A path-dependent option in which the final outcome (payoff) depends on the price events over the entire course of the option’s life, and up to its expiration date/ exercise date. Examples of this option include average options or Asian options or lookback options where the value depends on every step taken by the underlying asset during the life of the option. Strongly path-dependent options can be classified as dense path-dependent options and light path-dependent options.

This type of option is also known as a hard path-dependent option or a path-dependent option with strong memory.