I could need some help with deriving the put-call-parity for asian options.
Let $S_t$ be the price of the underlying asset at time $t$ and set $Y_t = \int_0^t S_t dt$. Then the payoff of an asian option at expiration date $T$ is
$Payoff = \left( \frac{Y_T}{T} - K \right)^+$.
Now let $C(t)$ be the asian call value, $P(t)$ the asian put value. Then, if i 'm correct, at expiration date i get the equation:
$C(T)-P(T) = \left( \frac{Y_T}{T} - K \right)^+ - \left( K - \frac{Y_T}{T} \right)^+ = \frac{Y_T}{T} - K$
Now if I discount this back to $t=0$ I get
$C(0)-P(0) = e^{-rT} \mathbb{E} \left[ \frac{Y_T}{T} - K \right]$
My Question: Is this correct? Can I say more about this expectation?