1. Speculative option valuation and the fractional diffusion equation Enrico Scalas (DISTA East-Piedmont University) www.econophysics.org www.fracalmo.org FDA04 - Bordeaux (FR) 18-21 July 2004

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

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

4. Tick-by-tick price dynamics

5. 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

6. 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

7. 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, 011107, 2004) Mittag-Leffler function Subordination: see Clark, Econometrica, 41, 135-156 (1973).

8. Theory (IV): Limit theorem, uncoupled case (II) (Scalas, Gorenflo, Mainardi, PRE, 69, 011107, 2004) This is the characteristic function for the Green function of the fractional diffusion equation. Scaling of probability density functions Asymptotic behaviour

9. Theory (V): Fractional diffusion (Scalas, Gorenflo, Mainardi, PRE, 69, 011107, 2004) Green function of the pseudo-differential equation (fractional diffusion equation): Normal diffusion for  =2,  =1.

10. Theory (VI): The coupled case (I)

11. 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!

12. 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/0310305, or preprint: www.maths.usyd.edu.au:8000/u/pubs/publist/publist.html?preprints/2004/scalas-14.html) Normal and anomalous diffusion in physical systems Subordinated processes Fractional calculus Diffusion processes Mathematics Physics Finance and Economics

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

14. Generalisations Perturbations of the NCPP: general waiting-time and log-return densities; (with R. Gorenflo, Berlin, Germany and F. Mainardi, Bologna, Italy, PRE, 69, 011107, 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/0310351 ); other forms of dependence (autoregressive conditional duration models, continuous-time Markov models); (work in progress in connection to bioinformatics activity).

15. Application to speculative option valuation Portfolio management: simulation of a synthetic market (E. Scalas et al.: www.mfn.unipmn.it/scalas/~wehia2003.html). 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: http://www1.fee.uva.nl/cendef/sce2004/sce_ams.htm).

16. Results (I) Fig. 1: Simulated log-price as a function of time. This simulation includes 10000 log-prices. It takes a few minutes to run on an old Pentium II processor at 349 MHz.

17. 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).

18. Results (III) Fig. 3: Theoretical PDF (solid line) and simulated PDF (circles). p(x,T) is computed for T=5000s,  0 =10s,  =0.005. The simulated PDF is computed from the histogram of 1000 realisations.

19. 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.

20. 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.