1. Nonlinear Filtering of Stochastic Volatility Aleksandar Zatezalo, University of Rijeka, Faculty of Philosophy in Rijeka, Rijeka, Croatia e-mail: zatezalo@pefri.hr

2. Abstract Stochastic volatility models for increments of logarithms of stock prices are considered. Given historic data of stock prices and a state model for volatility, we are applying nonlinear filtering methods to estimate conditional probability density function of stochastic volatility at time given sigma algebra generated by all stock prices up to time. Numerical methods for nonlinear filtering based on Strang splitting scheme are proposed and numerical simulations are presented. Method for evaluation and comparison of different schemes is proposed based on value at risk (VaR) calculations.

3. References [1] W. F. Ames, Numerical Methods for Partial Differential Equations, Academic Press, London, 1992. [2] R. Frey and W. J. Runggaldier, A nonlinear filtering approach to volatility estimation with a view towards high frequency data, preprint, to appear in International Journal of Theoretical and Applied Finance. [3] A. Kolmogoroff, Zufällige bewegungen, Annals of Mathematics, Volume 35, No. 1, 1934. [4] A. N. Shiryaev, Essentials of Stochastic Finance, World Scientific, New Jersey, 1999. [5] I. Karatzas and S.E. Shreve, Methods of Mathematical Finance, Springer, New York, 1998. [6] J. Ma and J. Yong, Forward-Backward Stochastic Differential Equations and Their Applications, Springer, New York, 1999. [7] J. Cvitanić, R. Liptser, and B. Rozovskii, Tracking Volatility, In Proc. 39th IEEE Conf. On Decision and Control, Sydney, 2000.

4. Continuous Model Let be probability space with complete filtration, Wiener process with respect to the filtration, and progressively measurable processes. Let price process (stock price) satisfies Black-Scholes type model that allows stochastic volatility i.e. for we have (see [7]) Further we consider stochastic process which represents amount of money investor puts into the stock with price and let represents interest rate of the riskless asset and let wealth process satisfies (see [6])

5. Assumptions We assume, and which for gives (using Itô’s formula) (for this model see [4]) where are the so called i.i.d. random variables. We can consider as observed data. Considering discrete observation we have that the change of wealth is also discrete i.e. we have where and

6. Assumptions (Cont.) We assume that we do not borrow the money and that we do not sell shares which we do not own at the moment i.e. we have constraint Generally we assume such that satisfies Itô’s differential equation i.e. we have that is solution of where is Wiener process with respect to filtration and and are suitable functions. We consider special case of the so called constant velocity tracking (in radar tracking) i.e. we have where

7. Simulation Examples vs. Real Data 050100150200250300 0.95 1 1.05 1.1 1.15 1.2 1.25 Simulation: Daily stock price in 1995 for ATT stock

8. Value at Risk (VaR) Let be the amount of money which the agent is willing to risk with probability at most under observed past data where represents the sets of measure zero. That is we are interested in estimating portfolios such that where i.e. the problem is to mathematically describe If we are able to calculate quantity (2) we should be able to determine the interval of desirable portfolios.

9. Value at Risk (Cont.) Let is conditional probability density of given Therefore we have Since and is strictly decreasing function on such that there exists a unique such that Then the optimal portfolio is

10. Value at risk (Cont.) The problem is to approximate i.e. the conditional probability density of given. The approximation of will be done at the grid points i.e. Let where we want to find the maximal such that

11. Predictor for Tracking Transitional probability density function of the process defined by (1) is the fundamental solution of the Fokker-Planck equation Therefore to give an estimate of the conditional probability density defined for any nonnegative or bounded Borel function by the following equality we approximate solution of (5) on interval with initial condition

12. Predictor for Tracking (Cont.) For model we have prediction conditional probability density function given by which is defined by This is the so called prediction step.

13. Correction for Tracking For correction in tracking Bayes rule can be applied i.e. Since we have In calculations instead of we take Therefore the problem is how to approximate the solution of (5) as simple and fast as possible.

14. Numerical Schemes Let and For the exact solution at we have where

15. Numerical Schemes (Cont.) We have the following two approximations The approximation in (6) corresponds to Strang scheme by separately solving in given order the following PDE’s

16. Numerical Schemes (Cont.) Generalization of Strang’s scheme is given by for prescribed we have The finite difference corresponding to (8) is given by where

17. Numerical Schemes (Cont.) Let for From the expression for the fundamental solution of (5) (see [3]) we have that the exact solution of (5), with initial condition expressed by and is given by

18. Numerical Schemes (Cont.) Checking Strang’s splitting vs. simple operator splitting. Both schemes converge with increasing m and refinement of space grid. vs. Strang’s scheme shows higher accuracy in time variable t. Strang’s scheme simple operator splitting scheme initial condition true solution

19. Simulation Example 1 Here we consider stochastic volatility model given with Stochastic Differential Equation (1) and Initial conditional probability density function is given by We consider two types of estimators: maximum probability estimator and standard conditional probability estimator (or only prob. estim.) We assume that 10 percent of wealth can be lost with probability at most 0.1 and we invest according to portfolio calculated by (3).

20. Simulation Example 1 (Cont.)

21. Simulation Example 2 For simulated observations we used where and we used the tracker from Example 1with

22. Real Data (from slide Simulation Examples vs. Real Data)

23. Conclusion We have the new method of estimating volatility of stock prices. The method is robust with respect to the model which governs the volatility i.e. it can perform well even though volatility fits better to different model. The method does not require huge historical data to estimate the volatility. It gives conditional probability distribution function of volatility at any time in future for given sigma algebra of observation. This gives us possibility to calculate value at risk (VaR) for any future investment with better precision and accuracy depending on how well model corresponds to reality. The method is satisfactory with respect to VaR calculated portfolio on given simulation and real data examples. It is derived and proposed method for evaluation and comparisons of different models and methods for estimation of volatility.

24. Further possibilities Application of the method to continuous SDE’s which are coming from already known discrete models (e.g. GARCH). Developing suitable correlation tracking techniques for accurate portfolio optimization i.e. possible application of our method to prediction of time-nonhomogeneous correlation coefficients for better assessing investments (diversification). Passively track a stock indexes (DAX). Optimal control of exchange rates and/or price of a stock (modeling diffusion of news into market). Optimal control for portfolio with transaction costs. Option pricing using partial differential equations and in more complex situations stochastic partial differential equations.