An elementary approach to the option pricing problem

TL;DR

This paper presents an accessible, elementary approach to option pricing using the binomial model, suitable for undergraduate students, covering European and American options, arbitrage, and portfolio optimization.

Contribution

It introduces a simple, calculus-based method for pricing options and explains key concepts like arbitrage and exercise criteria in an educational manner.

Findings

Derivation of put-call parity for European and American options

Bounds for option prices established

Decision criteria for exercising American options

Abstract

Our goal here is to discuss the pricing problem of European and American options in discrete time using elementary calculus so as to be an easy reference for first year undergraduate students. Using the binomial model we compute the fair price of European and American options. We explain the notion of Arbitrage and the notion of the fair price of an option using common sense. We give a criterion that the holder can use to decide when it is appropriate to exercise the option. We prove the put-call parity formulas for both European and American options and we discuss the relation between American and European options. We give also the bounds for European and American options. We also discuss the portfolio's optimization problem and the fair value in the case where the holder can not produce the opposite portfolio.