1. Option Pricing Montecarlo Method, Path Integrals A.Chiarugi, G. Cipriani, M.Rosa-Clot, S.Taddei Firenze E. Bennati Dip Scienze Econ. (Pisa) G.Lotti Dip. Matematica (Parma) M.Cerchiai, G. Einaudi, P. Rosa-Clot (Pisa) S.R. Amendolia, B.Golosio (Sassari)

2. Tossing a Coin:100, 1000, 10000, 50000 Trials

3. Montecarlo Method: a simple barrier test

4. Probability distribution How we can check a distribution law ? Looking at many trials ! How many ? A lot ! ! ! Tossing a coin we assume  x=  1   x =  w This is a Wiener process

5. Wiener Process In general we write  x =  x,t)  t +  (x,t)  w As a particular case we have  r =  a (b - r )  t +   w Vasicek or  r =  a (b - r )  t +   r  w CIR

6. General case: stochastic equation This equation can be solved with several techniques analytical methods differential equations (Fokker Plank) tree discretized steps Montecarlo method !!!! path integral approach !!!!

7. Analytical approach: the CIR model

8. A CIR model realisation

9. Why realistic models? Vasicek model has serious drawbacks (it allows negative interest rate values) CIR (Cox Ingersoll Ross) model looks more realistic and it allows analytical solutions. However the first target is to do without “analytical models” and to use real rates. The second target is to work with a general “functional” (???)

10. What “Functional” means? In the left plot, all the paths which overcome the black line are weighted with the corresponding interest rate Functionals can be very complicated especially for exotic options We have to deal with barriers, look back options, and with option price depending on past averaged quantities

11. A functional evaluation requires : To average on all the possible paths However a path can depend on the functional So we have to perform a huge number of trials Then => MONTECARLO To convert the continuos process in a set of finite steps To know the probability distribution at any time To integrate numerically on the prob. distributions Then => PATH INTEGRAL

12. Convolution and composition law The density  (y,t,x,0) gives the probability to get the y value at a time t’, once the distribution is known a time t=0. Such a density satisfies the convolution law

13. Composition law for a short time  t For a short time we get with

14. The numerical problem: N  t = T We have to perform N numerical convolutions i.e. N matrix products. Matrices are exponentials with a lagrangian L(x,v,t) as exponent where v= (y-x)/  t is the “velocity” of the system.

15. Some paths through finite steps We show 5 realisations of a stochastic process The transfer function  is given for each step  t

16. Feynman Approach: the P.I. Wiener creates the theory of stochastic processes in 1921 Feynman introduces the path integral concept in physics with his master thesis in 1942. The computational problems are too big. In fact only in 1981 Kreutz e Freedman are able to perform a first numerical calculation of the “Harmonic Oscillator” 90th Huge explosion of Montecarlo approaches to P.I. Recently: deterministic approaches (Rosa-Clot and Taddei). Very quick but low dimension (<4). Which is enough for financial markets.

17. Numerical and theoretical improvements Well based theory All the analytical cases are under control All the results in the literature are “easily” reproduced Approximation techniques are well known Numerically stable It is quick as and it looks like tree approach It allows the evaluation of functional of arbitrary complexity in one, two or more dimensions

18. The functional In the more general case it is necessary to evaluate quantities which depends on the process itself. Two typical examples are the barrier options and the put American options This is “impossible” with Montecarlo but “easy” with Path Integrals

19. The put American option This option is exercised when its value is below a minimum which depends on the path and on the market model

20. What is available A set of Montecarlo codes for different models 1-2-3 D with and without stochastic volatility and for vanilla, barrier, swap options A corresponding set of Path Integral codes A set of Path integral codes for path dependent options (American, look back and exotic)

21. What of new CPU time for PI codes is remarkably shorter than for MC. The reduction factor ranges between 10 and 1000. The PI can be extended to a larger set of functional. The CPU time does not depend on the model complexity

23. Open Problem: DATA ANALYSIS

24. METHODS Statistical analysis Autoregressive models Spectral test and wavelets Neural networks TARGETS Models for the Option Pricing Optimisation of Hedging strategy Short Forecasting (minutes) Long Forecasting (months)

25. A short look to FIB30

27.  price distribution with delay of 1 4 16 64 256 1024 tic

28. Real % Gaussian distribution

29. This work can be improved but NEVER ENDS See Amendolia Slides