= Running-maximum boundary for a lookback option
{title2=$v_m(t,m,m)=0$}
For running maximum $M_t=\max_{u\le t}S_u$, a smooth price $v(t,S_t,M_t)$ has an <Itô formula> term $v_m\,dM_t$. The maximum increases only when $S_t=M_t$. A <self-financing portfolio> in stock and bank account cannot supply an extra singular finite-variation term there, so the smooth price must satisfy the displayed boundary condition. Inside $0<s<m$, it obeys the <Black-Scholes equation> in the price variable. The condition need only hold before maturity, where terminal corners may destroy differentiability.
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