Risk-Neutral Binomial Pricing with Positive Interest Rates

Summary

This note extends the one-step binomial option model from zero interest rates to a positive continuously compounded risk-free rate. It bounds the stock’s possible up and down prices around risk-free growth, then chooses a risk-neutral probability that makes the expected stock value grow at that rate. The probability must lie between zero and one for the model to avoid arbitrage.

Because a risk-free bond grows over the step, asset values are compared after discounting rather than by requiring their undiscounted expectations to equal current prices. The option value is then the discounted risk-neutral expectation of its payoff, justified by forming a zero-cost portfolio whose risk-neutral expected discounted value is zero. The discussion introduces risk-neutral pricing and notes replication and hedging as alternative approaches, but does not work through those methods or a numerical example. Its scope is a single time step and a two-state stock model.

Key ideas

• A positive risk-free rate makes the bond grow over each binomial time step.
• The risk-neutral probability is chosen so the stock’s expected growth matches the risk-free rate.
• No-arbitrage requires the risk-neutral probability to lie strictly between zero and one.
• Derivative values are obtained from discounted risk-neutral expected payoffs.
• Replication and hedging are named as alternative pricing approaches but are not developed here.

Tags

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.