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.
Binomial means two. The model looks at one period only and assumes the underlying price ends in one of exactly two states, an up price or a down price. This simple set-up lets you copy the option's payoff exactly with a mix of the underlying and borrowing or lending. Since two payoffs can be matched with two instruments, the no-arbitrage price is a single number. A normal distribution is a different, continuous-price assumption.
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.
You can build a portfolio of the underlying and borrowing that pays exactly the same as the option in both the up and down states. This is called replication. Two things that pay the same in every state must cost the same today, whatever the chance of each state. Option value = Cost of the replicating portfolio. That cost depends only on the two payoffs, today's price and the interest rate, not on the real probability of an up move.
The hardest tier
Why does exact replication force the option's price?
- Because the exchange requires itWhy not: No exchange enforces a theoretical value; the discipline comes from the trade being available to anyone.
- Because the replicating portfolio is cheaper to tradeWhy not: Relative trading costs are irrelevant to the argument, which turns on the payoffs being identical.
- Because identical payoffs must cost the same - correct
Replication means building a portfolio that pays exactly the same as the option in every outcome. Rule: identical payoffs must cost the same. If the option were cheaper than the portfolio, you could buy the option, sell the portfolio and keep the difference with no risk. If it were dearer, you would do the reverse. Such risk-free profits are called arbitrage, and trading would remove them. The same argument underlies the option bounds and put-call parity.
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