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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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    $\begingroup$ Think of it this way: At any future time $t>t_0$, the value of the option is not only determined by what will happen from that point on into the future, but also by what has already happened in the past between $t_0$ and $t$. $\endgroup$ Sep 9, 2021 at 10:37
  • $\begingroup$ A prime example is the floating strike Asian option which has payoff $\left( \int_0^T S_t dt - S_T \right)_+$ $\endgroup$ Sep 9, 2021 at 11:22
  • $\begingroup$ @Kermittfrog Ok thanks.But why is I independent of S and t? in particular can we say that $\frac{\partial I}{\partial t}=0$ $\endgroup$
    – smbch
    Sep 9, 2021 at 12:11
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    $\begingroup$ My take: For the option, given $I(t)$, the valuation equation is independent of $S_u$ for $t_0\leq u \leq t$. So you do not have to "look back" on the whole history of $S$, but only on the current value of $I(t)$. It is only a state variable to be used for relevant for the option... $\endgroup$ Sep 9, 2021 at 12:13
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    $\begingroup$ @smbch I think Wilmott's point about these three (two spatial, one time) variables is that they need to be treated separately - as in each has it's own dimension, it needs to be tracked or modeled. In a PDE setting, the I dimension depends on S (larger values of S grows I more, etc) but S does not depend upon I. So pricing this option in PDE, you'd have a 2D grid of prices moving through time to make a 3D cube. But of the two spatial, one is diffusive (S) and the other is an auxiliary variable that partly determines the final or boundary conditions. Hope this is helpful. $\endgroup$ Sep 9, 2021 at 16:46

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