292 Frequently Asked Questions In Quantitative Finance
Hull and White The risk-neutral model
dr=(η(t)−u−γr)dt+cdX 1
and
du=−au dt+bdX 2
is a two-factor version of the one-factor Hull & White.
The functionη(t) is used for fitting the initial yield
curve.
All of the above, except for the Brennan & Schwartz
model, have closed-form solutions for simple bonds in
terms of the exponential of a linear function of the two
variables.
The market price of risk as a random factor Suppose that we
have the tworealrandom walks
dr=udt+wdX 1
and
dλ=pdt+qdX 2 ,
whereλis the market price ofrrisk. The zero-coupon
bond pricing equation is then
∂Z
∂t
+^12 w^2
∂^2 Z
∂r^2
+ρwq
∂^2 Z
∂r∂λ
+^12 q^2
∂^2 Z
∂λ^2
+(u−λw)
∂Z
∂r
+(p−λλq)
∂Z
∂λ
−rZ= 0.
Since the market price of risk is related to the slope
of the yield curve as the short end, there is only one
unobservable in this equation,λλ.
SABR
The SABR (stochastic,α,β,ρ) model by Hagan, Kumar,
Lesniewski & Woodward (2002) is a model for a forward
