A Green′s function for a convertible bond using the Vasicek model
Hindawi Publishing Corporation
Journal of Applied Mathematics
R. MALLIER
A. S. DEAKIN
We consider a convertible security where the underlying stock price obeys a lognormal random walk and the risk-free rate is given by the Vasicek model. Using a Laplace transform in time and a Mellin transform in the stock price, we derive a Green's function solution for the value of the convertible bond.
1. Introduction
A convertible bond is defined to be (e.g., Jorion [
10
]) a bond issued by
a corporation that can be converted into the equity of that corporation
at certain times using a predetermined exchange ratio. This entails the
creation of new shares issued by the corporation if and when conversion
occurs, and the existing shares are diluted by the creation of the new
shares. The option to convert is solely at the discretion of the bond holder
who will do so only if it is beneficial.
Typically, firms issue convertible bonds because they offer a lower
interest cost and less restrictive covenants than a nonconvertible bond, but
the drawback is that the issuer will be confronted with capital structure
uncertainty. Convertible bonds are often subordinated debentures, and
because of this, the bond rating agencies have usually rated convertibles
one class below that of a straight debenture (Dialynas et al. [
7
]), and
typically issuing convertibles will not affect a company?s rating. In return
for a reduced yield, an investor will receive a security with considerable
upside potential along with downside protection.
Conceptually, the behavior of a convertible bond can be segmented
into four regions (e.g., Dialynas et al. [
7
]). In the late 1990s, most new
issue convertibles were balanced converts, with around a 25%
conversion premium, where the conversion premium is the excess an investor
would pay to acquire the stock by buying the convertible and
immediately converting rather than buying the stock itself. Typically, the price of
a balanced convert responds materially to changes in both the
underlying stock price and the spot interest rate, with a correlation of about 55%
to 80% with changes in the stock price. A second category is equity
substitute converts, where the conversion premium is less than 15%, usually
because of rises in the price of the underlying. Typically, the price of an
equity substitute responds much more to changes in the stock price than
to interest rate changes. A third category is busted converts, where the
underlying stock price has declined so significantly that the conversion
option is worth very little and the value of the convertible approaches
that of an otherwise identical nonconvertible bond. A fourth category is
distressed converts, which are busted converts where the stock price has
fallen so much that there is a significant chance of bankruptcy.
As of 2000, the market value of convertible securities outstanding
globally was in excess of $400 billion US, with approximately $200
billion in the USA alone (Dialynas et al. [
7
]). Given the size of the market
for these securities, the pricing of convertible bonds is obviously an
important problem. Traditionally, convertibles were valued based on the
premise that buying a convertible is equivalent to buying the stock at
a premium and recouping that premium from the coupons on the
convertible, and the payback period is the time taken to recover the
premium. More recently, however, contingent claims analysis has been used
to value convertibles, which is the approach taken in the present study,
and this approach dates back to the work of Ingersoll [
9
] and Brennan
and Schwartz [
3
]. Brennan and Schwartz originally used the firm value
as the underlying variable, and later (Brennan and Schwartz [
4
])
extended their analysis to include stochastic interest rates and also to
include the value of the stock rather than that of the firm (McConnell and
Schwartz [
11
]). Almost all of this earlier work led to a numerical rather
than an analytical solution of the underlying equations for the value of a
convertible bond, typically using binomial trees; by contrast, the present
work in entirely analytical.
In our analysis, we consider a convertible bond, whose value depends
on both the price of the underlying stock, which is assumed to obey a
lognormal random walk with constant volatility, as in the Black-Scholes
option pricing model, and on the interest rate, which is assumed to
follow a random walk given by the Vasicek [
13
] model. We will say more
about this interest rate model and its advantages and disadvantages in
Section 3. By constructing a risk-free portfolio, it is possible to go from
the stochastic differential equations for the stock price and the spot rate
to a PDE for the value of the convertible (e.g., Wilmott [
14
]), and that
PDE is the starting point for our analysis in the next section. In our
analysis, we consider this PDE, and by using a double integral transform,
specifically a Laplace transform in time an (...truncated)