Price an option with a single-period binomial model
The idea
The one-period binomial model assumes the underlying can take only two values at expiry. A portfolio of the underlying and borrowing can be constructed to replicate the option's payoff in both states, and because the replication is exact, no arbitrage forces the option to cost what the portfolio costs. Equivalently, the option can be valued by taking the expected payoff under risk-neutral probabilities and discounting at the risk-free rate. The actual probability of an up move never enters the calculation, which is the model's most surprising and most important result.
Formulas
One-period binomial option value
option value = [π × value up + (1 − π) × value down] ÷ (1 + r)
- π
- the risk-neutral probability of an up move
- value up
- what the option pays if the underlying rises
- value down
- what it pays if the underlying falls
- r
- the risk-free rate for the one period
Expected payoff under the risk-neutral probabilities, discounted at the risk-free rate. The real probability of a rise never enters, which is the part that feels wrong until the replicating portfolio explains it.
A practice question
What does the one-period binomial model assume about the underlying at expiry?
- It follows a normal distributionWhy not: That is the continuous-time assumption behind the Black-Scholes framework; the binomial tree uses discrete states.
- It takes one of exactly two possible values — correct
- It remains at its current priceWhy not: Then there would be no uncertainty and nothing to value.
Two states is the simplifying assumption that makes exact replication with two instruments possible, and therefore makes the no-arbitrage price unique.
A harder one
In the binomial model, why does the true probability of an up move not affect the option's value?
- Because the payoff can be replicated exactly — correct
- Because the probability is assumed to be one halfWhy not: No such assumption is made; the actual probability drops out of the argument entirely, whatever it is.
- Because the option is valued at expiry rather than todayWhy not: The valuation is today's, discounted back from the two possible expiry payoffs.
The replicating portfolio's cost does not depend on that probability. If two positions have identical payoffs in every state they must cost the same today, regardless of how likely each state is, so pricing by replication removes the probability from the problem.
The hardest tier
An underlying at 100 can rise to 120 or fall to 90. A call struck at 100 is valued with a risk-free rate of zero. What is the risk-neutral probability of the up move, and the option's value?
- Probability 0.5, value 10Why not: Uses an even split rather than the probability that makes the expected underlying price equal today's 100. That requires one third, not one half.
- Probability 0.4, value 6Why not: Internally inconsistent as well as wrong: a probability of 0.4 against a payoff of 20 would give 8, and a value of 6 would need a probability of 0.3.
- Probability one third, value 6.67 — correct
With a zero rate the risk-neutral probability solves 120p + 90(1-p) = 100, giving p = 1/3. The payoff is 20 up and 0 down, so the value is 20/3 = 6.67.
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