Stochastic Calculus
of Variations to Hedge Contingent Claims
Maria Elvira Mancino - Università di Firenze
In a recent paper by Fournié et al. (1999 Finance and Stochastics)
it is shown how to use Malliavin calculus to devise efficient Monte
Carlo methods for the expected values of the payoff associated with
the no arbitrage price and their differentials.
They apply stochastic calculus of variations to
the determination of variations of the prices with respect to
perturbations of the initial data, the drift coefficient or the
volatility coefficient. Such perturbations, that is the directional
derivatives of the Wiener functional which describes the price of
a contingent claim, gives extended Greeks formulae.
In the procedure of Fournié et al. (1999) there are two
steps: first the evaluation of the variation, second the computation
of the differential of interest as the mean of
a product of the payoff function and of a weight which independepent
of the payof.
In this paper we consider some generalizations in the study of the
extended Greeks which take into account both steps.
First we show that all the particular cases of variations (with respect to
the initial data, drift coefficient and volatility coefficient) can be
obtained through a
unified approach using the concept of reduced variation introduced
in Malliavin (1976).
About the second step, the procedure of Fournié et al. (1999)
is based on the validity of an integration by parts formula satisfied by
the vector field in the direction of which
we are considering the variation. In this context we allow for
two generalizations:
we consider a general payoff function, that is a random variable F
measurable with respect to the completion of the $\sigma$-algebra generated
by the Brownian motion
$(w_t)_{t\in [0,1]}$ with some regularity in the sense of
Malliavin derivative, and we consider a general class of
admissible directions along which such representation through
an integration by parts formula holds.