294 Frequently Asked Questions In Quantitative Finance
A multi-factor version of this results in the following
risk-neutral process for the forward rate curve
dF(t,T)=
(N
∑
i= 1
νi(t,T)
∫T
t
νi(t,s)ds
)
dt+
∑N
i= 1
νi(t,T)dXi.
In this thedXiare uncorrelated with each other.
Brace, Gatarek and Musiela
The Brace, Gatarek & Musiela (BGM) model is a discrete
version of HJM where only traded bonds are modelled
rather than the unrealistic entire continuous yield curve.
IfZi(t)=Z(t;Ti) is the value of a zero-coupon bond,
maturing atTi,attimet, then the forward rate applic-
able betweenTiandTi+ 1 is given by
Fi=
1
τ
(
Zi
Zi+ 1
− 1
)
,
whereτ=Ti+ 1 −Ti. Assuming equal time period between
all maturities we have the risk-neutral process for the
forward rates are given by
dFi=
∑i
j= 1
σjFjτρij
1 +τFj
σiFidt+σiFidXi.
Modelling interest rates is then a question of the func-
tional forms for the volatilities of the forward ratesσi
and the correlations between themρij.
Prices as expectations
For all of the above models the value of fixed-income
derivatives can be interpreted as
EQt
[
Present value of cashflows
]
,
