Valuing a Derivative Using a One-Period Binomial Model
Refresher reading access
Introduction
Earlier lessons explained how the principle of no arbitrage and replication can be used to value and price derivatives. The put–call parity relationship linked put option, call option, underlying asset, and risk-free asset prices. This relationship was extended to forward contracts given the equivalence of an underlying asset position and a long forward contract plus a risk-free bond.
Forward commitments can be priced without making assumptions about the underlying asset’s price in the future. However, the pricing of options and other contingent claims requires a model for the evolution of the underlying asset’s future price. The first lesson introduces the widely used binomial model to value European put and call options. A simple one-period version is introduced, which may be extended to multiple periods and used to value more complex contingent claims. In the second lesson, we demonstrate the use of risk-neutral probabilities in derivatives pricing.
Learning Outcomes
- explain how to value a derivative using a one-period binomial model, and
- describe the concept of risk neutrality in derivatives pricing.
Summary
- The one-period binomial model values contingent claims, such as options, and assumes the underlying asset will either increase by Ru (up gross return) or decrease by Rd (down gross return) over a single period that corresponds to the expiration of the derivative contract.
- The binomial model combines an option with the underlying asset to create a risk-free portfolio in which the proportion of the option to the underlying security is determined by a hedge ratio.
- The hedged portfolio must earn the prevailing risk-free rate of return; otherwise, riskless arbitrage profit opportunities would be available.
- Valuing a derivative through risk-free hedging is equivalent to computing the discounted expected payoff of the option using risk-neutral probabilities rather than actual probabilities.
- Neither the actual (real-world) probabilities of underlying price increases or decreases nor the expected return of the underlying are required to price an option.
- The one-period binomial model can be extended to multiple periods as well to value more complex contingent claims.
0.75 PL Credit
If you are a CFA Institute member don’t forget to record Professional Learning (PL) credit from reading this article.