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A forward start option whose call option strike is fixed at time $T_1$ as $\lambda S_{T_1}$, with expiry $T_2>T_1$ and $\lambda >0$. Its terminal financial payoff depends on the intermediate stock price as well as the final price. Two paths ending at the same price can therefore have different payoffs, so a recombining stock tree must retain enough history to value the contract.

In a homogeneous dividend-free Black-Scholes model, let $c(1,\lambda ,T_2-T_1)$ be the European call option price with current stock one and strike $\lambda$ over the remaining duration. Homogeneity gives value $S_{T_1}c(1,\lambda ,T_2-T_1)$ when the strike is fixed. The discounted stock is a martingale, so risk-neutral pricing before that date givesThe same argument holds in a homogeneous binomial options pricing model with the corresponding remaining-step call value. It depends on the normalized future pricing rule being independent of the stock level at the fixing date.

A financial option grants a right, without a corresponding obligation to exercise, to transact at specified terms. A call option is a right to buy and a put option is a right to sell. The contract specifies its permitted exercise times and strike price; the value of that right differs from the exercise payoff at any particular time.