Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends (with interest rate r, stock drift $\mu$ and volatility $\sigma$). Let $c(t; St;Gt;K)$ and $p(t; St;Gt;K)$ be the prices at time t of the (continuous) Geometric Asian call option and put option with strike $K$. Find a put-call parity relation for Geometric Asian options. In other terms, and an explicit expression for $c(t; St;Gt;K)-p(t; St;Gt;K)$ (without using $N(.)$, the cumulative distribution function of the standard normal random variable).

So far, this is what I have: $G_T=\exp\{\frac{1}{T}\int_{0}^{T}\log S_udu\}\\ X_T=\frac{1}{T}\int_{0}^{T}\log S_udu\\ G_T=e^{X_T}$

Payoff functions are: $c_{fix}=(G_T-K)^+=(e^{X_T}-K)^+\\ p_{fix}=(K-G_T)^+=(K-e^{X_T})^+\\ c_{fix}-p_{fix}=G_{T}-K$

By risk neutral evaluation: $c_{fix}-p_{fix}=e^{-r(T-t)}E^{Q}[e^{X_T}-K]$.

Hoping to understand how to compute this without the standard normal variable.

Assume the risk-free bond $B_t$ and the stock $S_t$ follow the dynamics of the Black & Scholes model without dividends (with interest rate r, stock drift $\mu$ and volatility $\sigma$). Let $c(t; St;Gt;K)$ and $p(t; St;Gt;K)$ be the prices at time t of the (continuous) Geometric Asian call option and put option with strike $K$. Find a put-call parity relation for Geometric Asian options. In other terms, and an explicit expression for $c(t; St;Gt;K)-p(t; St;Gt;K)$.

So far, this is what I have: $G_T=\exp\{\frac{1}{T}\int_{0}^{T}\log S_udu\}\\ X_T=\frac{1}{T}\int_{0}^{T}\log S_udu\\ G_T=e^{X_T}$

Payoff functions are: $c_{fix}=(G_T-K)^+=(e^{X_T}-K)^+\\ p_{fix}=(K-G_T)^+=(K-e^{X_T})^+\\ c_{fix}-p_{fix}=G_{T}-K$

By risk neutral evaluation: $c_{fix}-p_{fix}=e^{-r(T-t)}E^{Q}[e^{X_T}-K]$.

Hoping to understand how to compute this without the standard normal variable.