A path-dependent contingent claim whose payoff depends on the running minimum or maximum of the underlying stock. Floating-strike calls and puts pay and , respectively. Fixed-strike versions use positive parts of the difference between a running extremum and a specified strike. A sufficient pricing state contains the current price and the relevant running extremum.
For running maximum , a smooth price has an Itô formula term . The maximum increases only when . A self-financing portfolio in stock and bank account cannot supply an extra singular finite-variation term there, so the smooth price must satisfy the displayed boundary condition. Inside , it obeys the Black-Scholes equation in the price variable. The condition need only hold before maturity, where terminal corners may destroy differentiability.
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A Lookback option is a type of exotic financial derivative that allows the holder to "look back" over a predefined period of time and exercise the option based on the optimal price of the underlying asset during that period. This unique feature distinguishes Lookback options from standard options. There are two main types of Lookback options: 1. **Lookback with a maximum:** The payoff is based on the difference between the final price of the underlying asset and its minimum price during the life of the option.