# Pricing of Asian-like option

I am considering an option which has payoff function $$\max\{S_T-\frac1\tau\int_0^\tau S_t\mathrm{d}t,0\}$$ for a fixed $$\tau$$ in the risk-neutral measure $$\mathrm{d}S_t/S_t=r_t\mathrm{d}t+\sigma_t\mathrm{d}W_t^\mathbb{Q}$$. I have a few questions:

1. What is the name of this kind of option? This looks like an arithmetic average floating strike Asian call, but if I recall correctly for usual Asian options the integral runs from $$0$$ to $$T$$ instead of $$\tau$$. (Please let me know if I have missed something on this SE, I’ll remove this question if it is redundant!)
2. Does the price of this option have a closed form solution? I know the conventional arithmetic Asian call does not, which is why I am quite hesitant to go through the potential rabbit hole to solve for $$\mathrm{e}^{-r(T-t)}\mathbb{E}^\mathbb{Q}(\max\{S_T-\frac1\tau\int_0^\tau S_t\mathrm{d}t,0\}|\mathcal{F}_t).$$ I'm assuming there should be different considerations for $$t\in[0,\tau)$$ and $$t\in[\tau,T)$$.

Any guidance is appreciated!

1. It would be an Asian strike call option, with the Asianing being computed over some period $$[0;\tau]$$. Not a big deal that the average is not computed from 0 to T. It even seems more natural to be done that way in practice.
2. Before $$\tau$$, pricing would be similar to your option Asianing until T. For $$t>=\tau$$, you already know the value of the Asian strike though (it is $$F_\tau$$-measurable), so its pricing would be like any European call.
• Can you plesse elaborate the case of $t \geq \tau$ case? Standing at time $t = 0$, how can I know the stake price fully? – Daniel Dec 24 '20 at 16:53
• At $t=0$ you can't. But after time $\tau$ the integral $\int_0^\tau S_tdt$ is known so it is just a constant. – Soumirai Dec 24 '20 at 17:54
• Thanks for the clarification. So at $t=0$, if I want to price this option then how should I use regular Asian valuation? Or other method? – Daniel Dec 24 '20 at 18:41
• What do you call "regular Asian valuation"? If you're talking about the Markovian technique of using an additional asset $A_t$ that represents the average, you could apply it to price the payoff $(S_T - A_\tau)^+$. Just adapt whatever you do to account for the fact that your terminal payoff contains $A_\tau$ not $A_T$ – Soumirai Dec 24 '20 at 19:42