Frequently Asked Questions In Quantitative Finance

Chapter 5: Models and Equations 293

rate,F, and its volatility,α, both of which are stochastic:

dF=αFβdX 1 and dα=ναdX 2.

There are three parameters,β,νand a correlationρ.
The model comes into its own because it is designed
for the special case where the volatilityαand volatility
of volatility,ν, are both small. In this case there are
relatively simple closed-form approximations (asymp-
totic solutions). The model is therefore most relevant
for markets such as fixed income, rather than equity.
Equity markets typically have large volatility making the
model unsuitable.

The models calibrates well to simple fixed-income instru-
ments of specified maturity, and if the parameters are
allowed to be time dependent then a term structure can
also be fitted.

Heath, Jarrow and Morton

In the Heath, Jarrow & Morton (HJM) model the evo-
lution of the entire forward curve is modelled. The
risk-neutral forward curve evolves according to

dF(t;T)=m(t,T)dt+ν(t,T)dX.

Zero-coupon bonds then have value given by

Z(t;T)=e−

∫T

tF(t;s)ds,

the principal at maturity is here scaled to $1. A hedging
argument shows that the drift of the risk-neutral process
forFcannot be specified independently of its volatility
and so

m(t,T)=ν(t,T)

∫T

t

ν(t,s)ds.

This is equivalent to saying that the bonds, which are
traded, grow at the risk-free spot rate on average.