Pricing weighted/average stock price claim

Question

In a market consisting of a bank account with a constant interest rate r and a non-dividend paying stock S, consider a T-claim that pays $X = S(T)/S(T_0)$ at time T, where $T_0 < T$.

a) Find a replicating strategy for X.

b) What is the arbitrage-free price of X at time 0?

Is there a name for this sort of problem, or a "general" approach that I can study? I am comfortable with replicating strategies for linear combinations, but I am not sure how to approach it with quotients and products.

Answer 1

It's not easy to give hints without giving away the whole solution. But here's a try:

1. Forget about Black Scholes, I see you tagged the question like that, but this is irrelevant. The problem is much simpler.
2. Try to divide the problem in periods. Here, there's basically three time points: $t=0$, $t=T_0$ and $t=T$ with $0<T_0<T$. You have to do something at each of those points such that at the end your portfolio has value $S(T)/S(T_0)$.
3. $S(T)/S(T_0)$ is a linear combination.
4. For the second part, once you found the price of your replication, argue how you have an arbitrage opportunity if the price of the claim $X$ is lower than the price of the replication. Again take into account the time points. Then do the same for the price of $X$ higher than the price of the replicating portfolio.

Hope this helps.