Let us derive a
closed form formula for the transition probability density function of the
Let R and R+ be the sets of real and of positive real numbers respectively and t be a real variable that denotes time.
Let St, Vt, t Î R+, be real stochastic processes that describe respectively the asset price and the associated stochastic variance as a function of time.
where r, m, x are real parameters. The processes Wt, Zt, t Î R+, are standard Wiener processes such that W0 = Z0 = 0, and dWt, dZt, t ÎR+, are their stochastic differentials. Moreover we assume that <dWt dZt >= rdt, t>0, where <· > denotes the expected value of · and rÎ(-1,1) is a constant known as correlation coefficient. Equations (1.1) and (1.2) are equipped with the initial conditions:
where , are random variables that we assume to be concentrated in a point with probability one. For simplicity we identify the random variables , with the points where they are concentrated. We assume , > 0. The assumption , > 0 with probability one and (1.1), (1.2) imply that St, Vt > 0 with probability one for t Î R+.
For later convenience we rewrite equations (1.1), (1.2) using the volatility process vt, t Î R+, instead of the variance process Vt, t Î R+. Recall that we have: Vt = vt2, t Î R+. Equations (1.1), (1.2) become:
where e = x/2. Note that when r = 0 and m = e2 the
Let us introduce the centered log-return xt = ln(Ste-r t/), t Î R+, and the quantity =(m/e2)-1. Equations (1.5), (1.6) can be rewritten as follows:
where , are random variables concentrated in a point with probability one. Note that is equal to zero with probability one.
Moreover Î R+ with probability one and (1.8) or (1.10) imply that vt > 0 with probability one for t Î R+.
Let be the transition probability density function of the Hull and White model (1.7), (1.8), (1.9), (1.10), that is let be the probability density function of having , when , , t, t’³0, t-t’>0. Note that when t’=0 we must choose x’=, v’=. In  starting from the backward Kolmogorov equation associated to the stochastic processes defined by (1.7), (1.8) the following formula is deduced:
where i is the imaginary unit, and the function g is given by:
In (1.12) sinh denotes the hyperbolic sine and denotes the Whittaker function of indices a, w.
When r=0 formula (1.11) reduces to: