Explanation

In the context of derivatives and arbitrage-free pricing, a hedge portfolio that is constructed to be risk-free (by combining underlying assets and derivatives in appropriate proportions) should earn the risk-free rate of return in the absence of arbitrage opportunities.

Key Concepts:

1. Arbitrage-Free Pricing: In efficient markets, risk-free portfolios should earn the risk-free rate. If they earned more, arbitrageurs would exploit this by borrowing at the risk-free rate and investing in the hedge portfolio, earning risk-free profits.
2. Hedge Portfolio Construction: A hedge portfolio combines underlying assets and derivatives (like options) in such a way that the portfolio's value becomes insensitive to changes in the underlying asset price, making it risk-free.
3. Risk-Free Return: Since the portfolio is constructed to eliminate all risk (through perfect hedging), it should earn the risk-free rate, not zero return.

Why Other Options Are Incorrect:

• Option A (zero return): Incorrect because even risk-free investments should earn at least the risk-free rate, not zero.
• Option C (weighted average return): Incorrect because a properly hedged portfolio eliminates the risk associated with the underlying assets, so it shouldn't earn their weighted average return.

Real-World Application:

This principle is fundamental to derivative pricing models like the Black-Scholes model, where the risk-neutral valuation approach assumes that a hedged portfolio earns the risk-free rate.