3.  Hull and White model with nonzero correlation: option price formulae

Let us derive the formulae in the Hull and White model of the prices at time t = 0 of European call and put options having maturity time T > 0 and strike price E > 0. These formulae express the option prices as three dimensional integrals of explicitly known integrands.

To this aim we rewrite the transition probability density function computed in Section 1 as follows:

 

 

 

p(x,v,t,x,v,t) = e-c(x-x)

1


2p

 


+

-
 

dk  e-i k(x-x)gc(t-t,k,v,v),

 

 

 

(x,v),  (x,v) RR+,  t,t 0,  t < t,

(3.1)

 

 

 

where c is a constant and gc is a suitable function. The function gc satisfies the following partial differential equation:

with the initial condition:

 

 

 

gc(0,k,v,v) = d(v-v),  k R,  v,v R+,

(3.3)

 

 

where d is the Diracs delta function.

Recall that s = t-t R+. Proceeding as done in [7] to deduce the formulae shown in Section 1 it is possible to deduce the following formula:

 

 

where the functions , ac(k), k R, are given by:

Note that in order to guarantee that for k R the function gc does not diverge when v goes to plus infinity and that the function ac(k), k R, is well defined we must require that for k R the real part of is positive. An easy computation shows that this last condition (i.e. Re() > 0, k R) implies that c must satisfy the following inequalities:

 

 

 

0 c <

1


1-r2

,  when  r (-1,0)(0,1),

(3.7)

 

 

or

 

 

 

0 c 1,    when  r = 0.

(3.8)

 

 

Let us choose c as follows:

c = cg = 1+g 

r2


(1-r2)

=

1


(1-r2)

-(1-g)

r2


(1-r2)

,   g [0,1), r (-1,1).

(3.9)

Note that the choice of c made in (3.9) when g [0,1) satisfies conditions (3.7), (3.8). We rewrite the transition probability density function p defined in (3.1) as follows:

 

where the function is given by:

 

 

and the functions , , k R, g [0,1), are given by:

The price at time t = 0 of a European call option having maturity time T > 0 and strike price E > 0 is the expected value of the discounted payoff with respect to a risk neutral measure. As shown in [7] the risk neutral measures of the Hull and White model are obtained replacing in the physical measure (whose density is given by (3.10), (3.11)) the parameter r with the risk free interest rate r* and the parameter with the parameter m* = -(2l/e2) where l is a new parameter used to price options called risk premium parameter. That is, we have:

 

 

 

where is the asset price at time t = 0, (  )+ = max(  ,0) is the maximum between and zero and p is a risk neutral transition probability density function. That is in (3.14) the function p is given by formula (3.10) with the parameters r* and m* instead of the parameters r and respectively. Note that the initial stochastic volatility is not observable and must be determined in the calibration process.

From formulae (3.10) and (3.14) we have:

where is given by (3.11), and, as already said, in (3.11) r* and m* replace r and respectively. Note that on the right hand side of (3.15) we have g [0,1), however the prices on the left hand side of (3.15) do not depend on g [0,1). In the numerical experiments presented in Section 4 we choose g = 1/2.

The price at time t = 0 of a European put option having maturity T > 0 and strike price E > 0 can be obtained from the call option price using the put call parity relation. That is using the relation:

(see [7] for further details).