HW4_BENGALIM

# HW4_BENGALIM - Math 623 Computational Finance Homework...

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Math 623 Computational Finance - Homework Assignment 4 Murtuza Bengali * Department of Mathematics,Rutgers University, 110 Frelinghuysen Road,Piscataway, NJ 08854 This report delves into explaining the working of a C++ program which prices a Vanilla European Call option using four techniques. Firstly, using the closed form implementation of the Black Scholes equation. This price can be considered as the benchmark price to which all the rest of the prices should come close to. Secondly, we use the Park Miller random number generator once with and other without Antithetic variates which reduces variance to price the option. Lastly the Sobol sequence generator is used for the pricing. Given, stock price (s=100),volatility (v=30%), interest rate (r=5%), maturity (T=1 year). I. INTRODUCTION The evolution of the stock price is considered to be Geometric Brownian Motion which determines the value of the option price under various circumstances. Initially in earlier assignments we used to price the option using various different random number generators such as Park Miller and LEcuyer methods. These were then passed through the normalization process such as Inverse Cumulative Normal method among others. Here we instead use the Quasi Random Sequences instead of the pseudo random numbers thus using Quasi Monte Carlo simulation. The Quasi Monte Carlo simulation is used to reduce the computation time and provide higher accuracy with regard to the option pricing. Here we price a Vanilla European Call option using various methods as explained above. The Quasi Random sequences such as Sobol Sequence are more evenly spread across the plane thus rendering a more accurate prediction of the option price using Monte Carlo integrands. Throughout the work, we shall assume to be working in the risk neutral probability measure. II. PRICING THE OPTION USING CLOSED FORM METHODOLOGY The model in a setting of constant volatility, σ , and risk free interest rate, r , is dS ( t ) = rS ( t ) dt + σS ( t ) dW ( t ) (1) where W is a Brownian motion. A European Call is defined as an option which has the payoff given by ( S ( T ) - K ) + (2) K is the strike price and S ( T ) is the terminal stock price at the payoff date. Eq. (1) has a solution of the following form at terminal time, T , [1] S ( T ) = S (0) e r - 1 2 σ 2 T + σW ( T ) (3) The following equation gives the solution with regard to the Closed Form equation for an Vanilla European Call option by [1] c ( t, x ) = xN ( d + ( T, x )) - e - rT KN ( d - ( τ, x )) (4) In the above equations N () stands for Normal Cumulative Distribution whereas d + and d - are as given below: d + = d - + σ T = 1 σ T log x K + r + 1 2 σ 2 T . (5) To calculate the Closed Form solution we wrote a C++ routine which solved 4. For the parameter values provided in the Abstract, we got the value of the European Call as 10.0201. The Normal Cumulative Distribution was used using [3] * [email protected]

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2 III. PRICING THE OPTION USING PARK MILLER RANDOM NUMBER GENERATOR The part of the C++ code deals with generating random numbers using the Park Miller generator. This part of the assignment specifically mentioned no use of the Antithetic variance reduction method to be employed. We know
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• Spring '16
• Math, Randomness, Monte Carlo method, Monte Carlo methods in finance, Quasi-Monte Carlo method, park miller, Sobol Generators

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