callable CB, and provided solutions using the finite difference

method in their later article (Brennan & Schwartz, 1977b), Bernini (2001) used the binomial tree method to obtain the

solution. Yagi and Sawaki (2005) priced the callable CBs with the utilization of the game options defined by Kifer

(2000). On the other hand, there are also a few references on pricing puttable CBs in the literature. For instance,

Nyborg (1996) presented the boundary condition of puttable CBs, while Lvov et al. (2004) obtained the numerical

solution by using Monte Carlo simulations.

In this paper, two types of CBs mentioned above are combined together to form a new type of CBs, called

callable–puttable CBs, which should be considered on behalf of both the issuer and the holder. An integral equation

(IE) formulation is presented to price callable–puttable CBs under the Black–Scholes model with the method of

incomplete Fourier transform (Chiarella et al., 2004; He & Chen, 2021,2022) and Green's function (Duffy, 2015). One

may argue that it is more practical to adopt stochastic interest rate models (Brennan & Schwartz, 1980; Cox et al., 1985;

Vasicek, 1977) for pricing CBs, as CBs are usually designed for a long time period, during which the interest rate itself

may be subject to changes. However, we still assume a constant interest rate in this study, since the pricing exercise is

already very complicated even under this simple model, resulting from the tangled presence of callability, puttability, as

well as conversion, which have led to possible coexistence of two moving boundaries at the same time, depending on

the values of the call price, the put price, and the conversion ratio.

If a callable–puttable CB needs to be priced at a time sufficiently far away from the expiry, only the moving

boundary associated with the puttability needs to be dealt with. For this situation, the partial differential equation

(PDE) system governing the price of a callable–puttable CB is presented. When the pricing time is closer to expiry

beyond a critical value, it is then possible to have two distinct cases. While the two moving boundaries associated with

conversion and puttability coexist in one case, they may both disappear in another with callability remaining to be the

only issue that needs to be dealt with. The former case can be solved through one of the PDE systems presented in Zhu

et al. (2018), while the PDE system for the latter case can be built without the presence of any free boundaries.

Furthermore, there exists another critical value, beyond which the callable–puttable CB can be treated as the vanilla

counterpart, solving which requires the utilization of the PDE system presented in Zhu (2006). In summary, the pricing

problem for our issue should be designed with three different scenarios, and in each case, there are three or two PDE

systems governing the price of a callable–puttable CB.

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Of course, the proposed PDE approach can be extended to more complicated cases with other features such as

credit risk and discrete coupon payments being included as Zhang and Liao (2014) and Coonjobeharry et al. (2016) did

in the literature, although the revised IE may end up in different forms. As the current paper is already very long,

containing plenty of new information, such extensions will be left for future research.

This paper is organized as follows. In Section 2, the pricing problem is divided into three cases, and the PDE

systems governing the price of a callable–puttable CB are established for each case, and also the form of IE is derived.

In Section 3, we compared our results with the known benchmark. Numerical examples are presented in Section 4,

followed by some concluding remarks given in Section 5.

2|MODELS AND RESULTS

In this section, the PDE systems are established to price callable–puttable CBs under the Black–Scholes model, and the

IE formulations are obtained by solving these systems. Before we present the details in constructing these systems,

some preliminary knowledge of the callable–puttable CB should be presented first.

2.1 |Preliminary analysis

As mentioned above, the pricing problem should be divided into three scenarios according to the relationship between

the call price,

K

, and the put price,

M

. In particular, due to the fact that two additional rights are actually added into

the vanilla CB, there will be two critical moments in the callable–puttable CB, corresponding to the time instance

tK

when the callability disappears and the time instance

tM

when the puttability disappears. Throughout the paper, we

also assume the bond issuer or holder would exercise their “call,”“put,”or “conversion”rights rationally, when it is

“optimal”to exercise them, that is, when the corresponding optimal exercise price is reached (Yagi & Sawaki, 2010;

Zhu et al., 2018). In reality, however, this may not happen as these American‐style rights would entitle them to not

exercise even if a theoretical threshold has been reached. For example, the well‐known late calls are a phenomenon

associated with not exercised callability (J. Ingersoll, 1977a).

To establish the pricing PDE systems of the callable–puttable CB, it is important to figure out the similarities

among the three scenarios. First, to make both the puttability and callability effective at least for some time during

the lifetime of the bond, both features should be possible at the beginning of the bond, that is,

t

tt[0, min(,)]

KM

∈

.

This is because the maximum and minimum values of the bond are a decreasing and an increasing function of the

time, respectively, and they will reach neither the call price nor the put price if initially the maximal and minimal

bond value is lower and higher than the call price and the put price, respectively. Second, it is not an optimal

choice to call or put the bond when the time is sufficiently close to expiry, that is,

t

tt T[max( ,),]

KM

∈

,sinceduring

this time period the maximal bond value is smaller than the call price, and the minimal bond value is larger

than the put price. Therefore, the PDE systems corresponding to these two time intervals are the same for three

scenarios.

On the other hand, one should also be noted that these two critical moments, that is,

tK

and

tM

, can separate the

time zone into two or three parts, and this means that the difference between these three cases is only the middle part.

In the following, these three cases are discussed one by one.

In Case 1, when the call price is sufficiently small, the callability disappears later, that is,

t

t>

K

M

, since a small

value of the call price makes it harder for the maximal price of the bond to drop down below the call price

compared with the case that the minimal value of the bond hits the put price. Considering the property of a

callable CB, the moment when the callability disappears is also the time when the optimal conversion boundary

gets to the call price divided by the conversion ratio, K

n,where

n

is the conversion ratio. Thus, for this case, the first

part

2

actually consists of the time to expiry period when the optimal put boundary is equal to zero and the optimal

conversion boundary is less than a certain value, that is,

t

TtT[−,]

K

∈, implying that the PDE system for this part

is actually as same as that for the vanilla CB. The second part of this case represents the time to expiry period when

2

Under the classical treatment of the financial mathematics, we consider the pricing problem with the increase of the time to expiry directly.

Therefore, the first part in this paper means the time is sufficiently close to the expiry.

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