Firstly, we will discuss Taylor approximations. The Taylor approximation is a vital tool in the world of mathematics, serving a purpose quite similar to a ruler used to map a winding road. Imagine you’re out hiking and you come across winding trail, full of twists and turns. Describing its intricate bends would be challenging if all you had was a straight ruler. However, by drawing a series of short straight lines, you can begin to capture the essence of that winding path.
Similarly, in the vast landscape of mathematical functions, many functions resemble winding trails with their complex curves and undulations. Directly dealing with these functions can be daunting. Enter the Taylor approximation: this mathematical “ruler” allows us to represent these complex, curvy functions as a combination of simpler, straighter entities called polynomials. These polynomials, formed of basic algebraic terms, are much easier to handle.
Just as the accuracy of our trail sketch improves the more short lines we draw, the accuracy of the Taylor approximation improves with the inclusion of more polynomial terms. The magic lies in the balance: using just enough terms to capture the essence of the function while keeping calculations manageable. As you delve deeper into this page, you’ll discover the intricate beauty of the Taylor approximation and how it bridges the gap between simple algebra and the complexities of advanced functions.
One-variable Taylor Approximation
Firstly, the one-variable Taylor approximation is discussed. The one-variable Taylor approximation offers a structured way to capture the behavior of a function around a specific point, allowing for effective predictions based on simpler mathematical terms. For a given function \(f(x)\), its Taylor approximation around the point \(a\) is:
\( f(x) \approx f(a) + f'(a)(x – a) + \frac{f”(a)}{2!}(x – a)^2 + \dots \)
Here, the idea is that any function, regardless of its complexity, can be represented as a sum of simpler functions (polynomials). The approximation starts with the value of the function at the point \(a\), and then adds terms that account for changes in the function-first linearly, then quadratically, and so on. The terms \(f'(a)\) and \(f”(a)\) denote the first and second derivatives of \(f\) evaluated at the point \(a\). These derivatives represent the rate of change and the rate of change of the rate of change, respectively.
For practical applications, especially when an analytical derivative isn’t readily available, these derivatives can be approximated numerically. For instance, the first derivative at \(a\) can be approximated as:
\( \frac{f(a+\epsilon) – f(a-\epsilon)}{2\epsilon}\)
where \(\epsilon\) is a small number.
Multivariable Taylor Approximation
Secondly, the multivariable Taylor approximation is discussed. When we have a function that depends on more than one variable, like \(f(x, y)\), we need a way to approximate how it behaves. Think of \(f(x, y)\) as a wavy surface, not just a curve. Just as with our single-variable function, we want to use simpler shapes to represent it, but now in two dimensions. Around a point \((a, b)\), the function can be described as:
\(f(x,y) \approx f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) + \frac{1}{2}f_{xx}(a,b)(x-a)^2 + f_{xy}(a,b)(x-a)(y-b) + \frac{1}{2}f_{yy}(a,b)(y-b)^2\)
Here, terms like \(f_x\) and \(f_{yy}\) are called partial derivatives. They tell us how the surface \(f(x, y)\) tilts or changes direction when we move in the \(x\) or \(y\) direction. Mixed terms like \(f_{xy}\) explain how the surface bends when we move both in \(x\) and \(y\). If we deal with functions of even more variables or want a more precise approximation, the idea remains the same, but with more terms added.
Decomposition of Profit net Losses (PnL)
When we trade options, one of the most important things we want to know is: “How much money will I make or lose?” This is what we call Profit and Loss (PnL). But calculating PnL in options trading can be a bit like solving a puzzle because it’s influenced by many factors like the stock price, time, and even market volatility.
This is where the Taylor approximation comes to the rescue! It’s like a magical simplifier that helps us break down the complex puzzle of PnL into smaller, more understandable pieces. Together with some wisdom from the famous Black-Scholes model (a well-known theory in options trading), the Taylor approximation helps us get a clearer picture of our potential profits and losses. Here’s a simpler way to look at it:
\(PnL \approx \Delta \times \Delta S + \frac{1}{2} \Gamma \times (\Delta S)^2 + \Theta \times \Delta t + \nu \times \Delta \sigma + \rho \times \Delta r – \text{Q} \times \Delta q\)
These pieces, like Delta, Gamma, Theta, and so on, are known as the “Greeks,” and they are like helpful friends, each telling us a different story about our option:
- Delta (\( \Delta \)): Tells us how our option’s price might change when the stock price goes up or down.
- Gamma (\( \Gamma \)): Gives us an idea of how stable or unstable our profits might be.
- Theta (\( \Theta \)): Shows us how our option’s value decreases over time.
- Vega (\( V \)): Helps us understand how changes in market mood (volatility) can affect our option.
- Rho (\( R \)): Informs us about the impact of interest rate changes on our option.
- Q (\( Q \)): Sheds light on how borrowing costs can affect our option’s value.
By understanding each Greek’s story and putting them together using the Taylor approximation, we can get a pretty good estimate of our potential PnL. This way, we don’t have to scratch our heads over a complex puzzle; instead, we get a simpler, clearer view of our possible profits and losses, helping us make better trading decisions.
In this blog, we dive deep into the art of trading, focusing on trading only three factors mentioned above. We will explore:
- Gamma (\(\Gamma\)) and Theta (\(\Theta\)): Learn how to effectively trade these Greeks on the Realized Volatility page. Gamma and Theta are essential tools in understanding and managing the risks associated with options trading.
- Vega (\(V\)): Dive into the world of Vega on the Implied Volatility page. Vega is crucial for understanding how changes in market volatility impact an option’s price.
- Q (\(Q\)) and Theta (\(\Theta\)): Explore these intriguing aspects on the Synthetics page. Understanding Q and Theta can offer valuable insights into the nuances of owning syntetics instead of stocks.
Continue to the next part of this guide here.