1. Hull and White model with nonzero correlation: formulae for the transition probability density function

Let us derive a closed form formula for the transition probability density function of the Hull and White model  in presence of a (possibly) nonzero correlation between the stochastic differentials of the Wiener processes appearing on the right hand side of the model equations.

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.

The Hull and White stochastic volatility model assumes that St, Vt, t ÎR+, satisfy the following stochastic differential equations (see ): 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 Hull and White model (1.5), (1.6) reduces to the lognormal SABR model . The lognormal SABR model is a generalization of the Black model in the context of stochastic volatility and is widely used in the practice of the financial markets.

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: and the initial conditions (1.3), (1.4) become: 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: where  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: 