## 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 ó õ +¥ -¥ dk  e-i k(x¢-x)gc(t-t¢,k,v,v¢),

 (x,v),  (x¢,v¢) Î R×R+,  t¢,t ³ 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 Î R,  v¢,v Î R+,

(3.3)

where d is the Dirac’s delta function.

Recall that s = t-t¢ Î R+. Proceeding as done in  to deduce the formulae shown in Section 1 it is possible to deduce the following formula: where the functions , ac(k), k Î R, are given by:  Note that in order to guarantee that for k Î R the function gc does not diverge when v goes to plus infinity and that the function ac(k), k Î R, is well defined we must require that for k Î R the real part of is positive. An easy computation shows that this last condition (i.e. Re( ) > 0, k Î R) implies that c must satisfy the following inequalities:

 0 £ c < 1 1-r2 ,  when  r Î (-1,0)È(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 Î [0,1), r Î (-1,1).

(3.9)

Note that the choice of c made in (3.9) when g Î [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 ÎR, g Î [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  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 Î [0,1), however the prices on the left hand side of (3.15) do not depend on g Î [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  for further details).