Interest Rate Models - Computational Methods for Quantitative Finance - page 85

Chapter 7
Interest Rate Models

We consider options on interest rates and present commonly used short rate models
to model the time-evolution of the interest rate. Many interest rate derivatives in
fixed income markets can then be priced numerically using the computational tech-
niques described in the previous chapter, i.e. they can be interpreted as compound
options on bonds.

7.1 Pricing Equation

We model the short rate r t as a continuous Markovian process satisfying
d r t = b(t, r t ) d t + σ(t,r t ) d W t , 0 = r, (7.1)
where b and σ satisfy the assumptions of Theorem 1.2.6 and W is a Brownian
motion on (Ω, ) . We are interested in the computation of prices at time t

of a zero coupon bond B(t,r) . This instrument pays out one unit of currency at
maturity T ,i.e. B(T,r)

F , P , F

= 1. Therefore, its price satisfies the following relation

B(t,r) = E e − t r s d s

| r t = r ,
where the expectation is taken under the pricing measure

. We refer to [109,
Chap. 6] for details on the choice of measure. Similar as in Chap. 4, B(t,r) can
be shown to satisfy a deterministic partial differential equation. Here we consider a
general payoff g(r) . For a zero coupon bond, g(r)

Q

= 1.

C 1 , 2 (J C 0 (J C 1 ( Theorem 7.1.1 Let V ∈ × ( 0 , ∞ )) ∩ ×[ 0 , ∞ )) ∩ [ 0 ,T) ×[ 0 , ∞ ))

with bounded derivatives in r be a solution of

∂ t V + A V − rV = 0 in J × R + ,V T, ) = g(r) in R + , (7.2)

1

2 σ(t,r) 2 ∂ rr , where b(t, r) , σ(t,r) and g are suffi-

with A given as A = b(t, r)∂ r +

ciently smooth and σ(t,r) = 0 holds if and only if r = 0. Then , V(t,x)can also be

represented as

= E e − t r(s) d s g(r T ) r .

V(t,r) | r t =