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Journal of Applied Mathematics
Volume 2010, Article ID 263451, 5 pages
http://dx.doi.org/10.1155/2010/263451
Research Article

An Analytic Solution for a Vasicek Interest Rate Convertible Bond Model

1Department of Applied Mathematics, University of Western Ontario, London, ON, Canada N6A 5B7
2Department of Statistical & Actuarial Sciences, University of Western Ontario, London, ON, Canada N6A 5B7

Received 31 May 2009; Revised 5 November 2009; Accepted 6 January 2010

Academic Editor: Peter Spreij

Copyright © 2010 A. S. Deakin and Matt Davison. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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

𝜕𝑉=1𝜕𝜏2𝜎2𝑆2𝜕2𝑉𝜕𝑆2𝜕+𝜌𝜎𝑐𝑆2𝑉+1𝜕𝑆𝜕𝑟2𝑐2𝜕2𝑉𝜕𝑟2+𝑟𝑆𝜕𝑉𝜕𝑆+(𝑎𝑏𝑟)𝜕𝑉𝜕𝑟𝑟𝑉(1) in the log stock variables 𝑥=log𝑆 and 𝑆̃𝑥=log is

𝑉(𝑆,𝑟,𝜏)=𝑉0𝑒̃𝑥,̃𝑟𝐺(𝑟,̃𝑟,𝑥̃𝑥)𝑑̃𝑟𝑑̃𝑥,(2) where 𝑉=𝑉0(𝑆,𝑟) at 𝜏=0 and the Green's function (GF) is

𝐺(𝑟,̃𝑟,𝑥̃𝑥)=exp(𝐹)𝑁(𝑤,Ξ)𝑁(𝛼,Φ).(3) The normal distribution with variance 𝑤 and argument Ξ is here denoted by

𝑁(𝑤,Ξ)=(2𝜋𝑤)1/2Ξexp2(2𝑤),(4) and the coefficients are

𝑤=1𝑒2𝑏𝜏𝑐22𝑏,Ξ=̃𝑟𝑟𝑒𝑏𝜏𝐵𝑎𝐵𝑐22,(5)𝐹=𝐴𝐵𝑟,Φ=̃𝑥𝑥𝐷Ξ(2𝜌𝜎/𝑐+𝐵)1+𝑒𝑏𝜏,(6)𝛼=𝜏𝜎21𝜌2+𝑐𝑏+𝜌𝜎22𝜏𝑏tanh𝑏𝜏2𝑣,(7)𝐹+𝐷+2=0,𝐴=(𝐵𝜏)2𝑎𝑏𝑐22𝑏2𝑐2𝐵24𝑏,(8)𝐵=1𝑒𝑏𝜏𝑏,𝑣=𝜏𝜎2+(𝜏𝐵)(2𝜌𝜎𝑏+𝑐)𝑐𝑏2𝑐2𝐵2.2𝑏(9) In the case of the convertible bond, the initial condition 𝑉0 in (2) is independent of ̃𝑟. Integrating (2) in ̃𝑟, we obtain the simpler Green's function

𝐺(𝑟,𝜏,𝑥̃𝑥)=exp(𝐹(𝑟,𝜏))𝑁(𝑣(𝜏),̃𝑥𝑥𝐷(𝑟,𝜏)).(10) The parameters in the solution have the range of values: 𝜎>0, 𝑐>0, |𝜌|<1, while 𝑎 and 𝑏 are arbitrary since the solutions are analytic in 𝑎 and 𝑏.

To prove (3), we assume 𝑉 to be bounded as 𝑆0 and 𝑆𝑐0𝑉, where 𝑐𝑜 is a positive constant, is bounded as 𝑆 so that the Mellin transform of 𝑉 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., 𝑉0=max(𝑆,1)). In the derivation of the solution, the condition 𝑏>0 is assumed in (1).

To solve for 𝑉 in (1), the Mellin and Laplace transform 𝑉(𝑝)=[𝑉] and 𝑉(𝑧)=[𝑉] (equations (2.6),(2.7) in [1]) are applied to obtain the ODE

𝑐22𝑉𝑟𝑟+(𝑎𝜌𝑐𝜎𝑝𝑏𝑟)𝑉𝑟+21𝜎2𝑝𝑟(1+𝑝)𝑧𝑉𝑉=0(𝑆,𝑟).(11) The generalhomogeneous solution ([2, 3] Section V.I, page 249) of (11) is

𝑉=exp(1+𝑝)𝑟𝑏𝜈2,12,𝑢22𝑧,(12)𝜈=𝑏+2𝐸,𝑢(𝑟)=2𝑏3𝑐2𝑟𝑏2𝑎𝑏+𝑐2+𝑝𝑐𝑏𝜎𝜌+𝑐2,(13)𝐸=(1+𝑝)2𝑎𝑏𝑐2𝑏𝑝Λ34,Λ=(𝑐+𝑏𝜎𝜌)2+(𝑏𝜎)21𝜌2,(14) 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 𝐷𝜈(𝑢) ([2, 3] Section V.II, page 117), with arbitrary constants 𝐶1 and 𝐶2 (𝜈0,1,), is

𝑉=exp(1+𝑝)𝑟𝑏2𝜈/2𝑒𝑢2/4𝐶1𝐷𝜈(𝑢)+𝐶2𝐷𝜈(𝑢).(15) Replacing [𝑉0(𝑆,𝑟)] in (11) by the delta function 𝛿(𝑟̃𝑟) (c.f., (20) for details), the GF for (11) has the form

𝐺1(𝑟,̃𝑟)=2𝑐21(𝑟)2(̃𝑟)𝑊11(̃𝑟),2(̃𝑟),𝑟>̃𝑟,(16) where 𝑗 are appropriate homogeneous solutions in (15), 𝑊 is the Wronskian, and 𝐺1 for 𝑟<̃𝑟 is defined by interchanging 𝑟 and ̃𝑟 in 𝑗, but not in 𝑊.

For the existence and the evaluation of the inverse Laplace transform (ILT) of 𝐺1, the asymptotic expansion, valid for large (𝜈) in the sector |arg(𝜈)|<𝜋,

Γ(𝜈)𝐷𝜈(𝑣(𝑟))𝐷𝜈(𝑤(̃𝑟))𝜈2𝜋1/2exp(𝜈)1/2(𝑣(𝑟)𝑤(̃𝑟))(17) is required where 𝑣(𝑟)=±𝑢(𝑟) and 𝑤(̃𝑟)=±𝑢(̃𝑟). 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 (8)) and the general case is proved by applying the Method of Steepest Descent to the integral representation ([4, 5], page 349). The solutions 𝑖 in (16) must be chosen such that 𝐺1 has an ILT that exists for all 𝑟 and ̃𝑟. For the general case, we define 𝑖 in (15) by replacing 𝐶𝑗 by 𝐶𝑖𝑗. There are four terms in (16), only one for which the ILT exists: 𝐶12=𝐶21=0, 𝑣𝑤=(2𝑏)1/2|𝑟̃𝑟|/𝑐 in (17). Thus,

𝐺1=𝑔1(𝑟)𝑔2(̃𝑟)𝑐1(𝑏𝜋)1/2Γ(𝜈)𝐷𝜈(𝑢(𝑟))𝐷𝜈(𝑢(̃𝑟)),𝑟>̃𝑟,(18) where 𝑔𝑗(𝑟)=exp[(1)𝑗((1+𝑝)𝑟/𝑏𝑢2(𝑟)/4)]. For 𝑟<̃𝑟, 𝐺1 is defined by interchanging 𝑟 and ̃𝑟 in 𝐷𝜈. However, to explain the results in [1], we compare (2.16) to (16,20) so that 1𝒱2 and 2𝒱1 in (2.13) (change sign on RHS of (2.14), (2.16)). Consequently, 1 and 2 are defined in (15) by taking (𝐶1=0,𝐶2=1) and (𝐶1=1,𝐶2=1), respectively. The modified GF is 𝐺𝑚1=𝐺1+𝐺𝑠1 where 𝐺1 and 𝐺𝑠1 are defined from 𝐺1 by changing 𝑢 to 𝑢 and 𝑢(𝑟) to 𝑢(𝑟), respectively.

As outlined in [1], the ILT 𝐺2=1(𝐺1) ((2.17), [1]) is equal to the contributions from the simple poles of Γ(𝜈) at 𝜈=𝑛 (𝑛=0,1). 𝐺2 is equal to a sum involving Hermite polynomials ([2, 3] Section V.II, page 194 (22)) so that

𝐺2=𝑁(𝜂,̃𝑟𝑟)exp2𝑏4𝑐(𝑟̃𝑟)𝑠1𝑠2𝜆𝑏𝜏8+𝑏𝜏22𝑏𝐸𝜏(1+𝑝)𝑏(𝑟̃𝑟),(19) where 𝑠𝑚=𝑢𝑚(𝑟)+𝑢𝑚(̃𝑟), 𝜂=𝜏𝑐2sinh(𝑏𝜏)/(𝑏𝜏), 𝜆=(2/(𝑏𝜏))tanh(𝑏𝜏/2). The semicircle's contribution to 𝐺2 goes to zero as the radius goes to infinity follows from the approximation of 𝐺1 in (18) via (17). For the modified GF, 𝐺𝑚2=𝐺2+𝐺𝑠2 where 𝐺2 and 𝐺𝑠2 are formally defined by the contributions from the poles: 𝐺2=𝐺2, 𝐺𝑠2=𝐺2exp(𝑢(𝑟)𝑢(̃𝑟)/sinh(𝑏𝜏)).

The last step is to evaluate the inverse Mellin transforms (IMT; (2.18), [1]) 𝐺3=1𝐺2 and, for the modified GF, 𝐺𝑚3=𝐺3+𝐺𝑠3, where 𝐺𝑠3=1𝐺𝑠2. To do this, the argument of the exponential in 𝐺2 and 𝐺𝑠2 is expressed in the form 𝛼𝑝2/2+𝛽𝑝+𝛾, and formula (2.29) in [1] is applied. For 𝐺2, 𝛼 is given by (7). For 𝐺𝑠2, 𝛼=𝛼𝑠 is given by (7) where tanh is replaced by coth. Correcting the error in [1] (page 228, L.-4, (+) to ()), then 2𝛼+=𝛼 and 2𝛼=𝛼𝑠, where 𝛼± appear in (2.27) and (2.33). Assuming that (𝑐/𝑏+𝜌𝜎)0, then there is a positive number 𝜏𝑜 such that 𝛼<0 for 0<𝜏<𝜏𝑜. Thus the IMT of 𝐺𝑠2 does not exist for 0<𝜏<𝜏𝑜, and 𝐺1 in (18) is the correct Green's function. For 𝐺3, we have 𝐺3=exp𝛾𝑁(𝜂,̃𝑟𝑟)𝑁(𝛼,𝛽log𝑆). The variables (𝑉,𝑉0,𝐺1) and (𝑉,𝑉0,𝐺3) are connected by

𝑉=𝑉0𝐺(𝑆,̃𝑟)1𝑑̃𝑟,𝑉=1𝑉0𝐺3𝑑̃𝑟.(20) Using the convolution theorem ((2.30), [1]), the solution is (2), where

𝜎𝐺(𝑟,̃𝑟,𝑥̃𝑥)=exp(𝛾)𝑁(𝜂,̃𝑟𝑟)𝑁(𝛼,̃𝑥𝑥+𝛽),(21)𝛼=𝜏21𝜌2+𝑐𝑏2(1𝜆)𝜙2,𝜙=1+𝜌𝜎𝑏𝑐,(22)2𝛽=2(𝑟̃𝑟)𝜌𝜎𝑐𝑐+𝜏𝑏2(1𝜆)𝑑1𝜙+𝜎2(𝑟+̃𝑟)𝜙+2𝜎𝑎𝜌𝑐((23)2𝛾=𝑟̃𝑟)((𝑟+̃𝑟)𝑏2𝑎)𝑐2𝑎+𝜏𝑏𝑐2+(𝑐/𝑏)2(1𝜆)𝑑22(𝑟+̃𝑟)1𝑎𝑏𝑐2𝑟2+̃𝑟2(𝑏/𝑐)22,(24)𝑑𝑚=𝑞𝑚(𝑟)+𝑞𝑚(̃𝑟), and 𝑞(𝑟)=(𝑟𝑏2𝑎𝑏+𝑐2)/𝑐2. Extensive algebraic manipulations are required to express 𝐺 in (21) in the final form (3). The Green's function in (3) has the expected property: 𝐺𝛿(𝑟̃𝑟)𝛿(𝑥̃𝑥) and 𝑉(𝑆,𝑟,𝜏)𝑉0(𝑆,𝑟) as 𝜏0 in (2).

References

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