Advances in Risk Management

MANUEL MORENO 71

The solution of this equation, subject to the terminal condition given
by the payment to be received by the bondholder at maturity, allows us to
price discount bonds and, thereafter, infer the term structure of interest rates.
To solve this valuation equation, we must make some assumptions about
the market prices of risk and the dynamics of the state variables. Since a
constant market price of risk implies strong restrictions on the preferences
of investors, we establish the following:

Assumption 1 The market price of each state variable risk is linear in

this variable, that is,

λ 1 (.)=a+bs,λ 2 (.)=c+dL (4.3)

Assumption 2 Each of the state variables follows a diffusion process,

ds=k 1 (μ 1 −s)dt+σ 1 dw 1

(4.4)

dL=k 1 (μ 2 −L)dt+σ 2 dw 2

Under Assumptions 1 and 2, we can rewrite equation (4.2) as:

1

2

[

σ 12 Pss+σ 22 PLL

]

+q 1 [μˆ 1 −s]Ps+[μˆ 2 −L]PL

+Pt−(L+s)P= 0 (4.5)

subject to the terminal condition:

P(s,L,T,T)=1, ∀s,L (4.6)

where

q 1 =k 1 +bσ 1 ,μˆ 1 =

k 1 μ 1 −aσ 1

q 1

(4.7)

q 2 =k 2 +dσ 2 ,μˆ 2 =

k 2 μ 2 −cσ 2

q 2

Solving the PDE (4.5), we obtain the following proposition:

Proposition 1 The value at timetof a discount bond^6 that pays $1 at

timeT,P(s,L,t,T)≡P(s,L,τ), is given by:

P(s,L,τ)=A 1 (τ)e−B(τ)s−C(τ)L (4.8)

whereτ=T−tand

A(τ)=A 1 (τ)A 2 (τ)