Replicating a Claim on the Time Average of Squared Brownian Motion

# Dynamic hedging strategy example


# Dynamic hedging strategy example












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?

## Answer by Bravo (score 5, accepted)

https://quant.stackexchange.com/a/8190

Consider a dynamic hedging strategy where you invest $H_t$ in the stock at time $t$. To eliminate all risk, the value of the investment must be equal to the claim at time $T$. Using Ito's calculus, we could express $A_T$ as follows:

$$A_T=\frac{T}{2}+\int_0^T 2W_t \left(1-\frac{t}{T}\right) dW_t=\frac{T}{2}+\int_0^T H_tdW_t$$

Thus the strategy would be to start with an amount $T/2$ (fair price at $t=0$) and invest $H_t=2W_t(1-t/T)$ dynamically in the stock.

PS: This is a special case of the Black-Scholes setup where the interest rate $r=0$. If $X_t$ is the value of the holding and $S_t$ is the stock price, the value of $dX_t$ is $dX_t=H_tdS_t+(X_t-H_tS_t)rdt$. $H_t$ is the amount to be investment dynamically in the stock, and is also known as the delta of the option.

## Answer by IMK (score 0)

https://quant.stackexchange.com/a/8171

Your fair price formula is not general enough. You need to make W(0) appear, then differentiate wrt it. This will be your delta. You assumed W(0) = 0 and got rid of it, and now you're stuck with nothing to differentiate wrt.