Financial Modelling in Python

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Appendix C

Hull–White Model Mathematics

In this appendix we give a brief outline of the Hull–White model. For a more in-depth

discussion of the Hull–White model, readers are encouraged to consult [10] and [18]. The

Hull–White model belongs to a class of HJM models called extended Vasicek models. This

class of one-factor models has, in the risk-neutral measure denoted by Q, the following short

rate process

dr(t) =

(

m(t) −λ(t)r(t)

)

dt + σ (t)dW(t)(C.1)

with r(t), the short rate at time t, m(t),λ(t),σ(t):R

+

→ R

+

and W(t)isaQ-Brownian mo-

tion. The original Hull–White model makes the further simplification that both σ and λ are

constant in time. Introducing the auxiliary variables defined below

C(t):= σ (t)exp(λt)(C.2)

φ(t):=

1 −exp(−λt)

λ

(C.3)

M(t):= exp(−λt)r(0) +



t

0

exp(λ(s −t))m(s)ds (C.4)

it is straightforward to show that

R(t, T ):=



T

t

r(s)ds =



T

t

M(s)ds +



T

t

(

φ(T ) −φ(s)

)

C(s)dW(s). (C.5)

The stochastic discount factor and zero coupon bond can be expressed in terms of R(t, T)

as follows:

B(t)

−1

:= exp(−R(0, t)) (C.6)

P(t, T ):= E



exp(−R(t, T ))|F

t



(C.7)

where E denotes the expectation in the risk-neutral measure and F

t

the filtration at time

t. Before carrying out the above expectations we note that B(t) is called the money-market

account and is the numeraire in the risk-neutral measure. Performing the expectations we

obtain

B(t)

−1

= P(0, t)e