Explanation

In the context of derivatives and hedge portfolios, the fundamental principle of no-arbitrage states that a perfectly hedged portfolio should earn the risk-free rate of return. Here's why:

Key Concepts:

1. Hedge Portfolio: A portfolio constructed by combining underlying assets with derivatives (like options) to eliminate risk.
2. No-Arbitrage Principle: In efficient markets, there should be no opportunity to earn risk-free profits without any initial investment.
3. Risk-Free Rate: The theoretical return on an investment with zero risk, typically represented by government bonds.

Why Option B is Correct:

• When a portfolio is perfectly hedged, all market risk is eliminated through the derivative positions.
• Since there is no risk remaining, the portfolio should earn the risk-free rate of return.
• If it earned more than the risk-free rate, arbitrageurs would exploit this by borrowing at the risk-free rate and investing in the hedge portfolio, earning risk-free profits.
• If it earned less than the risk-free rate, arbitrageurs would short the hedge portfolio and invest at the risk-free rate.

Why Other Options are Incorrect:

• Option A (zero return): Incorrect because even risk-free investments earn some return. Zero return would create arbitrage opportunities.
• Option C (weighted average return): Incorrect because a hedge portfolio eliminates systematic risk, so it shouldn't earn the average return of the underlying assets, which includes risk premiums.

Mathematical Foundation:

This principle is fundamental to derivative pricing models like the Black-Scholes model, where the risk-neutral valuation approach assumes that all assets earn the risk-free rate in a risk-neutral world.

Real-World Application:

This concept is crucial for understanding how options and other derivatives are priced and for constructing arbitrage-free trading strategies.