Inria / Raweb 2004
Team: Mathfi

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## 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 ). 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 , , .

### 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 ).

### Fractional Brownian Motion (FBM).

The Fractional Brownian Motion BH(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 BH(t) coincides with the standard Brownian motion, which has independent increments. If then BH(t) has a long memory or strong aftereffect. On the other hand, if , then H(n)<0 and BH(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 Yn while the anti-persistence appears in turbulence and in the behavior of volatilities in finance.

For all H (0, 1) the process BH(t) is self-similar, in the sense that BH( t) has the same law as HBH(t), for all >0.

Nevertheless, if , BH(t) is not a semi-martingale nor a Markov process , , , 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 BH(t) with Hurst parameter . The interpretation of the integrals with respect to BH(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 . When the results converge to the corresponding (known) results for standard Brownian motion . Moreover, a stochastic maximum principle holds for the stochastic control of FBMs .