Here, we will derive formulas for European style power Asian call and put options when we are taking geometric average of the underlying’s price. We will put expectation and variance of the power geometric average into Black’s formula (generalized version), and will simplify it to obtain option formulas.

As we know that the expectation and variance of geometric average is

So, in case of Geometric power Asian option, we have

Where

and

Now, we need to put these two values in Black’s formula (generalized version)

First take

Where

Now, consider

Using

We have

For simplicity, we set

So, now

Hence,

Now, by putting all values in Black’s formula, we obtain the pricing formula for our geometric power Asian option.