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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?

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  • $\begingroup$ Did you try using general method where the portfolio has to be a martingale like the one I suggested here? $\endgroup$
    – SBF
    May 30, 2013 at 12:08
  • $\begingroup$ Could you at least disclose that this is some sort of homework. Its incredibly hard to come to any other conclusion. $\endgroup$
    – Matt Wolf
    May 30, 2013 at 13:58
  • $\begingroup$ @MattWolf: This is an exercise problem and not homework. $\endgroup$
    – Bravo
    May 31, 2013 at 16:36

2 Answers 2

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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.

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  • $\begingroup$ I have managed to solve the problem myself. Posting the answer here for reference. $\endgroup$
    – Bravo
    Jun 8, 2013 at 11:48
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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.

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  • $\begingroup$ Why should he differentiate wrt to the starting value - which is assumed to be $0$ ($W_0=0$)? He needs usual Ito calculus. Then in the value process $W_0$ could appear but it is usually zero ... in fact it does not matter that much in this theoretical problem what $W_0$ is. The interesting thing is $dA_t$ which can be derived by Ito calculus. $\endgroup$
    – Richi Wa
    Jun 6, 2013 at 7:17
  • $\begingroup$ But that's mistaking a symbol for its value. The symbolic equation for fair value should have W(0) and hence allow differentiation wrt it. Differentiating wrt to W(0) will give the delta to the traded asset (assumed to be W here). The bottom line is that delta is the change in expected value of the product wrt starting value of the asset. Hence it's always more instructive to derive it in that way. There is no doubt one gets the same answer with Ito, but it's less instructive from a practical standpoint. Maybe I'm missing the point of the exercise, which may be to do Ito after all. $\endgroup$
    – IMK
    Jun 7, 2013 at 14:01

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