Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 29K / b

The arbitrage-free value of the discounted claim is the conditional expectationConditional on the observed path $(S_0^1,\dots ,S_t^1)=(x_0,\dots ,x_t)$, the future multiplicative increments are independent of ${\mathcal{F}}_t$ and have the same law as a fresh stock path divided by its initial value. ThereforewhereThis is Risk-neutral valuation in the Cox--Ross--Rubinstein model for a path-dependent claim.