Speculative option valuation and the fractional diffusion equation Enrico Scalas (DISTA East-Piedmont University) FDA04 - Bordeaux (FR) July 2004

In collaboration with: Rudolf Gorenflo Francesco Mainardi Mark M. Meerschaert

Summary Continuous-time random walks as models of market price dynamics Limit theorems Link to other models Application to speculative option valuation Conclusions

Tick-by-tick price dynamics

Theory (I) Continuous-time random walk in finance (basic quantities) : price of an asset at time t : log price : joint probability density of jumps and of waiting times : probability density function of finding the log price x at time t

Theory (II): Master equation Marginal jump pdf Marginal waiting-time pdf Permanence in x,t Jump into x,t In case of independence: Survival probability

This is the characteristic function of the log-price process subordinated to a generalised Poisson process. Theory (III): Limit theorem, uncoupled case (I) (Scalas, Mainardi, Gorenflo, PRE, 69, , 2004) Mittag-Leffler function Subordination: see Clark, Econometrica, 41, (1973).

Theory (IV): Limit theorem, uncoupled case (II) (Scalas, Gorenflo, Mainardi, PRE, 69, , 2004) This is the characteristic function for the Green function of the fractional diffusion equation. Scaling of probability density functions Asymptotic behaviour

Theory (V): Fractional diffusion (Scalas, Gorenflo, Mainardi, PRE, 69, , 2004) Green function of the pseudo-differential equation (fractional diffusion equation): Normal diffusion for  =2,  =1.

Theory (VI): The coupled case (I)

Theory (VII): The coupled case (II) Basic message: Under suitable hypotheses, the fractional diffusion equation is the diffusive limit of CTRWs also in the coupled case!

Continuous-time random walks (CTRWs) CTRWs Cràmer-Lundberg ruin theory for insurance companies Compound Poisson processes as models of high-frequency financial data ( Scalas, Gorenflo, Luckock, Mainardi, Mantelli, Raberto QF, submitted, preliminary version cond-mat/ , or preprint: Normal and anomalous diffusion in physical systems Subordinated processes Fractional calculus Diffusion processes Mathematics Physics Finance and Economics

Example: The normal compound Poisson process (  =1) Convolution of n Gaussians The distribution of  x is leptokurtic

Generalisations Perturbations of the NCPP: general waiting-time and log-return densities; (with R. Gorenflo, Berlin, Germany and F. Mainardi, Bologna, Italy, PRE, 69, , 2004); variable trading activity (spectrum of rates); (with H.Luckock, Sydney, Australia, QF submitted); link to ACE; (with S. Cincotti, S.M. Focardi, L. Ponta and M. Raberto, Genova, Italy, WEHIA 2004!); dependence between waiting times and log-returns; (with M. Meerschaert, Reno, USA, in preparation, but see P. Repetowicz and P. Richmond, xxx.lanl.gov/abs/cond-mat/ ); other forms of dependence (autoregressive conditional duration models, continuous-time Markov models); (work in progress in connection to bioinformatics activity).

Application to speculative option valuation Portfolio management: simulation of a synthetic market (E. Scalas et al.: VaR estimates: e.g. speculative intra-day option pricing. If g(x,T) is the payoff of a European option with delivery time T: Large scale MC simulations of synthetic markets with supercomputers are being performed (with G. Germano, P. Dagna, and A. Vivoli:

Results (I) Fig. 1: Simulated log-price as a function of time. This simulation includes log-prices. It takes a few minutes to run on an old Pentium II processor at 349 MHz.

Results (II) Fig 2: Power to factorial ratio for T=5000s and  0 =10s. Only evidenced values of n have been used to compute p(x,T).

Results (III) Fig. 3: Theoretical PDF (solid line) and simulated PDF (circles). p(x,T) is computed for T=5000s,  0 =10s,  = The simulated PDF is computed from the histogram of 1000 realisations.

Results (IV) Fig. 4: Payoff histogram for a very short-term (T=5000 s) plain vanilla call European option with initial price S(0)=100 and strike price E=100. The evolution of the underlying has been simulated 1000 times times by means of a NCPP, with parameters given in Fig. 3.

Conclusions CTRWs are suitable as phenomenological models for high-frequency market dynamics. They are related to and generalise many models already used in econometrics. They can be helpful in various applications including speculative option valuation.