I am faced with the following problem. Let the standard Brownian motion $W_t$ be the price process of a traded asset in an economy with zero interest rate. Define $$A_T=\frac{1}{T}\int_0^T W_t^2 dt$$
I have two questions:
- What is the fair price at time $t=0$ of a contract that offers $A_T$?
- How do we form a dynamic hedging strategy that eliminates all risk in having to deliver this claim?
I answered part 1 by simply taking the expectation. The fair price is $E(A_T\mid \mathcal{F}_0)=\frac{T}{2}$. How could the dynamic hedging be strategised?