HW6 - Math 623 Homework Assignment 6 Shwetabh Singh...

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Math 623 - Homework Assignment 6 Shwetabh Singh * Department of Mathematics,Rutgers University, 110 Frelinghuysen Road,Piscataway, NJ 08854 We write a C++ program to calculate the value of European vanilla and double barrier options on an underlying asset, given certain standard input parameters for the volatility, initial stock price, risk free interest rate, dividend yield, strike price, and maturity. The volatility, interest rate, and dividend are assumed to be constant and the underlying model is assumed to be Black-Scholes. The program calculates the option prices using the closed form formula for the vanilla European call, Monte Carlo simulations, and Quasi Monte Carlo techniques for both the vanilla and double barrier call. For our Monte Carlo simulations we used the Park Miller random number generator with antithetic and stratified sampling from a Brownian bridge. For the Quasi Monte Carlo simulations, the Sobol sequence generator was employed. The results were benchmarked using the Numerix software and the results were in broad agreement with the ones generated by our methods and internally consistent in their agreement with the values obtained from closed form solutions for the vanilla European call. I. INTRODUCTION Stock price evolution is modeled as a geometric Brow- nian motion. This fundamental idea is extended to price options by the application of the fundamental theorems of asset pricing and the principle of no arbitrage [1]. An- alytical solutions to calculate the option prices exist only for the simplest of options like the Call and Put. Thus, from a practical standpoint it is imperative to design and implement numerical solutions and techniques to price options. A basic overview of these numerical techniques and their implementation can be found in [2] and [3]. In this homework, we calculate the price of a vanilla and double barrier European call using several different analytical and numerical techniques. In section II we use the closed form formula to calculate the value of a vanilla European call. We provide a brief overview of European double barrier options. We then proceed to calculate the price of a vanilla European Call and double barrier Eu- ropean using Monte Carlo simulations with the random numbers generated by the Park Miller algorithm with antithetic sampling in section IV[2]. In section V, we price these options using the Park Miller algorithm cou- pled with stratified sampling generated using a Brownian bridge. In section VI, we price the same options using Quasi Monte Carlo techniques and generate our sequence from the Sobol generator. Finally, we benchmark our so- lutions using the Numerix option pricer and discuss some of the implications of our results. Throughout the work, we shall assume to be working in the risk neutral proba- bility measure.
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