## Team : mathfi

## Section: Application Domains

**Keywords: ***stochastic volatility*, *stable law*, *fractional Brownian motion*.

## Modelisation of financial assets

**Participants:**V. Genon-Catalot, T. Jeantheau, A. Sulem.

### Statistics of stochastic volatility models.

It is well known that the Black-Scholes model, which assumes a constant volatility, doesn't completly fit with empirical observations. Several authors have thus proposed a stochastic modelisation for the volatility, either in discrete time (ARCH models) or in continuous time (see Hull and White [70]). The price formula for derivative products depend then of the parameters which appear in the associated stochastic equations. The estimation of these parameters requires specific methods. It has been done in several asymptotic approaches, e.g. high frequency [65], [66], [64].

### Application of stable laws in finance.

Statistical studies show that market prices do not follow diffusion prices but rather discontinuous dynamics. Stable laws seem appropriate to model cracks, differences between ask and bid prices, interventions of big investors. Moreover pricing options in the framework of geometric -stable processes lead to a significant improvement in terms of volatility smile. Statistic analysis of exchange rates lead to a value of around 1.65. A. Tisseyre has developped analytical methods in order to compute the density, the repartition function and the partial Laplace transform for -stable laws. These results are applied for option pricing in ``stable'' markets. (see [93]).

### Fractional Brownian Motion (FBM).

The Fractional Brownian Motion B_{H}(t) with Hurst parameter H has
originally been introduced by Kolmogorov for the study of turbulence.
Since then many other applications have been found.

If then B_{H}(t) coincides with the standard
Brownian motion, which has independent increments. If then B_{H}(t) has a *long memory* or *strong
aftereffect*. On the other hand, if , then
_{H}(n)<0 and B_{H}(t) is *anti-persistent*: positive values
of an increment is usually followed by negative ones and conversely.
The strong aftereffect is often observed in the logarithmic returns
for financial quantities Y_{n} while the
anti-persistence appears in turbulence and in the behavior of
volatilities in finance.

For all H(0, 1) the process B_{H}(t) is *self-similar*, in
the sense that B_{H}(t) has the same law as ^{H}B_{H}(t),
for all >0.

Nevertheless, if , B_{H}(t) is not a semi-martingale
nor a Markov process [68][55], [69], [19], and
integration with respect to a FBM requires a specific stochastic
integration theory.

Consider the classical Merton problem of finding the optimal
consumption rate and the optimal portfolio in a Black-Scholes market,
but now driven by fractional Brownian motion B_{H}(t) with Hurst
parameter . The interpretation of
the integrals with respect to B_{H}(t) is in the sense of Itô
(Skorohod-Wick), not pathwise (which are known to lead to arbitrage).
This problem can be solved explicitly by proving that the martingale
method for classical Brownian motion can be adapted to work for fractional Brownian motion as well [68]. When the results
converge to the corresponding (known) results for standard Brownian
motion [69]. Moreover, a stochastic maximum
principle holds for the stochastic control of FBMs [55].