The Hidden Cost of Volatility Drag: Understanding the Fundamentals of Arbitrage Pricing

The Black-Scholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages (the term will be defined shortly) do not exist in efficient markets. Although this is never completely true in practice, it is a useful basis for pricing theory, and we shall limit our attention (at least for now) to efficient (that is, arbitrage-free) markets.

Why Shouldn’t Arbitages Exist in Efficient Markets?

If one did, it would provide an investment opportunity with infinite rate of return. Investors could, and would, try to use it to make large amounts of money without putting up anything at time t = 0 and without any risk. Since the arbitrage entails buying certain assets at time t = 0, there would be, in effect, an infinite demand for such assets. Economists tell us that this would immmediately raise the prices of these assets, and the arbitrage opportunity would vanish.

Proposition 1: Forward Prices

In an arbitrage-free market, the forward price is F = S0er. This means that if you sell a forward contract at time t = 0, you will receive the underlying asset at time t = 1 for free. To prove this formally, we need to show that either of the alternative possibilities F < S0er or F > S0er leads to an arbitrage.

Suppose F < S0er. Consider the following strategy: Financed Forward Portfolio: At time t = 0, sell 1 share of Stock short. Invest the proceeds S0 in the riskless asset MoneyMarket, and simultaneously enter into a forward contract to buy 1 share of Stock at time 1 at the forward price F. Use the share of Stock obtained from the forward contract at time 1 to settle the short position.

Example: Call Options

A (European) call option is a contract between two agents, a Buyer and a Seller, that gives the Buyer the right to buy one share of an asset Stock at a pre-specified future time t = 1 (the expiration date) for an amount K (called the strike price, or the strike) in Cash.

The call option has a payment at time t = 0: the Buyer pays the Seller an amount V0 in cash at time t = 0. If the Buyer behaves rationally (as we shall assume all agents in the economy do) he/she will exercise the option at expirat

The Fundamentals of Arbitrage Pricing Remain Relevant Today

Despite the absence of arbitrage, the fundamental theorem of arbitrage pricing remains relevant today. This theorem states that in an arbitrage-free market, the market imposes a probability distribution, called a risk-neutral or equilibrium measure, on the set of possible market scenarios.

This probability measure determines market prices via discounted expectation. In other words, investors can determine the fair price of any security by using the expected value of its future cash flows. This insight has far-reaching implications for investment strategies and portfolio management.

What Does it Mean to Hedge?

Hedging involves taking a position in an asset that is opposite in direction to the original asset but with the same magnitude. By hedging, investors can mitigate potential losses or gains from price movements.

In the context of derivatives, hedging means using options, futures, or other instruments to manage risk. For example, if you buy a call option on a stock, you are essentially hedging against potential losses if the stock price falls below the strike price.

Practical Implementation

When it comes to implementing arbitrage strategies, there are several key considerations:

1. Timing: Timing is critical when executing arbitrage trades. Investors need to be aware of market conditions and adjust their strategies accordingly. 2. Entry/Exit Strategies: Investors must carefully select entry and exit strategies to minimize losses or maximize gains. 3. Risk Management: Investors should have a comprehensive risk management strategy in place, including stop-loss orders and diversification.

By understanding the fundamentals of arbitrage pricing and implementing effective hedging strategies, investors can potentially profit from price movements in derivatives markets. However, it is essential to approach these strategies with caution and careful consideration of market conditions and potential risks.

Conclusion

The analysis presented here highlights the importance of understanding the fundamental theorem of arbitrage pricing in the context of derivatives markets. By recognizing the hidden costs associated with volatility drag and implementing effective hedging strategies, investors can potentially profit from price movements.

As we continue to navigate the complexities of derivatives markets, it is essential to stay informed about the latest developments and insights. This analysis provides a comprehensive overview of the subject matter, and we hope that it will prove informative and helpful to our readers.