Derivatives

CHAPTER 21 BOPM: Introduction

Aims

• To demonstrate how the binomial option pricing model (BOPM), is used to determine option premia by establishing a risk-free arbitrage portfolio consisting of a position in stocks and the option – this is delta hedging.
• To show how delta hedging and the no-arbitrage approach can be interpreted in terms of risk-neutral valuation (RNV), which allows us to price options using a simple backward recursion.
• To demonstrate how RNV leads to other useful approaches to pricing options such as Monte Carlo simulation.

We present a detailed account of the BOPM for pricing options using the no-arbitrage principle – the option is priced so that traders faced with this ‘BOPM price’ cannot undertake trades which result in risk-free profits. We construct a risk-free portfolio from two risky assets, namely calls and stocks. Using the principle of delta hedging, whereby the proportions held in stocks and the option gives a risk-free portfolio (over small intervals of time) – we obtain the BOPM formula for the price of the option. We then interpret the BOPM formula in terms of risk-neutral valuation (RNV). Using insights from RNV we can then price an option without going through all the details involved in delta hedging and forming a ‘risk-free arbitrage portfolio’ – instead we price the option directly by using the BOPM formula and ‘backward recursion’.

21.1 ONE-PERIOD BOPM

To understand delta hedging and risk-neutral valuation we first price a European call ...