Section: Application Domains
Risk modelling in finance
A striking feature of many financial logs is that they are both irregular in the Hölder sense and display jumps. Furthermore, the local roughness as well as the size of jumps typically vary in time. This hints that multifractional multistable processes may provide well-adapted models. As a first step, we shall investigate the simple case of multistable Lévy motions and concentrate on understanding how a time-varying function translates in terms of risk, in particular for VaR computation. This will require both a deeper understanding of the stochastic properties of these processes and a fine analysis of the microstructure of financial logs.
In another direction, we will study whether multifractional Brownian motion (mBm) and SRP provide useful models in the frame of financial modeling. Fractional Brownian motion-based option pricing and portfolio selection has attracted a lot of interest in recent years. This process is certainly a more adequate model than pure Brownian motion, as many studies have shown. However, it is also clear that it suffers various limitations. One of the most obvious is that the local regularity of financial logs is not constant, as is apparent on any sufficiently long sample. The most direct way of generalizing fractional Brownian motion to account for this fact is to consider mBm, as we have done in  , using the theory of stochastic calculus with respect to mBm that we have recently developed in  ,  . Another possibility is to use SRP. This requires to extend both the theoretical results (mainly those related to stochastic calculus) and their applications (pricing, portfolio selection) beyond the case of fractional Brownian motion. A disadvantage of mBm is that, in order to price for instance, one has to know the regularity function ahead of time, which usually requires additional assumptions, or to build a model for its evolution. This problem is not present for the SRP: no further information is required once the function relating the amplitude and the regularity has been identified. On the other hand, stochastic integration with respect to SRP (which is neither a Gaussian process nor a semi-martingale) does not seem to be within reach at present, since little is known indeed about this process. This nevertheless constitutes one of our long term goals.