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.
  • $\begingroup$ Can you plesse elaborate the case of $t \geq \tau$ case? Standing at time $t = 0$, how can I know the stake price fully? $\endgroup$ – Daniel Dec 24 '20 at 16:53
  • $\begingroup$ 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. $\endgroup$ – Soumirai Dec 24 '20 at 17:54
  • $\begingroup$ 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? $\endgroup$ – Daniel Dec 24 '20 at 18:41
  • $\begingroup$ 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 $A_T$ $\endgroup$ – Soumirai Dec 24 '20 at 19:42

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