Solution

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

{c}

For a payoff $h(S_T,M_T)$ with $M_t=\max_{s\leq t}S_s$, the pair $(S_t,M_t)$ is a sufficient Markov state for valuation. If an independent restarted stock path has initial value $y$, terminal value $z$, and running maximum $w$, then the continued payoff is
$$
g(x,m,y,z,w)=h\left(\frac{xz}{y},\max\left\{m,\frac{xw}{y}\right\}\right).
$$


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 Incoming links: Running-maximum state reduction in the Cox--Ross--Rubinstein model