Running-maximum state reduction in the Cox--Ross--Rubinstein model
= Running-maximum state reduction in the Cox--Ross--Rubinstein model
{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
$$
g(x,m,y,z,w)=h\left(\frac{xz}{y},\max\left\{m,\frac{xw}{y}\right\}\right).
$$