Path dependency

In some instances, the future valuation function will only depend on the marketstate
vector at the target time T. When we can write the future valuation function as
a function of mT alone, we say that the future valuation function is path-independent.
For example, consider a zero-coupon bond maturing after time T. The value of
the bond at time T will depend solely on the term structure as of time T and is thus
path independent. It should be recognized that many instruments that are not
commonly thought of as being path dependent are path dependent in the context of
future valuation. For example, consider a standard reset-up-front, pay-in-arrears
swap that resets at time tr and pays at time tp, with t0\tr\T\tp. To perform future
valuation at time T, we need to know the term-structure at time T, to discount the
cashflow, as well as the term-structure as of time tr, to fix the reset. Thus, we see
that the swap is path dependent. Similar reasoning shows that an option expiring at
time te and paying at time tp, with t0\te\T\tp, is, possibly strongly, path dependent.