Why Path-Dependent Option Pricing Adds an Integral State Variable

Summary

The document asks why a strongly path-dependent option’s value is written as a function of the asset price, time, and an accumulated quantity. The payoff depends on a path summary defined as the time integral of a function of the asset price and time. Since this accumulated quantity reflects the history up to the current time, the option’s value generally needs to track it alongside the current asset price.

Calling the integral a state variable means treating its current value as an additional coordinate in the pricing problem. It is not independent in the sense of being unrelated to time or the asset path: its value is determined by the path so far. Instead, it is an additional variable needed to describe the information relevant to future payoff, beyond the current price and clock time alone. The question provides no specific option payoff, derivation, or numerical example, so it introduces the state-variable idea without showing the resulting pricing equation or boundary conditions.

Key ideas

• A path-dependent payoff can depend on an accumulated quantity computed over the asset’s history.
• The current integral summarizes path information accumulated up to the present.
• A state variable is an additional coordinate needed to describe the option’s value.
• The integral is not unrelated to time or price; it is tracked separately from current price and time.
• The document poses the concept but does not derive a pricing equation for a particular contract.

Tags

# Independent variable in pricing of strongly path dependent options


# Independent variable in pricing of strongly path dependent options












I am reading Paul Wilmott on quantatative finance where he discuss the pricing of strongly path dependent options.The payoff at expiry T depends on the path taken by the asset in the sense that it depends on the path dependent quantity I which can be represented by the integral

$$I(T)=\int_0^Tf(S,\tau)d\tau$$

Therefore he says that the value of the option at any time t is not only a function of S and t but also a function of

$$I(t)=\int_0^tf(S,\tau)d\tau$$

and so we can think of I as a new independent variable called the state variable.I dont understand how is I independent of t and S as from the expression of I it clearly depends on them.

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