seminario presso CNR-IMATI, Milano

[Alessandra Guglielmi <alessan@mi.imati.cnr.it>]

Il giorno 28 aprile 2004, alle h. 11.30, si terra' presso il CNR-IMATI,
sezione di Milano, Via Bassini, 15, Milano, in aula convegni (piano terra),
il seminario

         A STOCHASTIC VOLATILITY MODEL FOR EXCHANGE RATE DERIVATIVES


                              BARBARA TRIVELLATO
                    Dip. Matematica - Politecnico di Torino


Abstract.

It is well known that the Black and Scholes model (1973) produces a convenient
formula to evaluate European options but misprices them in many cases. This
fact has many reasons, the most relevant ones being the assumption of
normally distributed returns on the asset and its constant volatility. The
famous smile effect can be observed on practically all derivatives markets
and obviously weakens the use of the Black and Scholes formula as a sound
pricing tool.
Implementation of more complex frameworks for evaluation requires, then, a
more flexible workbench for volatility. During the last decades many authors
have tried to overcome the Black and Scholes hyphothesis. Heston (1993), for
instance, proposed a stochastic volatility model in which two correlated
diffusions represent the dynamics of both the underlying asset and the
volatility. A closed-form solution is provided for the evaluation of a
European call option. The resulting price captures biases, such as return
skewness and kurtosis, of the Black and Scholes formula.
All refinements of the basic Black and Scholes model enhance accuracy and
pricing but, unfortunately, broaden the computational burden and require fast
algorithms to compute the parameters. This can be a problem expecially when
pricing derivatives that are not plain vanilla such as barrier and Asian
options.
Due to the smile effect, we need to find a process that is representative of
the dynamics of the underlying price and that reproduces the volatility
behaviour observed in the market.
In this work we present an alternative model where the dynamics of the
underlying "lies" between the Black-Scholes  and Heston's one. The volatility
variance follows a square-root process and has a stochastic correlation with
spot returns. According to Heston, we obtain a closed-form solution for the
European option price with a number of free parameters belonging to the
dynamics. These parameters have to be chosen so that the model gives the
"best" possible fit to market data.
The reason of our choice lays in the fact that Heston's continuous-time
dynamics do not have a discrete-time  counterpart that can be easily managed
in pricing exotic derivatives when using binomial or trinomial trees. Our aim
is then to build a computationally simple approximation of the diffusions
through recombining trees,  according to Nelson and Ramaswamy (1990)
technique.
Our model has a general setting. However, we have decided to focus our
attention on the analysis of forex markets, since these have been steadily
growing during the last years. Our data refer to the euro/dollar exchange
rate and to its derivatives market.

The talk is based on joint work with Enrico Moretto, Università  di Parma,
and Sara Pasquali, CNR-IMATI.

-- 
Alessandra Guglielmi             e-mail: alessan@mi.imati.cnr.it
CNR-IMATI (ex IAMI)              tel.  : ++39.02.23699529 
via Bassini, 15                  fax   : ++39.02.23699538 
20133 Milano (ITALIA)

web page: www.mi.imati.cnr.it

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