Risk-Neutral Pricing of a Call in a Two-State Model
Summary
The article derives risk-neutral pricing for a call option in a one-period, two-state stock model. The stock starts at 100 and can finish at 110 or 90, while the call has a strike of 100. Using the option value established by a preceding no-arbitrage hedge, it identifies an up-state probability of 0.5 that makes the expected option payoff agree with that price. Under this probability, the stock’s expected future value equals its current value, assuming a zero interest rate.
The discussion distinguishes this pricing probability from investors’ actual beliefs: a risk-neutral valuation does not require real buyers to be indifferent to risk. It argues that a replicating hedge removes the risk premium for the hedged position, and notes that a probability of 1 for the up move would permit a risk-free profit in the stated setup. The result depends on the simplified two-state model, zero rates, and the earlier hedging argument; the article points to replication as a further pricing method.
Key ideas
- Risk-neutral pricing chooses probabilities so discounted expected payoffs match no-arbitrage prices.
- In the stated two-state example with zero interest rates, the matching up-state probability is 0.5.
- Under that probability, the stock’s expected future value equals its current value.
- Risk-neutral probabilities are pricing devices and need not represent actual investor beliefs.
- The example relies on a hedgeable two-state market and does not model more complex market conditions.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.