TITLE:
Convolution Rather Than Monte Carlo Simulation to Price a Barrier Option
AUTHORS:
Daniel T. Cassidy
KEYWORDS:
Option Pricing, Convolution, Truncation, Student’s t Distribution, Fat Tails, S&P 500 Returns
JOURNAL NAME:
Journal of Mathematical Finance,
Vol.15 No.3,
August
18,
2025
ABSTRACT: Repeated convolution and truncation of a truncated fat-tailed distribution, instead of Monte Carlo simulation, for pricing a discrete, simple barrier option is presented. The parameters for the truncated fat-tailed distribution are obtained by fitting the sum of a Student’s t distribution plus a normal distribution to the one-day returns obtained from the adjusted closing values for the S&P 500. It is argued that truncation of the fat-tailed, one-day returns distribution is a physically reasonable action and evidence to support this truncation is provided. The steps to price a discrete up-and-out barrier on an European call option through repeated convolution and truncation are given.