Forward-start option

Let the reset date be $T_1$, the expiry $T_2$, and the ratio $\kappa$. The call pays $(S_{T_2} - \kappa S_{T_1})^+$ at expiry. If $\kappa$ is 1, the strike is the price on the reset date, and an at-the-money option starts on that date.

Consider a hypothetical example in which the price on the reset date is 100 yen and $\kappa$ is 1. If the price at expiry is 108 yen, the payoff is 108 − 100 = 8 yen. If the price on the reset date is 120 yen, the strike becomes 120 yen, and a price of 129.6 yen at expiry (an 8% rise from the reset date) gives a payoff of 9.6 yen. If the rise from the reset date is the same, the payoff is proportional to the price on the reset date.

In the Black-Scholes model, the price corresponds to that of a standard call whose term equals the period from the reset date to expiry, but this does not hold in more general models such as stochastic volatility models. One period of a cliquet can be expressed as a combination of these options.