DER 10 – Valuing a Derivative Using a One-Period Binomial Model
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A forward commitment can be priced without any view on where the underlying will trade at expiration, because its payoff is symmetric: both counterparties are locked into the same agreed price whatever the market does. An option is different. Its payoff is asymmetric, since the holder transacts only when it pays to do so, so the value at expiration depends on which future prices are reachable and how far apart they sit. Pricing a contingent claim therefore needs a model of how the underlying price can move, and the binomial model is the simplest one available.
Everything rests on the law of one price. If two positions deliver identical payoffs in every future state, they must carry the same value today; otherwise a trader could buy the cheaper, sell the dearer, and lock in a riskless profit. We will use that principle to solve for the single unknown, the option price today.
Over one period running to the option’s expiration, the model lets the underlying price S0 reach exactly two values: it rises to S1u above S0, or falls to S1d below S0, as the outcome of a single Bernoulli trial. Each move is described by its gross return, the ratio of the new price to the old.
Let q be the probability of an up move and 1 minus q the probability of a down move; with only two outcomes those must add to one. It seems as though q ought to matter for the option value, and yet it does not. All the model needs is the two reachable prices S1u and S1d (equivalently RuS0 and RdS0). The gap between them fixes the spread of outcomes, which is how this model expresses the volatility of the underlying, and volatility is precisely what an option responds to. The real-world probability of each move, and the expected return that it implies, never enter the pricing.
At expiration the option value is a known function of the underlying price. A European call struck at X pays Max(0, ST minus X); a European put pays Max(0, X minus ST). Writing the two reachable call values as c1u and c1d, and the two put values as p1u and p1d, gives everything needed to value the option today.