HW1_ROSSJ - COMPUTATIONAL FINANCE ASSIGNMENT 1 JOSEPH R...

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COMPUTATIONAL FINANCE ASSIGNMENT 1 JOSEPH R. ROSS, JR. JANUARY 24, 2012 1. Discussion The purpose of this homework assignment was to determine the price of European Call/Put Options using several different analytical methods including the closed-form solution and those of Euler and Milstein. More specifically, the prices were determined using algorithms designed in C++. 1.1. C++ Algorithms. As can be seen in the submitted *.cpp file, the implementation is fairly simple with the user providing the following information at the prompt: Time to Expiry (T) Time of Pricing (t) Strike (K) Spot Price at Time-t (S(t)) Volatility ( σ ) Risk-Free Rate (r) Dividend Rate ( δ ) Furthermore, for implementation of the Monte-Carlo methods, the user was also prompted for the number of price paths as well as the number of time steps. Interaction with the program is directed by a ’switch’ loop, which is user-directed and then calls the necessary member function depending on the selection and whether the price for a call or a put is desired. Using C++, the following algorithms were implemented for determining present value option prices: Closed-Form Solution Path-Independent Monte-Carlo Solution of SDE Euler Monte-Carlo Solution of SDE Euler Monte-Carlo Solution of log SDE Milstein Monte-Carlo Solution of SDE 1.1.1. Closed-Form Solution. Using the closed-form solution to the Black-Scholes equation, func- tions were developed using C++ to price either a European Call or Put, depending on the user input. The functions are identical in methodologies and differ only algebraically. The values for Φ( d 1 ) and Φ( d 2 ) were computed using the Hastings approximation of the Cumulative Distribu- tion Function for standard normal random variables, which was written in C++ by Joshi in the file ’Normals.cpp’. All other computations were completed using the math functions associated with ’cmath’. The following formulas summarize the calculations implemented: V C ( S, t ) = Se - δ ( T - t ) Φ( d 1 ) - Ke - r ( T - t ) Φ( d
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