1. A Finite Differencing Solution for Evaluating European Prices Computational Finance ~cs 757 Project # CFWin03-33 May 30, 2003 Presented by: Vishnu K Narayanasami vishnun@cs.umanitoba.ca

2. May 30, 2003cs757--Winter 20032 Outline Introduction Problem Statement Background Methodology Experimental Analysis and Results Future Work

3. May 30, 2003cs757--Winter 20033 Introduction European Call Options  The buyer pays a one-time premium for buying a particular stock at a particular rate.  The option can be exercised only at maturity. Finite Differencing  The basic method of solving differential equations in a computer.  Allows analysis of all kinds and shapes of objects.

4. May 30, 2003cs757--Winter 20034 Problem Statement Evaluating prices of options is of practical importance and is a tedious task. Complicated finance problems lead to complex coupled equations. Closed form solutions of these equations are almost impossible. With finite differencing techniques, it is possible to model the problem and achieve better computational results. Implementing Crank-Nicholson scheme for evaluating European options

5. May 30, 2003cs757--Winter 20035 Background Black-Scholes model: Calculate theoretical call price based on five parameters: strike price, stock price, volatility, time of expiration and short-term risk free interest. Black-Scholes equation is given as

6. May 30, 2003cs757--Winter 20036 Background Crank-Nicholson Finite Differencing Technique: Implicit method Uses Central-differencing System of linear equations Unconditionally stable Values of unknowns are assigned to the grid points.

7. May 30, 2003cs757--Winter 20037 Major Difference Many finite differencing schemes: Explicit methods, Classical Implicit, Implicit, etc. Crank-Nicholson method is fully implicit. Second order accurate in time whereas other schemes are first order accurate Unconditionally stable; uses central differencing; space derivative at time level n = ½ (mean of other methods).

8. May 30, 2003cs757--Winter 20038 Methodology

9. May 30, 2003cs757--Winter 20039 Methodology (2) Discretized Black-Scholes equation: Whereis the timestep andis the distance between the nodes.

10. May 30, 2003cs757--Winter 200310 Methodology- Pseudo Code (3)

11. May 30, 2003cs757--Winter 200311 Methodology (4) pmpdpu Option Value Nj N Maturity values

12. May 30, 2003cs757--Winter 200312 Challenges To bridge finite differencing concepts to suit finance problems. I have already worked in finite differencing on a fluid mechanics problem, and it was complicated to let go my fixed mindset on that field and switch to finance. Implementing the boundary conditions for this project. Experienced memory problems during execution of the code.

13. May 30, 2003cs757--Winter 200313 Experimental Analysis – testbed The code was developed in Java and tested in the Linux machines in the Department of Computer Science with the following configuration: Model NameIntel ® Pentium III CPU CPU MHz1000.065 Cache Size256 KB Memory526 MB

14. May 30, 2003cs757--Winter 200314 Experimental Analysis – Results (2) I could observe the general trend from the results that as the Strike Price (K) increases, the Option value decreases for European options, as in our assignment problem. I am still refining the code to achieve other results such as: varying strike price over option values, increasing no. of levels, time steps, analyzing execution time.

15. May 30, 2003cs757--Winter 200315 Continuing Work Making the code work completely to achieve satisfactory results Testing the code for differing parameters.

16. May 30, 2003cs757--Winter 200316 Future Work Comparison with other finite differencing schemes Parallel implementation

17. May 30, 2003cs757--Winter 200317 Thank you! To add value to the capital-raising and asset- management process by providing the highest- quality and most cost- effective self-regulated marketplace for the trading of financial instruments, promote confidence in and understanding of that process, and serve as a forum for discussion of relevant national and international policy issues.