The payoff depends on the optimal (maximum or minimum) underlying asset's price occurring over the life of the option.
The option allows the holder to "look back" over time to determine the payoff.
As the name introduces it, the option's strike price is floating and determined at maturity.
The floating strike is the optimal value of the underlying asset's price during the option life.
The payoff is the maximum difference between the market asset's price at maturity and the floating strike.
The difference is that the option is not exercised at the price at maturity: the payoff is the maximum difference between the optimal underlying asset price and the strike.
For the call option, the holder chooses to exercise at the point when the underlying asset price is at its highest level.
For the put option, the holder chooses to exercise at the underlying asset's lowest price.
Using the Black–Scholes model, and its notations, we can price the European lookback options with floating strike.
The pricing method is much more complicated than for the standard European options and can be found in Musiela.
[1] Assume that there exists a continuously-compounded risk-free interest rate
Finally, set that Then, the price of the lookback call option with floating strike is given by: where and where
Similarly, the price of the lookback put option with floating strike is given by: Partial lookback options are a subclass of lookback options with the same payoff structure, but with the goal of reducing its fair price.
[2] Thus the payoff is: Selecting specific dates is a more intricate way of creating partial lookback options and other partial path-dependent options.
The principle lies in selecting a subset of monitoring dates, so that the lookback condition is less strong and thus reducing the premium.
Examples include the partial lookback option proposed by Heynen and Kat,[3] and the amnesiac lookback option proposed by Chang and Li.
[4] Discrete partial path-dependent options are overpriced under continuous assumptions; their pricing is complex and is typically performed using numerical methods.