Risk Analysis in Finance and Insurance, 2nd Edition

Fixed Income Securities: Modeling and Pricing 159

and hence,

B(t, T )=exp



−



T

t

f(0,s) ds −

σ

2

2

tT (T − t)+σ (T − t) W

t

%

=

B(0,T)

B(0,t)

exp



−

σ

2

2

tT (T − t)+σ (T − t) W

t

%

.

We can also rewrite it in the form

B(t, T )=

B(0,T)

B(0,t)

exp



(T − t) f(0,T) −

σ

2

2

t (T − t)

2

− (T − t) r

t

%

.

Note that this model is a particular case of the Heath-Jarrow-Morton

model, and it is not difficult to check that the initial probability is a mar-

tingale probability.

Now we proceed to a detailed study of Vasiˇcek model. According to this

model, the interest rate oscillates around α/β: r

t

has positive drift if r

t

<α/β,

and negative if r

t

>α/β.Ifα/β =0,then r

t

is a stationary (Gaussian)

Ornstein-Uhlenbeck ...