= Forward-start call option
{title2=$(S_{T_2}-\lambda S_{T_1})^+$}
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 gives
$$
V_t=S_t c(1,\lambda,T_2-T_1),\qquad t<T_1.
$$
The 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.
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