A Valuation Model for Perpetual Convertible Bonds with Markov Regime-switching Models

Abstract

This paper develops a valuation model for a perpetual convert-ible bond when the price dynamics of the underlying share are governed by continuous-time Markovian regime-switching models. We suppose that the ap-preciation rate and the volatility of the underlying share are modulated by a continuous-time, finite-state, observable Markov chain. The states of this chain are interpreted as the states of an economy. Here the valuation problem of the perpetual convertible bond can be viewed as that of valuing a perpetual stock loan, or a perpetual American option with time-dependent strike price. With the presence of the regime-switching effect, the market in the model is, in gen-eral, incomplete. To provide a convenient method to determine a price kernel for valuation, we employ the regime-switching Esscher transform introduced in Elliott, Chan and Siu (2005) [4]. We then adopt the differential equation [7] to solve the optimal stopping problem associated with the valuation of the perpetual convertible bond. Numerical ex-amples are presented to illustrate the practical implementation of the proposed model.

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International Journal of Pure and Applied Mathematics
————————————————————————–
Volume 53No. 42009, 583-600
A VALUATION MODEL FOR PERPETUAL CONVERTIBLE
BONDS WITH MARKOV REGIME-SWITCHING MODELS
Na Song1, Yue Jiao2, Wai-Ki Ching3 §, Tak-Kuen Siu4, Zhen-Yu Wu5
1,2,3Advanced Modeling and Applied Computing Laboratory
Department of Mathematics
The University of Hong Kong
Pokfulam Road, HONG KONG, P.R. CHINA
1e-mail: smynmath@163.com
2e-mail: jiaoyue@hkusua.hku.hk
3e-mails: wching@hkusua.hku.hk, wkc@maths.hku.hk
4Department of Mathematics and Statistics
Curtin University of Technology
Perth, W.A. 6845, AUSTRALIA
e-mail: ktksiu2005@gmail.com
5Department of Finance and Management Science
N. Murray Edwards School of Business
University of Saskatchewan
25 Campus Drive, Saskatoon, SK S7N 5A7, CANADA
e-mail: wu@edwards.usask.ca
Abstract:
ible bond when the price dynamics of the underlying share are governed by
continuous-time Markovian regime-switching models. We suppose that the ap-
preciation rate and the volatility of the underlying share are modulated by a
continuous-time, finite-state, observable Markov chain. The states of this chain
are interpreted as the states of an economy. Here the valuation problem of the
perpetual convertible bond can be viewed as that of valuing a perpetual stock
loan, or a perpetual American option with time-dependent strike price. With
the presence of the regime-switching effect, the market in the model is, in gen-
eral, incomplete. To provide a convenient method to determine a price kernel
for valuation, we employ the regime-switching Esscher transform introduced
in Elliott, Chan and Siu (2005) [4]. We then adopt the differential equation
This paper develops a valuation model for a perpetual convert-
Received:
§Correspondence author
May 20, 2009
c ? 2009 Academic Publications

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584N. Song, Y. Jiao, W.-K. Ching, T.-K. Siu, Z.-Y. Wu
approach in Guo and Zhang (2004) [7] to solve the optimal stopping problem
associated with the valuation of the perpetual convertible bond. Numerical ex-
amples are presented to illustrate the practical implementation of the proposed
model.
AMS Subject Classification:
Key Words:
ching, Markov chain, incomplete market, optimal stopping
93A30, 62L15, 91B28
perpetual convertible bonds, Esscher transform, regime-swit-
1. Introduction
A convertible bond is a kind of bond, which gives its holder an option to convert
the bond in shares of the stock issued by the same firm at a certain pre-specified
ratio. It is also called a convertible debenture and is a hybrid financial instru-
ment with both equity and bond features. The contractual structure of a con-
vertible bond may be more complex than a vanilla warrant in the sense that the
convertible bond pays a periodic coupon and involves a dual option at the same
time. On one hand, the bondholder has the option to convert the bond in shares
of common stock at his discretion. On the other hand, the issuing firm has the
right to call back the bond for redemption and the bondholder retains the right
to convert the bond or to redeem it (see, for example, Brennan and Schwartz
(1977) [1]). A common restriction on the call option is usually that the bond
may not be called for five years. Since the bondholders’ conversion strategy
depends on the firm’s call strategy, and vice versa, these two optimal strategies
must be solved simultaneously. Brennan and Schwartz (1977) [1] showed that
the optimal call strategy of an issuer of a convertible is to call the bond when
the value of the bond if it is called is identical to the value of the bond if it is not
called. So, the optimal call strategy and the optimal conversion strategy can be
analyzed from a perspective of a two-person zero-sum game. The value of the
bond may be related to an equilibrium of the game. A more simple approach to
deal with the pricing problem of a convertible bond is through the relationship
between the convertible bond and an American option. Indeed, under certain
assumptions about the contractual structure of a convertible bond, the pric-
ing problem of the convertible bond can be transformed into the pricing of an
American option with time-dependent strike price, which has its similarity with
the fair valuation of a stock loan, or a security load, considered in the paper
by Xia and Zhou (2007) [12]. However, as pointed out in Xia and Zhou (2007)
[12], the valuation of an American option with time-dependent strike price and

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A VALUATION MODEL FOR PERPETUAL CONVERTIBLE... 585
a finite maturity is rather complicated and intractable. To simplify the prob-
lem and make the problem more mathematically tractable, they assume the
maturity of the stock loan to be infinite and consider the valuation problem
of a perpetual American option with time-dependent strike price. By estab-
lishing the relationship between a perpetual convertible bond and a perpetual
American option with time-dependent strike price, the valuation of a convert-
ible bond can be further simplified to that of valuing a perpetual American
option with time-dependent strike price or a perpetual stock loan. Due to the
relatively long maturity of a convertible bond, the perpetual convertible bond
cannot only provides a mathematical idealization of the convertible bond, but
also provides a reasonably good proxy for a convertible bond.
In this paper, we develop a valuation model for a perpetual convertible
bond when the price dynamics of the underlying share are governed by Marko-
vian regime-switching models. We suppose that the appreciation rate and the
volatility of the underlying share are modulated by a continuous-time, finite-
state, observable Markov chain. We interpret the states of the chain as the
states of an economy. The valuation of the perpetual convertible bond can be
viewed as the valuation of a perpetual stock loan, or a perpetual American
option with time-dependent strike price. With the regime-switching effect in-
corporated, the market, in general, is incomplete. Hence, there is more than
one price kernel or stochastic discount factor. To provide a convenient way
to determine a price kernel, we therefore employ the regime-switching Esscher
transform introduced in Elliott, Chan and Siu (2005) [4]. Indeed, the use of
the Esscher transform to specify a price kernel can be justified by the Minimal
Entropy Martingale Measure (MEMM) in the context of the regime-switching
model (see Elliott, Chan and Siu (2005)[4]). Instead of adopting the probabilis-
tic approach to solve the valuation problem, which has been fully explored in
Xia and Zhou (2007) [12], we employ the differential equation approach in Guo
and Zhang (2004) [7] to solve the optimal stopping problem associated with the
valuation problem. We illustrate the practical implementation of the proposed
model via some numerical examples.
The next section establishes the relationship between a perpetual convert-
ible bond and a perpetual American option with time-dependent strike price.
Section 3 presents the asset price dynamics governed by the Markovian regime-
switching model. In Section 4, we illustrate how to use the regime-switching
Esscher transform to determine a price kernel in the incomplete market. Section
5 provides a solution to the valuation problem using the differential equation
approach in Guo and Zhang (2004) [7]. Numerical examples are then presented
in Section 6. The final section summarizes the paper.

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586 N. Song, Y. Jiao, W.-K. Ching, T.-K. Siu, Z.-Y. Wu
2. Perpetual American Option with Time-Dependent Strike Price
In this section, we establish the link between a perpetual convertible bond
and a perpetual American option with time-dependent strike price. Firstly, we
establish the link between a convertible bond and an American option with
time-dependent strike price with finite maturity. Then the perpetual case can
be obtained by letting the maturity tend to infinite.
Assume that a bond has a maturity T, with an annual interest rate c payable
at the end of each year. Thus the interest rate c is referred to as the coupon
rate in finance, and the market discount rate is r determined by the market.
The face value is $1000, and it will be paid back to bondholders at the maturity
T if they do not want to convert the bond in a share of the issuing firm. If
there is no possibility to be converted to a share, but remains a regular bond,
the current selling price of the bond can be easily calculated using MS Excel or
a financial calculator. When it is a convertible bond, however, the bondholders
can maximize their benefits by choosing to become shareholders.
In a special case that the coupon rate c = 0, it is clear that the current
selling price of zero-coupon bond should be B0= Fe−rT, where F is the face
value of the bond and is a constant.
As a convertible bond, however, the current selling price should be CB0=
Fe−rT+ C, where C > 0 is the price of the American call option embedded in
the convertible bond. This option gives the bondholders the right, but not the
obligation, to convert bonds into shares at any time point t before the maturity
T of the convertible bond. In other words, when the bondholders exercise this
option, they are converted to shareholders of the issuing firms. Therefore, the
strike price of this American call option, the number of shares converted to, is
time-dependent.
At a time point t, t ∈ (0,T], the payoff of this American call option with
time-dependent strike price B0ertis Y (t) = (S(t) − B0ert)+, assuming that
one bond can be converted to one share. Here (x)+ = max{x,0}. Y (t) is
called the intrinsic value of the American call option at time t. By letting T →
∞, the relationship between the perpetual convertible bond and the perpetual
American option becomes clear.

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A VALUATION MODEL FOR PERPETUAL CONVERTIBLE...587
3. The Model
We consider a continuous-time economy with two primitive securities, namely,
a risk-free money market account B and a risky share S. These securities can
be traded continuously over time. We also assume a perfect market where there
is no transaction cost, income tax and capital gain tax; there is no restriction
on short-selling and assets are divisible. We fix a complete probability space
(Ω,F,P), where P represents a real-world probability measure. Let T denote
the time index set [0,∞) of the economy. We assume that the states of the
economy are modeled by a continuous-time, finite-state, observable Markov
chain X := {Xt}t∈T on (Ω,F,P) with a finite state space X := (x1,x2...,xN).
Following Elliott, Aggoun and Moore (1994) [6], we can write X as its canonical
representation with loss of generality; that is, the state space of X is a finite
set of unit vectors {e1,e2,...,eN}, where ei = (0,...,1,...,0) ∈ RN. Let
Q(t) for the Q-matrix or the generator [qij(t)]i,j=1,2,...,N of the Markov chain
X. Then, under the canonical representation for X, Elliott (1993) [5] and
Elliott, Aggoun and Moore (1994) [6] derived the following semi-martingale
representation theorem for X:
?t
where {Mt}t∈Tis an RN-valued martingale with respect to the right-continuous,
P-completed, filtration generated by X.
Xt= X0+
0
QXsds + Mt,(3.1)
Let r be the constant instantaneous market interest rate of the money mar-
ket account, where r > 0. Then, the evolution of the bond price is governed
by: Bt= ertand t ∈ T .
We supposethat the appreciation rate {µt}t∈Tand the volatility rate {σt}t∈T
of S also depend on X and are given by: µt := µ(t,Xt) = ?µ,Xt?,
σ(t,Xt) = ?σ,Xt? where µ := (µ1,µ2,...,µN) and σ := (σ1,σ2...,σN) with
σi> 0 for each i = 1,2,...,N.
Let W := {Wt}t∈T denote a standard Brownian motion on (Ω,F,P). We
suppose that X and W are stochastically independent. The share price process
S is then governed by the following Markov-modulated geometric Brownian
motion (GBM):
σt :=
dSt= µtStdt + σtStdWt,S0= s . (3.2)