Exponentials in Stochastic Volatility Models

From: Osher Doctorow (mdoctorow_at_comcast.net)
Date: 08/25/04 

Date: Wed, 25 Aug 2004 21:37:26 +0000 (UTC)

 From Osher Doctorow mdoctorow@comcast.net

COPYRIGHT NOTICE
Exponentials in Stochastic Volatility Models
Copyright by Owner Osher Doctorow Ph.D.
First Published 2004.

The important role of exponentials and their rational functions and
linear polynomials and their rational functions in the Riccati
equation appears to be mirrored by their important role in Stochastic
Volatility Models, which have become extremely important with the
Nobel Prize to Granger and Engel in 2003 for GARCH and Cointegration
related work. Sometimes exponentials are more prominent, sometimes
their "diagonal inverses" or composition inverses logarithms are.

In "Indifference pricing and hedging in stochastic volatility models,"
by M. R. Grasselli and T. R. Hurd of McMaster U. (Canada), logarithms
are more common in their equations, but exponentials appear to be
more fundamental in their reasoning and development.

 From page 3 to the end of their paper (a rather long paper), they
specialize to an exponential utility:

1) U(t) = -exp(-gx)

with constant risk aversion parameter g > 0. This specialization
has an advantage of factorizing the value function which plays a
big role in subsequent simplifications - just as factorizing
plays a key role in various differential equation solutions by
setting "independent" factors equal to constants when they occur
on opposite sides of an equations. A key martingale turns out to
an exponential with differences in (random) variables in its
exponent, and the key result ((58), p. 19) of the market price
of volatility risk induced by an exponential utility in the
reciprocal affine model turns out to be a vector which equals the
product of the column vector (1, (b/(sqrt(2)D)H(u - r)/sqrt(Yt-bar)
where H = 1 - exp(D(t - T) with D being the square root of a
certain weighted su of squared parameters. This is said to have
three "pleasant implications":

A. Long enough from maturity both components are constant multiples
of the usual risk market price for complete markets with time vary-
ing volatility o_t or ot = sqrt(Yt).

B. For typical parameter values, most of the volatility risk is
accounted for by dependence of the model on the first Brownian
motion (W_t)^1.

C. The process Yt is the reciprocal of a CIR process under both
the exponential risk adjusted measure Q and the economic measure
P, which is a consistency result that is difficult to obtain for
other stochastic volatility models if at all. Here CIR is the
Cox, Ingersoll, and Ross (1985) model and Wt^1, Wt^2 [1 and 2 are
superscripts only] are a pair of independent one-dimensional P-
Brownian motions for a probability measure P.

Osher Doctorow