Path dependency is a characteristic of certain financial derivatives whose value depends not only on the underlying asset’s price at expiration but also on the path of the underlying asset’s price throughout the life of the derivative. In other words, path-dependent options take into account the entire price history of the underlying asset rather than just its final price at expiration. This means that the value of such derivatives can be influenced by the sequence of price movements and the specific events that occur during the contract’s lifetime.
One common example of a path-dependent option is the Asian Option. An Asian Option, also known as an average price option, is an option whose payoff is determined by the average price of an underlying asset over a predefined period of time. The average can be calculated as either an arithmetic or geometric mean, and the averaging period can be set daily, weekly, monthly, or any other customized period depending on the contract.
Let’s consider a simple example of a European-style Asian call option:
Let St denote the price of the underlying asset at time t. Suppose we have a European-style Asian call option on this asset with expiration at time T, and the strike price is K. The averaging period is from t = 0 to T, and the average price is computed using the arithmetic mean:
$$A_T = \frac{1}{T} \int_0^T S_t \ dt$$
At expiration, the payoff of the Asian call option is determined by the difference between the average price AT and the strike price K, similar to a regular call option. However, if the difference is negative, the payoff is set to zero (since it’s a call option):
Payoff = max(AT − K, 0)
Because the payoff depends on the entire price path of the security through the averaging period, Asian options are path dependent. For instance, the value of the option will be influenced by events, such as price jumps or extreme price movements, that happen during the contract’s lifetime.
In contrast, a European call option’s payoff depends only on the underlying asset’s price at expiration, not on the path of prices between purchase and expiration:
Payoff = max(ST − K, 0)
As a result, European options are not path dependent.
Path-dependent options generally require more complex models and computation techniques (e.g., Monte Carlo simulations) for pricing and risk management, since they incorporate additional information about the price path of the underlying asset.