Introduction to Arbitrage in Options

Arbitrage is a critical concept in the world of financial trading, particularly in operations involving options. This article explores a scenario where we examine the potential for arbitrage between a European call and a put option on the same security, both with an expiration of 3 months and a strike price of $20. Let's delve into the complexities and implications of this scenario.

Understanding the Scenario: European Call and Put Options

To begin, let's define the scenario: You have a European call and a European put option with the same strike price ($20), both expiring in 3 months, and currently trading at $3 each. The nominal continuously compounded interest rate is 10%, and the current stock price is $25. This scenario raises questions about potential arbitrage opportunities, and we need to explore these questions carefully.

Identifying Implied Forward and Dividend Yield

When both calls and puts trade at the same price, it implies an underlying forward price. Using the provided details, the implied forward price is $20. This means that if one were to purchase the underlying stock today and hold it until expiration, the expected future value of the stock would be $20, reflecting the interest rate and time value.

Via mental calculation, we can derive the implied dividend yield as follows: Since the stock price is currently $25 and the strike price is $20, with a forward price of $20, the implied dividend yield is roughly 100%. This high dividend yield suggests that if one were to buy the option for $3 and hope to exercise it in 3 months at $20, the expected yield is 100%. This is a rough calculation and provides a regulatory framework to compare the options against purchasing the stock directly.

However, the key question remains: what is the volatility of the stock? If the stock price ends up at $0 in 3 months, you might have been better off purchasing the put option. The point is to consider the risk profile associated with both strategies.

Why the Dividend Yield Could Be 100%

The idea of a 100% dividend yield suggests that the company might be highly profitable or the stock might be undervalued. In such a scenario, the call might indeed be a better investment. Conversely, it could also imply a Ponzi scheme scenario where dividends are artificial, and the stock price might collapse.

Given the high dividend yield, it is crucial to assess the underlying factors contributing to this yield. A company with stable profits or an undervalued stock might justify such a high yield, but if it is artificially maintained, the risk associated with the stock's volatility increases significantly.

Exploring the Potential for Arbitrage

Now, let's explore the potential for arbitrage in this scenario. The scenario also illustrates that shorting the security, buying a call, and writing a put is a common strategy. Here's how it works:

Short the security at $25, buying a call option at $3, and selling a put option at $3. Net income is $25 - $3 - $3 $25 - $6 $19. Depending on your margin arrangements, you will have to post between 5% and 45% margin. With good terms, you’ll earn close to 10% interest on your margin, growing your $25 to nearly $25.60 by expiry. With poor terms, you won't earn any interest, and you might have to borrow at high interest rates, reducing your earnings. At expiry, if the price is above $20, you exercise the call and buy the security. If below $20, the put is exercised, and you buy the security. Either way, you close out the short position, ending up with a net income of $4.40 to $25.60 minus any distributions you had to pay.

This simple strategy outlines a possible arbitrage opportunity. It's important to note that such strategies are subject to various market conditions and risk factors, including interest rates, margin costs, and stock volatility.

Conclusion

The scenario involving a European call and put option with the same strike price and expiration date presents a rich landscape for exploring arbitrage. Understanding the implied forward and the potential for high dividend yields is crucial. However, the volatility and risk associated with the underlying stock are key factors that must be considered.

For those interested in deepening their knowledge of options and arbitrage, further exploration of concepts like implied volatility, forward prices, and volatility skews is essential. Financial markets are complex, and a well-informed approach is necessary to navigate these opportunities successfully.