Pricing a Call Option by Replication in a Two-State Binomial Model
Summary
The article derives a no-arbitrage value for a call by constructing a portfolio that combines a long position in the underlying stock with a short call. In its example, the stock starts at 100 and can finish at either 110 or 90; a call with a strike of 100 pays 10 in the up state and zero in the down state.
Equating the portfolio’s values in both states gives a hedge ratio of one half share per written call and a riskless terminal value of 45. With the interest rate assumed to be zero, the initial portfolio must also be worth 45, which implies a call price of 5. The argument does not require the probabilities of the two outcomes; it relies on replication and absence of arbitrage. The result is limited to this one-period, two-state illustration and its stated assumptions, including frictionless fractional holdings and zero interest. The article notes that risk-neutral valuation is a subsequent method that will bring probabilities into the discussion.
Key ideas
- A stock-and-short-call portfolio can be hedged by choosing a quantity that equalizes its value across future states.
- In the example, a hedge ratio of one half makes the terminal portfolio worth 45 in either state.
- Under the zero-interest assumption, the replicated riskless payoff implies a call price of 5.
- The hedge argument derives the price without using probabilities for the up and down outcomes.
- The conclusion depends on the simple one-period model and its idealized trading assumptions.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.