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

πœ•π‘‰=1πœ•πœ2𝜎2𝑆2πœ•2π‘‰πœ•π‘†2πœ•+πœŒπœŽπ‘π‘†2𝑉+1πœ•π‘†πœ•π‘Ÿ2𝑐2πœ•2π‘‰πœ•π‘Ÿ2+π‘Ÿπ‘†πœ•π‘‰πœ•π‘†+(π‘Žβˆ’π‘π‘Ÿ)πœ•π‘‰πœ•π‘Ÿβˆ’π‘Ÿπ‘‰(1) in the log stock variables π‘₯=log𝑆 and 𝑆̃π‘₯=log is

𝑉(𝑆,π‘Ÿ,𝜏)=βˆžβˆ’βˆžπ‘‰0𝑒̃π‘₯,Μƒπ‘ŸπΊ(π‘Ÿ,Μƒπ‘Ÿ,π‘₯βˆ’Μƒπ‘₯)π‘‘Μƒπ‘Ÿπ‘‘Μƒπ‘₯,(2) where 𝑉=𝑉0(𝑆,π‘Ÿ) at 𝜏=0 and the Green's function (GF) is

𝐺(π‘Ÿ,Μƒπ‘Ÿ,π‘₯βˆ’Μƒπ‘₯)=exp(𝐹)𝑁(𝑀,Ξ)𝑁(𝛼,Ξ¦).(3) The normal distribution with variance 𝑀 and argument Ξ is here denoted by

𝑁(𝑀,Ξ)=(2πœ‹π‘€)βˆ’1/2ξ‚Έβˆ’Ξžexp2ξ‚Ή(2𝑀),(4) and the coefficients are

𝑀=1βˆ’π‘’βˆ’2π‘πœξ€Έπ‘22𝑏,Ξ=Μƒπ‘Ÿβˆ’π‘Ÿπ‘’βˆ’π‘πœξ‚΅βˆ’π΅π‘Žβˆ’π΅π‘22ξ‚Ά,(5)𝐹=π΄βˆ’π΅π‘Ÿ,Ξ¦=Μƒπ‘₯βˆ’π‘₯βˆ’π·βˆ’Ξž(2𝜌𝜎/𝑐+𝐡)1+π‘’βˆ’π‘πœ,(6)𝛼=𝜏𝜎2ξ€·1βˆ’πœŒ2ξ€Έ+𝑐𝑏+𝜌𝜎2ξ‚€2πœβˆ’π‘ξ‚€tanhπ‘πœ2𝑣,(7)𝐹+𝐷+2ξ€·=0,𝐴=(π΅βˆ’πœ)2π‘Žπ‘βˆ’π‘2ξ€Έ2𝑏2βˆ’π‘2𝐡24𝑏,(8)𝐡=1βˆ’π‘’βˆ’π‘πœπ‘,𝑣=𝜏𝜎2+(πœβˆ’π΅)(2πœŒπœŽπ‘+𝑐)𝑐𝑏2βˆ’π‘2𝐡2.2𝑏(9) In the case of the convertible bond, the initial condition 𝑉0 in (2) is independent of Μƒπ‘Ÿ. Integrating (2) in Μƒπ‘Ÿ, we obtain the simpler Green's function

𝐺(π‘Ÿ,𝜏,π‘₯βˆ’Μƒπ‘₯)=exp(𝐹(π‘Ÿ,𝜏))𝑁(𝑣(𝜏),Μƒπ‘₯βˆ’π‘₯βˆ’π·(π‘Ÿ,𝜏)).(10) The parameters in the solution have the range of values: 𝜎>0, 𝑐>0, |𝜌|<1, while π‘Ž and 𝑏 are arbitrary since the solutions are analytic in π‘Ž and 𝑏.

To prove (3), we assume 𝑉 to be bounded as 𝑆→0 and 𝑆𝑐0𝑉, where π‘π‘œ is a positive constant, is bounded as π‘†β†’βˆž 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)∢=β„³[𝑉] and 𝑉(𝑧)∢=β„’[𝑉] (equations (2.6),(2.7) in [1]) are applied to obtain the ODE

𝑐22ξ‚Άπ‘‰π‘Ÿπ‘Ÿ+(π‘Žβˆ’πœŒπ‘πœŽπ‘βˆ’π‘π‘Ÿ)π‘‰π‘Ÿ+2ξ€Ίξ€·βˆ’1𝜎2ξ€Έξ€»π‘βˆ’π‘Ÿ(1+𝑝)βˆ’π‘§ξ€Ίπ‘‰π‘‰=βˆ’β„³0ξ€»(𝑆,π‘Ÿ).(11) The generalhomogeneous solution ([2, 3] Section V.I, page 249) of (11) is

π‘‰β„Žξ‚΅βˆ’=exp(1+𝑝)π‘Ÿπ‘ξ‚Άβ„±ξ‚΅βˆ’πœˆ2,12,𝑒22𝑧,(12)βˆ’πœˆ=𝑏+2𝐸,𝑒(π‘Ÿ)=2𝑏3𝑐2ξ€·π‘Ÿπ‘2βˆ’π‘Žπ‘+𝑐2ξ€·+π‘π‘π‘πœŽπœŒ+𝑐2ξ€·ξ€Έξ€Έ,(13)𝐸=(1+𝑝)2π‘Žπ‘βˆ’π‘2ξ€Έπ‘βˆ’π‘Ξ›βˆ’34,Ξ›=(𝑐+π‘πœŽπœŒ)2+(π‘πœŽ)2ξ€·1βˆ’πœŒ2ξ€Έ,(14) and β„± 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(𝑒) ([2, 3] Section V.II, page 117), with arbitrary constants 𝐢1 and 𝐢2 (πœˆβ‰ 0,1,…), is

π‘‰β„Žξ‚΅βˆ’=exp(1+𝑝)π‘Ÿπ‘ξ‚Ά2βˆ’πœˆ/2𝑒𝑒2ξ€Έ/4𝐢1𝐷𝜈(𝑒)+𝐢2π·πœˆξ€Έ(βˆ’π‘’).(15) Replacing β„³[𝑉0(𝑆,π‘Ÿ)] in (11) by the delta function 𝛿(π‘Ÿβˆ’Μƒπ‘Ÿ) (c.f., (20) for details), the GF for (11) has the form

𝐺1(π‘Ÿ,Μƒπ‘Ÿ)=2π‘βˆ’2β„Ž1(π‘Ÿ)β„Ž2(Μƒπ‘Ÿ)π‘Šβˆ’1ξ€Ίβ„Ž1(Μƒπ‘Ÿ),β„Ž2ξ€»(Μƒπ‘Ÿ),π‘Ÿ>Μƒπ‘Ÿ,(16) where β„Žπ‘— are appropriate homogeneous solutions in (15), π‘Š is the Wronskian, and 𝐺1 for π‘Ÿ<Μƒπ‘Ÿ is defined by interchanging π‘Ÿ and Μƒπ‘Ÿ in β„Žπ‘—, but not in π‘Š.

For the existence and the evaluation of the inverse Laplace transform (ILT) of 𝐺1, the asymptotic expansion, valid for large (βˆ’πœˆ) in the sector |arg(βˆ’πœˆ)|<πœ‹,

Ξ“(βˆ’πœˆ)𝐷𝜈(𝑣(π‘Ÿ))π·πœˆξ‚€βˆ’(βˆ’π‘€(Μƒπ‘Ÿ))∼𝜈2πœ‹ξ‚βˆ’1/2ξ€·expβˆ’(βˆ’πœˆ)1/2ξ€Έ(𝑣(π‘Ÿ)βˆ’π‘€(Μƒπ‘Ÿ))(17) is required where 𝑣(π‘Ÿ)=±𝑒(π‘Ÿ) and 𝑀(Μƒπ‘Ÿ)=±𝑒(Μƒπ‘Ÿ). 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 β„Žπ‘– in (16) must be chosen such that 𝐺1 has an ILT that exists for all π‘Ÿ and Μƒπ‘Ÿ. For the general case, we define β„Žπ‘– in (15) by replacing 𝐢𝑗 by 𝐢𝑖𝑗. There are four terms in (16), only one for which the ILT exists: 𝐢12=𝐢21=0, π‘£βˆ’π‘€=(2𝑏)1/2|π‘Ÿβˆ’Μƒπ‘Ÿ|/𝑐 in (17). Thus,

𝐺1=𝑔1(π‘Ÿ)𝑔2(Μƒπ‘Ÿ)π‘βˆ’1(π‘πœ‹)βˆ’1/2Ξ“(βˆ’πœˆ)𝐷𝜈(𝑒(π‘Ÿ))𝐷𝜈(βˆ’π‘’(Μƒπ‘Ÿ)),π‘Ÿ>Μƒπ‘Ÿ,(18) where 𝑔𝑗(π‘Ÿ)=exp[(βˆ’1)𝑗((1+𝑝)π‘Ÿ/π‘βˆ’π‘’2(π‘Ÿ)/4)]. For π‘Ÿ<Μƒπ‘Ÿ, 𝐺1 is defined by interchanging π‘Ÿ and Μƒπ‘Ÿ in 𝐷𝜈. However, to explain the results in [1], we compare (2.16) to (16,20) so that β„Ž1βˆπ’±2 and β„Ž2βˆπ’±1 in (2.13) (change sign on RHS of (2.14), (2.16)). Consequently, β„Ž1 and β„Ž2 are defined in (15) by taking (𝐢1=0,𝐢2=1) and (𝐢1=βˆ’1,𝐢2=1), respectively. The modified GF is πΊπ‘š1∢=βˆ’πΊβˆ—1+𝐺𝑠1 where πΊβˆ—1 and 𝐺𝑠1 are defined from 𝐺1 by changing 𝑒 to βˆ’π‘’ and 𝑒(π‘Ÿ) to βˆ’π‘’(π‘Ÿ), respectively.

As outlined in [1], the ILT 𝐺2∢=β„’βˆ’1(𝐺1) ((2.17), [1]) is equal to the contributions from the simple poles of Ξ“(βˆ’πœˆ) 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

𝐺2ξƒ¬βˆš=𝑁(πœ‚,Μƒπ‘Ÿβˆ’π‘Ÿ)exp2𝑏4𝑐(π‘Ÿβˆ’Μƒπ‘Ÿ)𝑠1βˆ’π‘ 2πœ†π‘πœ8+π‘πœ2βˆ’2π‘πΈπœβˆ’(1+𝑝)𝑏(π‘Ÿβˆ’Μƒπ‘Ÿ),(19) where π‘ π‘š=π‘’π‘š(π‘Ÿ)+π‘’π‘š(Μƒπ‘Ÿ), πœ‚=πœπ‘2sinh(π‘πœ)/(π‘πœ), πœ†=(2/(π‘πœ))tanh(π‘πœ/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, πΊπ‘š2∢=βˆ’πΊβˆ—2+𝐺𝑠2 where πΊβˆ—2 and 𝐺𝑠2 are formally defined by the contributions from the poles: πΊβˆ—2=𝐺2, 𝐺𝑠2=𝐺2exp(βˆ’π‘’(π‘Ÿ)𝑒(Μƒπ‘Ÿ)/sinh(π‘πœ)).

The last step is to evaluate the inverse Mellin transforms (IMT; (2.18), [1]) 𝐺3∢=β„³βˆ’1𝐺2 and, for the modified GF, πΊπ‘š3∢=βˆ’πΊ3+𝐺𝑠3, where 𝐺𝑠3∢=β„³βˆ’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, π›ΌβˆΆ=𝛼𝑠 is given by (7) where tanh is replaced by coth. Correcting the error in [1] (page 228, L.-4, (+) to (βˆ’)), then 2𝛼+=𝛼 and 2π›Όβˆ’=𝛼𝑠, where 𝛼± appear in (2.27) and (2.33). Assuming that (𝑐/𝑏+𝜌𝜎)β‰ 0, then there is a positive number πœπ‘œ such that π›Όβˆ’<0 for 0<𝜏<πœπ‘œ. Thus the IMT of 𝐺𝑠2 does not exist for 0<𝜏<πœπ‘œ, and 𝐺1 in (18) is the correct Green's function. For 𝐺3, we have 𝐺3=exp𝛾𝑁(πœ‚,Μƒπ‘Ÿβˆ’π‘Ÿ)𝑁(𝛼,π›½βˆ’log𝑆). The variables (𝑉,𝑉0,𝐺1) and (𝑉,𝑉0,𝐺3) are connected by

ξ€œπ‘‰=βˆžβˆ’βˆžβ„³ξ€Ίπ‘‰0𝐺(𝑆,Μƒπ‘Ÿ)1ξ€œπ‘‘Μƒπ‘Ÿ,𝑉=βˆžβˆ’βˆžβ„³βˆ’1ℳ𝑉0ℳ𝐺3ξ€»ξ€»π‘‘Μƒπ‘Ÿ.(20) Using the convolution theorem ((2.30), [1]), the solution is (2), where

ξ‚»πœŽπΊ(π‘Ÿ,Μƒπ‘Ÿ,π‘₯βˆ’Μƒπ‘₯)=exp(𝛾)𝑁(πœ‚,Μƒπ‘Ÿβˆ’π‘Ÿ)𝑁(𝛼,Μƒπ‘₯βˆ’π‘₯+𝛽),(21)𝛼=𝜏2ξ€·1βˆ’πœŒ2ξ€Έ+𝑐𝑏2(1βˆ’πœ†)πœ™2ξ‚Ό,πœ™=1+πœŒπœŽπ‘π‘,(22)2𝛽=2(π‘Ÿβˆ’Μƒπ‘Ÿ)πœŒπœŽπ‘ξ‚»ξ‚€π‘+πœπ‘ξ‚2(1βˆ’πœ†)𝑑1πœ™+𝜎2βˆ’(π‘Ÿ+Μƒπ‘Ÿ)πœ™+2πœŽπ‘ŽπœŒπ‘ξ‚Ό((23)2𝛾=π‘Ÿβˆ’Μƒπ‘Ÿ)((π‘Ÿ+Μƒπ‘Ÿ)π‘βˆ’2π‘Ž)𝑐2ξƒ―ξ‚€π‘Ž+πœπ‘βˆ’π‘ξ‚2+(𝑐/𝑏)2(1βˆ’πœ†)𝑑22ξ‚€βˆ’(π‘Ÿ+Μƒπ‘Ÿ)1βˆ’π‘Žπ‘π‘2ξ‚βˆ’ξ€·π‘Ÿ2+Μƒπ‘Ÿ2ξ€Έ(𝑏/𝑐)22ξƒ°,(24)π‘‘π‘š=π‘žπ‘š(π‘Ÿ)+π‘žπ‘š(Μƒπ‘Ÿ), and π‘ž(π‘Ÿ)=(π‘Ÿπ‘2βˆ’π‘Žπ‘+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: 𝐺→𝛿(π‘Ÿβˆ’Μƒπ‘Ÿ)𝛿(π‘₯βˆ’Μƒπ‘₯) and 𝑉(𝑆,π‘Ÿ,𝜏)→𝑉0(𝑆,π‘Ÿ) as πœβ†’0 in (2).