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

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

May 30, 2003cs757--Winter 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.

May 30, 2003cs757--Winter 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

May 30, 2003cs757--Winter 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

May 30, 2003cs757--Winter 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.

May 30, 2003cs757--Winter Major difference

May 30, 2003cs757--Winter Methodology

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

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

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

May 30, 2003cs757--Winter Challenges

May 30, 2003cs757--Winter 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 MHz Cache Size256 KB Memory526 MB

May 30, 2003cs757--Winter 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: ………. ……… ……… I have used the graph (from the assignment) as an example to show the above result.

May 30, 2003cs757--Winter Experimental Analysis (3) Assumptions: S=25, sigma=0.34, T=1, volatility=30%, dx=0.2, N,Nj= 3

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

May 30, 2003cs757--Winter What more could be done Comparison with other finite differencing schemes Parallel implementation

May 30, 2003cs757--Winter 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.