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 [10] 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 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 R+, satisfy the following stochastic differential equations (see [10]):

where r, m, x are real parameters. The processes Wt, Zt, t R+, are standard Wiener processes such that W0 = Z0 = 0, and dWt, dZt, t R+, are their stochastic differentials. Moreover we assume that <dWt dZt >= rdt, t>0, where < > denotes the expected value of and r(-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 R+.

For later convenience we rewrite equations (1.1), (1.2) using the volatility process vt, t R+, instead of the variance process Vt, t R+. Recall that we have: Vt = vt2, t 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 [4]. 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 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 R+ with probability one and (1.8) or (1.10) imply that vt > 0 with probability one for t 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, t0, t-t>0. Note that when t=0 we must choose x=, v=. In [7] 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: