Lookback option

Lookback options, in the terminology of finance, are a type of exotic option with path dependency, among many other kind of options. 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. There exist two kinds of lookback options: with floating strike and with fixed strike.

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. For the call, the strike price is fixed at the asset's lowest price during the option's life, and, for the put, it is fixed at the asset's highest price. Note that these options are not really options, as they will be always exercised by their holder. In fact, the option is never out-of-the-money, which makes it more expensive than a standard option. The payoff functions for the lookback call and the lookback put, respectively, are given by:

where ${\displaystyle S_{max}}$ is the asset's maximum price during the life of the option, ${\displaystyle S_{min}}$ is the asset's minimum price during the life of the option, and ${\displaystyle S_{T}}$ is the underlying asset's price at maturity ${\displaystyle T}$.

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