Pricing an Asian Call with an Average Fixed at an Earlier Date

# Pricing of Asian-like option


# 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:

- 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!)

- 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!

## Answer by Soumirai (score 2)

https://quant.stackexchange.com/a/60131

- 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.

- 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.