Frequently Asked Questions In Quantitative Finance

Chapter 5: Models and Equations 289

In all of the spot rate models below we have

dr=u(r,t)dt+w(r,t)dX

as the real process for the spot interest rate. The risk-
neutral process which governs the value of fixed-income
instruments is

dr=(u−λw)dt+wdX

whereλis the market price of interest rate risk. In each
case the stochastic differential equation we describe is
for the risk-neutral spot rate process, not the real.

The differential equation governing the value of non-
path-dependent contracts is

∂V

∂t

+^12 w^2

∂^2 V

∂r^2

+(u−λw)

∂V

∂r

−rV= 0.

The value of fixed-income derivatives can also be inter-
preted as

E

Q

t

[

Present value of cashflows

]

,

where the expectation is with respect to the risk-neutral
process

Vasicek In this model the risk-neutral process is

dr=(a−br)dt+cdX,

witha,bandcbeing constant. It is possible forrto go
negative in this model.

There is a solution for bonds of the form exp(A(t;T)−
B(t;T)r).

Cox, Ingersoll and Ross In this model the risk-neutral pro-
cess is

dr=(a−br)dt+cr^1 /^2 dX,