One of my first jobs after graduating was with a trading options firm, where I developed mathematical strategies for trading options. So, first things first: financial options are these nifty derivative instruments that give you the right (but not the obligation) to buy or sell an asset at a predetermined price within a specific timeframe. Sounds simple, right? Well, it’s a bit more complicated than that. Let’s dive into a basic scenario to break it down.
The Risk-Free Profit Puzzle
To understand this better, let’s consider a simple scenario. Suppose the market price of an asset is $1000, and an option to buy this asset at the market price costs $50. For simplicity, we’ll assume the asset price in the future could be either $1200 or $800, but we don’t know the probability of each outcome. You can choose to buy or sell the asset based on this information.
If we examine this scenario closely, we might realize a way to earn $100 in both possible future price outcomes, effectively creating a risk-free profit. This can be done using the following strategy:
Sell the Asset: Sell the asset at the current market price of $1000.
Buy Two Call Options: Buy two call options with a strike price of $1000 for $50 each. These options give you the right to buy the asset for $1000, regardless of the future market price.
This way, you have $900 in cash, two call options, and an obligation to buy the underlying asset.
Outcome Analysis:
Here’s how the strategy works in both future price scenarios:
If the price rises to $1200:
- The call options are exercised, and you sell the asset at $1000. Namely, you make $400.
- You need to buy the underlying asset at $1200.
- Your profit is $400 plus the $900 in your hand, minus $1200 for the underlying asset. Namely, you made $100.
If the price falls to $800:
- Sell the underlying asset at $800.
- The options would not be exercised.
- Your profit is $900 in your hand minus the $800 for the underlying asset. Again, you make $100.
Regardless of whether the price rises or falls, you pocket $100. This, my friends, is arbitrage — a risk-free profit.
Now, before you rush off to try this, remember: true arbitrage opportunities are as rare as hen’s teeth. Why? Because everyone would jump on them, driving up prices until the opportunity vanishes. This adjusted price, where no risk-free profit exists, is called the arbitrage-free price.
The Black-Scholes-Merton Framework
Taking this principle, Fischer Black, Myron Scholes, and Robert Merton derived their famous differential equation for option pricing. Although they employed sophisticated tools such as Ito’s Lemma and other stochastic analysis techniques, the foundational financial assumption they used is based on the principle of no arbitrage.
In this discussion, we won’t delve further into the equation itself. Moreover, we will sidestep the controversial debates surrounding other assumptions used in their analysis. Our focus remains on the intuitive understanding of arbitrage and its implications for financial markets. By grasping these fundamentals, you can better appreciate the complexities and elegance of financial models and strategies, even without getting into the heavy mathematical details. Understanding these principles is crucial for anyone looking to navigate the financial markets effectively.
The Black-Scholes model assumes known volatility for pricing options, but this is rarely the case in practice. However, when the option price is known, we can back-calculate the volatility required for the model to hold true. This calculated value is known as implied volatility. It is important to note that, by using the Black-Scholes model, we can see that the strike price is directly associated with implied volatility.
Observations on Option Pricing
As a mathematician in the trading options firm, one of my tasks was to analyze the option pricing curve. Here are two key observations from that analysis:
Option Price Relative to Underlying Asset: The option price is always less than the price of the underlying asset. This might seem obvious, but it’s important. If the option price were ever higher than the asset price, traders would simply buy the underlying asset and sell the option, guaranteeing a profit. If the underlying asset price exceeds the strike price, the option would be exercised, and the trader would not incur a loss. Conversely, if the option is not exercised, the trader still owns the underlying asset.
Option Price and Strike Price Relationship: The price of an option is a monotonic decreasing function of the strike price. This is based on the fact that if an option is exercised, the profit is the difference between the underlying asset price and the strike price. As the strike price increases, the potential profit decreases, leading to a lower option price. Thus, higher strike prices correlate with lower option premiums.
Avoiding Butterfly Arbitrage
Now, let’s flutter into the world of butterfly arbitrage. This strategy involves exploiting mispriced options to create a risk-free profit using a combination of options with different strike prices. It’s named after the shape of the profit diagram, which resembles a butterfly’s wings.
Here’s how it works:
Imagine three call options with strike prices K₁, K₂, and K₃, where K₁ is the average of K₂ and K₃. To avoid butterfly arbitrage, the price of the option with strike K₁ should not be more than the average price of the options with strikes K₂ and K₃. Mathematically, we express this as:
C(K1)≤ ½ [C(K2)+C(K3)]
where C(K1), C(K2), and C(K3) are the price of a call option with strike price K1, K2, and K3.
If this condition is violated, traders could construct a “butterfly spread” and capture a risk-free profit. The market ensures that option pricing adheres to these conditions to prevent such opportunities.
The assumption that K1 is the average of K2 and K3 can be relaxed under suitable adjustments.
A special case is K1=K2+h=K3−h, so we have:
0≤C(K1−h)−2C(K1)+C(K1+h).
Observing that C(K1−h)−2C(K1)+C(K1+h) is the stencil of the second derivative, leads us to conclude that the price curve is convex. This convexity aligns with our earlier observation that for high strike prices, the option price approaches zero as the probability of exercise at maturity becomes very small.
Moreover, one can point out that the first derivative is monotonic increasing. These observations align with the fact that for high strike prices, the option price is near zero because the probability of maturity is very small.
Similarly, one can find an upper bound on the first and the second derivative. These mathematical relationships aren’t just abstract concepts — they’re the guardrails that keep option markets efficient and arbitrage-free. Taking these observations, one can construct a probability measure called the arbitrage-free measure. This measure is not the probabilities of future events nor what traders think is the future.
The Volatility Smirk
An interesting phenomenon observed in options markets is the volatility smirk. Typically, if you plot implied volatility against different strike prices, you might expect to see a flat line, indicating that volatility is constant across strikes. However, in reality, the curve often shows a “smirk” or “smile” shape. This deviation suggests that out-of-the-money options tend to have higher implied volatilities than at-the-money options.
The volatility smirk can be attributed to several factors, with one key reason being the market’s anticipation of significant price movements, often influenced by historical events like market crashes. As the saying goes, “A war breaks out quickly, but peace takes a long process.” Similarly, there’s an old adage: “The bulls (bull market) climb the stairs, while the bears (bear market) jump out the window, and the pig gets slaughtered.” This colorful metaphor captures the idea that markets tend to rise gradually but fall abruptly.
Traders expect higher volatility for strikes further from the current price due to the perceived risk of sudden, drastic market moves. Consequently, implied volatility is adjusted to reflect these expectations, leading to the smirk pattern. This adjustment ensures that option pricing accounts for the higher perceived risk associated with significant price swings, resulting in the characteristic volatility smirk observed in options markets.
Calendar Arbitrage
Calendar arbitrage, also known as horizontal spread arbitrage, involves exploiting discrepancies in the pricing of options with different expiration dates. Traders use calendar spreads by simultaneously buying and selling options with the same strike price but different maturities. The goal is to profit from the mispricing between the options, which might occur due to differences in implied volatility or other market factors.
For example, if a near-term option is underpriced relative to a longer-term option, a trader could buy the near-term option and sell the longer-term option. If the market corrects the pricing discrepancy over time, the trader can close the position for a profit. This strategy requires careful analysis of the volatility and pricing dynamics of options across different expiration dates.
Calendar arbitrage can be complex, as it involves predicting changes in implied volatility and managing the risk associated with the time decay of options. However, when executed correctly, it can provide a low-risk opportunity to profit from market inefficiencies.
Exploring the Option Pricing Surface
When adding the time axis to the option pricing curve, the price curve evolves into a surface. This surface can provide deeper insights into the pricing dynamics of options. It is worth exploring the mean and the Gaussian curvature of this surface. The mean curvature provides information on the average bending of the surface, while the Gaussian curvature helps in understanding the intrinsic geometry. These curvatures can reveal how different factors, such as time to maturity and strike price, interact to affect the option’s price. Studying these geometric properties can enhance our understanding of the option pricing model and improve the accuracy of pricing and hedging strategies.
Closing Thoughts
Understanding financial options and the principles of arbitrage has been a fascinating journey for me, and I hope it has sparked some curiosity in you too. Reflecting on my early days at the trading options firm, I realize how vital it is to recognize and take advantage of arbitrage opportunities, even though they are rare and fleeting. Grasping these foundational concepts, like the no-arbitrage principle and the dynamics of option pricing, has been crucial for my growth in the financial markets.
The field of options trading is as complex as it is varied, and we’ve only scratched the surface in this post. As a mathematician, I’ve focused primarily on the mathematical perspective, but there’s so much more to explore.
For instance, we haven’t delved into the fundamental differences between call and put options. Calls give you the right to buy an asset, while puts give you the right to sell. Each has its own strategies and use cases that savvy traders leverage to their advantage.
We also haven’t touched on the distinction between American and European options. American options can be exercised at any time before expiration, while European options can only be exercised at expiration. This seemingly small difference can have significant implications for pricing and strategy.
Even within the mathematical realm, there are more sophisticated tools waiting to be discovered. Stochastic differential equations, for example, provide a powerful framework for modeling the randomness inherent in financial markets. The Heston model, which uses these equations to model volatility as a random process, offers a more nuanced approach to option pricing than the classic Black-Scholes model.
I encourage you to delve deeper into these topics. Each one opens up new avenues of understanding and can provide fresh insights into the world of options trading. Whether you’re interested in the practical applications or the theoretical underpinnings, there’s always more to learn.
As we wrap up this journey through the world of financial options and arbitrage, I hope you’ve gained some valuable insights. From basic arbitrage principles to the intricacies of butterfly spreads and volatility smirks, we’ve covered a lot of ground. But remember, this is just the beginning.
The beauty of this field lies in its complexity and the constant interplay between theory and practice. As markets evolve and new mathematical models are developed, there’s always something new to discover.
Whether you’re a seasoned trader, a curious student, or somewhere in between, I hope this post has sparked your interest in the fascinating world of options. Keep exploring, keep learning, and who knows? You might just uncover the next big insight in financial mathematics.
Thank you for joining me on this mathematical adventure through the world of options trading. What aspect intrigues you most? What would you like to learn more about? Share your thoughts in the comments below — I’d love to hear from you!
Disclaimer: As always, this post is for educational purposes only and does not constitute financial advice. Options trading involves significant risk, and you should always consult with a qualified financial advisor before making investment decisions.
