Running-maximum boundary for a lookback option ($v_m(t,m,m)=0$)

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 $v_{m}d{M}_t$. 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.