Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2010, Article ID 263451, 5 pages
doi:10.1155/2010/263451
Research Article
An Analytic Solution for a Vasicek Interest Rate
Convertible Bond Model
A. S. Deakin1 and Matt Davison1, 2
1
2
Department of Applied Mathematics, University of Western Ontario, London, ON, Canada N6A 5B7
Department of Statistical & Actuarial Sciences, University of Western Ontario, London,
ON, Canada N6A 5B7
Correspondence should be addressed to A. S. Deakin, asdeakin@uwo.ca
Received 31 May 2009; Revised 5 November 2009; Accepted 6 January 2010
Academic Editor: Peter Spreij
Copyright q 2010 A. S. Deakin and M. 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.
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
∂2 V
1
1 ∂2 V
∂2 V
∂V
∂V
σ 2 S2 2 ρσcS
c2 2 rS
a − br
− rV
∂τ
2
∂S∂r 2 ∂r
∂S
∂r
∂S
1
in the log stock variables x log S and x log S is
V S, r, τ ∞
−∞
V0 ex , r Gr, r, x − x
dr dx,
2
where V V0 S, r at τ 0 and the Green’s function GF is
Gr, r, x − x
expFNw, ΞNα, Φ.
3
2
Journal of Applied Mathematics
The normal distribution with variance w and argument Ξ is here denoted by
−1/2
Nw, Ξ 2πw
Ξ2
exp −
2w
,
4
and the coefficients are
Bc2
Ξ r − re − B a −
,
2
Ξ 2ρσ/c B
F A − Br,
Φ x − x − D −
,
1 e−bτ
2 c
bτ
2
2
2
ρσ
,
τ − tanh
α τσ 1 − ρ b
b
2
B − τ 2ab − c2
v
c2 B 2
F D 0,
A
,
−
2
2
4b
2b
τ − B 2ρσb c c c2 B2
1 − e−bτ
2
B
,
v τσ .
−
b
2b
b2
1 − e−2bτ c2
w
,
2b
−bτ
5
6
7
8
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
Gr, τ, x − x
expFr, τNvτ, x − x − Dr, 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 S → 0 and Sc0 V , 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 maxS, 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 : MV and V z :
LV equations 2.6, 2.7 in 1 are applied to obtain the ODE
c2
2
V rr a − ρcσp − br V r 2−1 σ 2 p − r 1 p − z V −MV0 S, r.
11
Journal of Applied Mathematics
3
The general homogeneous solution 2, 3 Section V.I, page 249 of 11 is
1p r
ν 1 u2
F − , ,
,
Vh exp −
b
2 2 2
12
z
2 2
2
2
−ν 2E, ur rb
,
−
ab
c
p
cbσρ
c
b
b3 c2
1 p 2ab − c2 − pΛ b−3
2
E
, Λ c bσρ bσ2 1 − ρ2 ,
4
13
14
and F 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
0, 1, . . ., is
V.II, page 117, with arbitrary constants C1 and C2 ν /
1p r
2
2−ν/2 eu /4 C1 Dν u C2 Dν −u.
Vh exp −
b
15
Replacing MV0 S, r in 11 by the delta function δr − r c.f., 20 for details, the GF for
11 has the form
G1 r, r 2c−2 h1 rh2 r W −1 h1 r , h2 r ,
r > r,
16
where hj are appropriate homogeneous solutions in 15, W is the Wronskian, and G1 for
r < r is defined by interchanging r and r in hj , 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−νDν vrDν −wr ∼
ν2 −1/2
exp −−ν1/2 vr − w
−
r π
17
is required where vr ±ur and wr ±ur . 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 hi in 16 must be chosen such that G1 has an ILT that exists for all r and r. For
the general case, we define hi in 15 by replacing Cj by Cij . There are four terms in 16, only
one for which the ILT exists: C12 C21 0, v − w 2b1/2 |r − r|/c in 17. Thus,
G1 g1 rg2 r c−1 bπ−1/2 Γ−νDν urDν −ur ,
r > r,
18
where gj r exp−1j 1pr/b−u2 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 h1 ∝ V2
and h2 ∝ V1 in 2.13 change sign on RHS of 2.14, 2.16. Consequently, h1 and h2 are
defined in 15 by taking C1 0, C2 1 and C1 −1, C2 1, respectively. The modified
4
Journal of Applied Mathematics
s
s
∗
∗
GF is Gm
1 : −G1 G1 where G1 and G1 are defined from G1 by changing u to −u and ur to
−ur, respectively.
As outlined in 1, the ILT G2 : L−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, r − r exp
1p
2b
λbτ bτ
− 2bEτ −
r − rs1 − s2
r − r ,
4c
8
2
b
√
19
where sm um r um r , η τc2 sinhbτ/bτ, λ 2/bτtanhbτ/2. The semicircle’s
contribution to G2 goes to zero as the radius goes to infinity follows from the approximation
s
s
∗
∗
of G1 in 18 via 17. For the modified GF, Gm
2 : −G2 G2 where G2 and G2 are formally
s
∗
defined by the contributions from the poles: G2 G2 , G2 G2 exp−urur / sinhbτ.
The last step is to evaluate the inverse Mellin transforms IMT; 2.18, 1 G3 : M−1 G2
s
s
−1 s
and, for the modified GF, Gm
3 : −G3 G3 , where G3 : M G2 . To do this, the argument of the
s
2
exponential in G2 and G2 is expressed in the form αp /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
s
τo such that α− < 0 for 0 < τ < τo . Thus the IMT of G2 does not exist for 0 < τ < τo , and G1 in
18 is the correct Green’s function. For G3 , we have G3 exp γNη, r − rNα, β − log S. The
variables V , V0 , G1 and V, V0 , G3 are connected by
V ∞
−∞
MV0 S, rG1 dr ,
V ∞
−∞
M−1 MV0 MG3 dr .
20
Using the convolution theorem 2.30, 1, the solution is 2, where
Gr, r, x − x
exp γ N η, r − r N α, x − x β ,
c 2
ρσb
α τ σ 2 1 − ρ2 ,
1 − λφ2 , φ 1 b
c
2
2r − rρσ
2σaρ
c
τ
2β 1 − λd1 φ σ 2 − r rφ c
b
c
2γ r − rr rb − 2a
c2
2
2
r r2 b/c2
ab
a
c/b2 1 − λd2
− r r 1 − 2 −
,
τ b−
c
2
2
c
21
22
23
24
dm qm r qm r , and qr rb2 − ab 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 → δr − rδx − x
and V S, r, τ → V0 S, r as τ → 0 in 2.
Journal of Applied Mathematics
5
References
1 R. Mallier and A. S. Deakin, “A Green’s function for a convertible bond using the Vasicek model,”
Journal of Applied Mathematics, vol. 2, no. 5, pp. 219–232, 2002.
2 A. Erdélyi, W. Magnus, F. Oberhettinger, et al., Higher Transcendental Functions. Vol. I, Robert E. Krieger,
Melbourne, Fla, USA, 1981.
3 A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, et al., Higher Transcendental Functions. Vol. II,
Robert E. Krieger, Melbourne, Fla, USA, 1981.
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.