I am trying to price the Asian lookback option at time $t$ with time-$T$ (European) payoff $\max\{M_T-A_T,0\}$, where $$M_t=\max_{u\in[0,t]}S_u,\quad A_t=\frac1t\sqrt{\int_0^tS_u^2\mathrm{d}u},$$ and $S_u$ is the price of an asset following GBM.

How does one price such a formula? What can we do to show that $V_t(t, s, m, i)$ is a deterministic function of $M_t$ and $I_t=\int_0^tS_u\mathrm{d}u$? Is it because we expect $V_t$ to follow some form of Black-Scholes-like equation due to the European-style payoff, and how can I prove it? Also, I am told that $V_t(t, s, m, i)=sf(t,\frac{m}s,\frac{s}i)$, where should I begin to show this?



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