2.  Hull and White model with non zero correlation: formulae for the moments of the logarithm of the asset price - an interactive application

The processes xt = ln(St e-r t/), vt, t R+, satisfy equations (1.7), (1.8) (1.9), (1.10) and as a consequence the processes xt = ln(St), vt, t R+, satisfy the equations:

with the initial conditions:

Recall that and that the parameters m and e are respectively the drift and the volatility of the variance process that appear in equation (1.2). As already done in Section 1 the stochastic differentials dWt, dZt, t R+ are assumed to satisfy the condition , that is they are assumed to be correlated with correlation coefficient r(-1,1).

For n = 0,1, let Ln, be the n-th moment with respect to zero of xt, t R+, we have:

 

 

 

Ln(t,x,v,t) =


+

- 

dx xn 


+

0 

dv p*(x,v,t,x,v,t),

 

 

 

x R, v R+,  t,t 0, t-t > 0,   n = 0,1, ,

(2.5)

 

 

where p* is the transition probability density function associated to the stochastic processes xt, vt, t R+, implicitly defined by (2.1), (2.2), (2.3), (2.4). The function p* can be written as follows:

 

 

 

p*(x,v,t,x,v,t) =

1


2p

 


+

- 

dx  e-i k(x-x)g*(t-t,k,v,v),

 

 

 

(x,v),  (x,v) RR+,  t,t 0, t < t,

(2.6)

 

 

and the function g* can be determined proceeding as done in Section 1 to determine the function g. Note that p* depends on s = t-t and not on t and t separately, t > t 0, this implies that we can rewrite the moments of xt, t R+, defined in (2.5) as follows:

 

 

 

L*n(s,x,v) = Ln(t,x,v,t) =

n

j = 0 

 



 

n

j

 



(x)n-j ij Dj(s,v),

 

 

 

s = t-t R+,  x R,  v R+,   n = 0,1,,

(2.7)

 

 

where

Dj(s,v) =


+

0 

dv 

dj


dkj

g*(s,k,v,v)





k = 0 

,  s,v R+, j = 0,1,.

(2.8)

 

 

The expressions of the functions Dj , j=0,1,2, are (see [7] for further details):

 

D0(s,v) = 1,  s,v R+,

(2.9)

 

 

 

 

 

 

 

Let us choose t = 0, we have x =, v = in (2.9), (2.10), (2.11). It follows that s = t and that the first three moments of xt, t R+, are given by:

 

 

In a similar way using elementary computations it is possible to obtain the expressions of the moments L*n(), t R+, R, R+, for n > 2 [7]. These expressions become more and more involved when n increases. Note that formulae (2.13), (2.14) are closed form formulae containing only elementary functions of quantities that can be observed in the financial markets. In particular the formulae (2.13), (2.14) do not contain integrals. These formulae can be used in the formulation of calibration problems for the Hull and White model. Thank to the closed form character of the formulae for the moments the resulting calibration problems can be solved by ad hoc very efficient numerical algorithms. In [5], [6] these ideas have been exploited to calibrate the normal and the lognormal SABR models.

 

We conclude this section with an interactive application that given a time value t=t1 >0 makes possible to compare the first order moment L*1 at time t=t1 computed using formula (2.13) with the first order moment obtained from numerical simulation of the Hull and White model (2.1), (2.2), (2.3), (2.4). That is given a time value t=t1>0 the numerically simulated first order moment is obtained from a sample of the random variable xt1 generated integrating numerically (2.1), (2.2), (2.3), (2.4). The first moment of this sample is computed. In the interactive application the user can choose the parameters of the Hull and White model and the size of sample used to compute the moment. A set of default parameters is proposed.

 

INTERACTIVE APPLICATION