Journal of Applied Mathematics

Journal of Applied Mathematics / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 263451 | 5 pages | https://doi.org/10.1155/2010/263451

An Analytic Solution for a Vasicek Interest Rate Convertible Bond Model

Academic Editor: Peter Spreij
Received31 May 2009
Revised05 Nov 2009
Accepted06 Jan 2010
Published01 Feb 2010

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).

References

  1. R. Mallier and A. S. Deakin, β€œA Green's function for a convertible bond using the Vasicek model,” Journal of Applied Mathematics, vol. 2, no. 5, pp. 219–232, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. A. ErdΓ©lyi, W. Magnus, F. Oberhettinger et al., Higher Transcendental Functions. Vol. I, Robert E. Krieger, Melbourne, Fla, USA, 1981. View at: Zentralblatt MATH | MathSciNet
  3. A. ErdΓ©lyi, W. Magnus, F. Oberhettinger, F. G. Tricomi et al., Higher Transcendental Functions. Vol. II, Robert E. Krieger, Melbourne, Fla, USA, 1981. View at: Zentralblatt MATH | MathSciNet
  4. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, New York, NY, USA, 2nd edition, 1927.
  5. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge University Press, New York, NY, USA, 4th edition, 1962. View at: MathSciNet

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.


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