Replicating a Stock-Price Ratio Claim Across Two Periods
Summary
The document presents a derivative that pays the ratio of a stock’s future price to its price at an earlier date. It frames the problem as constructing a replicating portfolio across three times: today, the intermediate observation date, and maturity. The suggested approach is to divide the strategy into periods and choose actions at the intermediate date so the terminal portfolio value matches the ratio payoff.
For pricing, the answer points to the no-arbitrage replication principle: if the claim’s price differs from the replicating portfolio’s cost, a trading strategy can exploit the discrepancy. The response offers hints rather than a worked solution, and it does not state the explicit holdings or price. Its setup assumes a constant-rate bank account and a non-dividend-paying stock; no broader pricing framework or empirical evidence is supplied.
Key ideas
- Break the replication problem into the dates when the payoff’s denominator and numerator are observed.
- Choose the intermediate portfolio so its terminal value produces the required stock-price ratio.
- Use the replication cost as the no-arbitrage benchmark for the claim’s price.
- The document gives a conceptual hint rather than explicit holdings or a computed price.
Tags
Full text
# Pricing weighted/average stock price claim # Pricing weighted/average stock price claim > 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 by Raskolnikov (score 1, accepted) https://quant.stackexchange.com/a/37575 It's not easy to give hints without giving away the whole solution. But here's a try: - Forget about Black Scholes, I see you tagged the question like that, but this is irrelevant. The problem is much simpler. - 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 $S(T)/S(T_0)$. - $S(T)/S(T_0)$ is a linear combination. - 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.
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