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 BlackScholes. 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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 Spring '16
 Math, Normal Distribution, Randomness, Cumulative distribution function, Monte Carlo methods in finance, park miller, double barrier

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