The Taylor Expansion in Simples Terms

In quantitative finance, where complex models are the norm, Taylor's expansion offers a way to approximate these models with simpler forms. This technique is vital for quick decision-making in trading, sensitivity analysis, pricing exotic instruments, and in numerical methods.

Consider a European call option, where the price depends on factors like volatility, interest rates, and time to maturity. Traders often need to approximate the option price's sensitivity to these factors, known as Greeks, such as delta and gamma.

The Taylor expansion allows us to locally approximate the option price function by considering changes in the underlying asset price and its derivatives:

0th order : The first, or zeroth, term of the Taylor series is just the value of the function at that point. This is equivalent to assuming the option price remains constant if the underlying asset price doesn't change.

1st order : The first-order term involves the first derivative, delta (\( \Delta \)), which measures the rate of change of the option price with respect to the underlying price. This is like approximating the option price by considering its slope at the current point.

2nd order : The second-order term considers the second derivative, gamma (\( \Gamma \)), which measures the curvature of the option price. This accounts for changes in the slope (delta) as the underlying price moves.

Higher orders : As you include higher and higher order terms, you account for more complex sensitivities, such as vega (sensitivity to volatility) and theta (sensitivity to time decay).

The Taylor expansion is all about locally approximating a function using polynomials (which are easier to work with), starting from the most basic approximation (constant price) and adding complexity (slopes, curvatures, etc.) as we incorporate higher-order terms. The higher the order of the term, the more subtle and detailed the behavior it captures.

Let's take the function \( f(x) = e^{-x^2} \), which represents the bell curve or Gaussian function. This function is fundamental in probability theory and statistics, but it's notoriously difficult to integrate.

Let's approximate the function near \( x = 0 \) using the Taylor expansion:

1. Zero_th Order :

\( f(0) = e^0 = 1 \)

2. First Derivative :

\( f'(x) = -2x e^{-x^2} \)

At \( x=0 \), \( f'(0) = 0 \). This indicates that there's neither growth nor decline at \( x = 0 \), as expected for the peak of a bell curve.

3. Second Derivative :

\( f''(x) = (4x^2 - 2) e^{-x^2} \)

At \( x=0 \), \( f''(0) = -2 \). A negative second derivative indicates that the function is concave down at \( x = 0 \).

From the Taylor expansion, the approximate function near \( x=0 \) is:

\( f(x) \approx 1 - x^2 \)

This is a parabola and gives a simplified representation of our bell curve near \( x=0 \). The higher you go in derivatives, the more refined and detailed information you get about the function's behavior close to that point.