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


where c is a constant and g_{c} is a suitable function. The
function g_{c} satisfies the
following partial differential equation:
_{}


where d is the Dirac’s
delta function.
Recall that s = tt¢ Î 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 _{}, a_{c}(k), k Î R, are given by:
_{}
_{}
Note that in order
to guarantee that for k Î R the
function g_{c} does not diverge when v goes to plus infinity and that
the function a_{c}(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:


or

(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 [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/e^{2}) 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 [7] for further details).