This paper provides the analytic solution to the partial differential equation for the value of a convertible bond. The equation assumes a Vasicek model for the interest rate and a geometric Brownian motion model for the stock price. The solution is obtained using integral transforms.
This work corrects errors in the original paper by Mallier and Deakin [1] on the Green's function for the Vasicek convertible bond equation. One error involves subtle points of the inverse Laplace transform. We show that the solution of
in the log stock variables and is
where at and the Green's function (GF) is
The normal distribution with variance and argument is here denoted by
and the coefficients are
In the case of the convertible bond, the initial condition in (2) is independent of . Integrating (2) in , we obtain the simpler Green's function
The parameters in the solution have the range of values: , , , while and are arbitrary since the solutions are analytic in and .
To prove (3), we assume to be bounded as and , where is a positive constant, is bounded as so that the Mellin transform of exists. Once the solution is determined, the initial condition may be extended to include the more general case where the integral (2) exists (e.g., ). In the derivation of the solution, the condition is assumed in (1).
To solve for in (1), the Mellin and Laplace transform and (equations in [1]) are applied to obtain the ODE
The generalhomogeneous solution ([2, 3] Section V.I, page 249) of (11) is
and is the general solution of the confluent hypergeometric equation ([2, 3] Section V.I). The general solution (12) in terms of the parabolic cylinder function ([2, 3] Section V.II, page 117), with arbitrary constants and (), is
Replacing in (11) by the delta function (c.f., (20) for details), the GF for (11) has the form
where are appropriate homogeneous solutions in (15), is the Wronskian, and for is defined by interchanging and in , but not in .
For the existence and the evaluation of the inverse Laplace transform (ILT) of , the asymptotic expansion, valid for large in the sector ,
is required where and . The expansion for the Gamma function is given in ([2, 3] Section V.I, page 47). The expansion with a restricted domain for the parabolic cylinder function appears in [2, 3] (Section V.I, page 249 ) and the general case is proved by applying the Method of Steepest Descent to the integral representation ([4, 5], page 349). The solutions in (16) must be chosen such that has an ILT that exists for all and . For the general case, we define in (15) by replacing by . There are four terms in (16), only one for which the ILT exists: , in (17). Thus,
where . For , is defined by interchanging and in . However, to explain the results in [1], we compare to so that and in (change sign on RHS of , ). Consequently, and are defined in (15) by taking and , respectively. The modified GF is where and are defined from by changing to and to , respectively.
As outlined in [1], the ILT (, [1]) is equal to the contributions from the simple poles of at (). is equal to a sum involving Hermite polynomials ([2, 3] Section V.II, page 194 ) so that
where , , . The semicircle's contribution to goes to zero as the radius goes to infinity follows from the approximation of in (18) via (17). For the modified GF, where and are formally defined by the contributions from the poles: , .
The last step is to evaluate the inverse Mellin transforms (IMT; [1]) and, for the modified GF, where . To do this, the argument of the exponential in and is expressed in the form , and formula in [1] is applied. For , is given by (7). For , is given by (7) where is replaced by . Correcting the error in [1] (page 228, L.-4, (+) to ), then and where appear in and Assuming that , then there is a positive number such that for . Thus the IMT of does not exist for , and in (18) is the correct Green's function. For , we have . The variables and are connected by
Using the convolution theorem (, [1]), the solution is (2), where
and . Extensive algebraic manipulations are required to express in (21) in the final form (3). The Green's function in (3) has the expected property: and as in (2).