An elementary approach to the option pricing problem
For undergraduate students, this is a pedagogical exposition of known material; it is entirely incremental.
This paper presents an elementary calculus-based derivation of European and American option pricing using the binomial model, aimed at first-year undergraduates. It covers arbitrage, fair pricing, exercise criteria, put-call parity, and bounds, but provides no new results or numerical comparisons.
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.