16:642:612-02 Selected Topics in Applied Mathematics – Computational Finance (Spring 2007)

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           Prof. Paul M. N. Feehan course 16:642:621

Notation:
   QMDF - Domingo Tavella, Quantitative Methods in Derivatives Pricing: An Introduction to Computational Finance, Wiley 2002, ISBN 0471394475.
   IDM  - L. Clewlow and C. Strickland, Implementing Derivative Models, Wiley, 1998
   MDC - J. London, Modeling Derivatives in C++, Wiley, 2004 
  GL - P. Glasserman, Monte Carlo Methods in Financial Engineering, Springer, 2003 

Topic 2

Pricing american options using a finite difference approach.

GOAL: Using Crank-Nicholson or BDF2 method write a code to price an american call and put options within a time-dependent Black-Scholes model. You consider your free interest rate and volatility to be a given fanction of time. Take them to be piece-wise constant and consider them to change 3 times within the life of the option. So, for instance, from t=0 to t=T/3 r_1 = 5%, from t=T/3 to t=2T/3 r_2 = 6%, from t=2T/3 to t=T r_3 = 7%. Similar to volatility. Implement continuous and discrete dividends. Compare the results obtained with known analytical and numerical solutions, namely. If there are no dividends, the call option value has to concide with the Black-Scholes European call option value, where you as r and \sigma you use an average of the given parameters over time, namely

,

For call and put options an online calculator is available here.

CODE: Matlab or C++

REFERENCES:

1. Samuli Ikonen, Jari Toivanen Operator Splitting Methods for Pricing American Options with Stochastic Volatility. (This is a two-dimensional case, but you can use this technique in one-dimensional case as well. Just assume the second variable to be constant).
2. Recommended papers: paper 1, paper 2, Paul Wilmott book