Interest Rate Models - Computational Methods for Quantitative Finance - page 86

The converse of Theorem 7.1.1 is also true. The proof is given in [63]. Note that
under certain assumptions on the coefficients, i.e. if r

0 can be reached with a
positive probability, boundary conditions for V(t, 0 ) have to be imposed in Theo-
rem 7.1.1 . We refer to [63] for details on this topic. There are many examples of
short rates models. We only state two: the Vasicek model [156] which was one of
the first short rate models and the Cox-Ingersoll-Ross (CIR) model.

=

Example 7.1.2 For the Vasicek model, let α , β , r and σ be positive constants. We
consider the following SDE

d r t = (α − βr t ) d t + σ d W t , 0 = r.
The corresponding generator is

1

2 σ 2 ∂ rr . Note that negative in-
terest rates occur with a positive probability at any time t . This model was applied
to modeling in commodity markets by Lucia and Schwartz [116]. For the Cox-
Ingersoll-Ross (CIR) model, the sport rate is given by

A = (α − βr)∂ r +

σ √ r t d W t , 0 =

d r t = (α − βr t ) d t + r, (7.3)

1

2 σ 2 r∂ rr .
Note that the diffusion coefficient of the CIR model, σ √ r t is non-Lipschitz. Ex-
istence and uniqueness of a strong solution is nevertheless ensured by Yamada's
Theorem [89]. We have

which yields the generator A = (α − βr)∂ r +

Theorem 7.1.3 For any standard Brownian motion W on [ 0 , ∞ ) and r ≥ 0, there

exists a unique , continuous , adapted stochastic process r t ⊂ R + such that

σ √ r t d W t

d r t = (α − βr t ) d t + in [ 0 , ∞ ), r 0 = r. (7.4)

For a proof, see [89], p. 221. We give some properties of r t .Let r t denote the

(unique, strong) solution of ( 7.4 ), and define
τ 0 =

r t inf { t ≥ 0 | = 0 } , (7.5)

with inf ∅:=∞ . Then it can be shown that

σ 2 / 2in( 7.4 ), then (τ 0 =∞

1. If α ≥ P ) = 1, ∀ r> 0.

α<σ 2 / 2 (τ 0 < 2. If 0 ≤ ∧ β ≥ 0, P ∞ ) = 1, ∀ r> 0.

α<σ 2 / 2 (τ 0 < 3. If 0 ≤ ∧ β< 0, P ∞ ) ∈ ( 0 , 1 ) , ∀ r> 0.

The weak formulation on a bounded domain G , R> 0, and written in

time-to-maturity, for the bond price in the CIR model reads:

=[ 0 ,R ]

L 2 (J H 1 (J Find v ∈ ; W 1 / 2 ,μ ) ∩ ; H μ ) such that

(∂ t v,w) μ + a CIR

1 / 2 ,μ (v, w) =

0 , ∀ w ∈ W 1 / 2 ,μ , a.e. in J, (7.6)

v( 0 ) = 1 ,

where we denote by ( · , · ) μ the inner product in H μ (G) and