Financial data analysis
The concept of “risk” is central to many applications in finance, e.g. portfolio management or option pricing.
An essential preliminary step for measuring the risk is to dispose of adequate models for the various assets. A huge literature is devoted to this issue. Popular recent models include certain Lévy processes, fractional Brownian motion and multifractional Brownian (on this last aspect, see some articles here).
We contribute in this area along the following lines:
- a striking feature of many financial logs is they are both irregular 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 investigate the simple case of multistable Lévy motions and concentrate on understanding how a time-varying α function translates in terms of risk, in particular for portfolio selection. This requires a deeper understanding of the stochastic properties of these processes,
- in another direction, we study whether mBm and SRP provide useful models in the frame of financial modelling. Fractional Brownian motion-based option pricing and portfolio selection has attracted a lot of interest in recent years. Taking into account a varying regularity requires to generalize either to multifractional Brownian motion or to SRMP. Stochastic integration with respect to mBm is mentioned here, and is defined by extending the tools developed for fBm. However, mBm has the disadvantage that, in order to price, 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 SRMP: 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 SRMP (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.
Some of these studies (specially the modelling with multistable processes) are performed in the frame of a collaboration with SMA-BTP and H-W Conseil.