Taylor series: Meaning, Criticisms & Real-World Uses

What Is the Taylor Series?

The Taylor series is a mathematical tool that represents a function as an infinite sum of terms, calculated from the values of the function's derivatives at a single point. It serves as a powerful method for approximation and analysis in various scientific and engineering disciplines, including quantitative finance. In essence, a Taylor series allows complex, smooth functions to be modeled and understood through simpler polynomials.

History and Origin

The concept behind the Taylor series was formally introduced by the English mathematician Brook Taylor in his 1715 work, "Methodus Incrementorum Directa et Inversa" (Method of Direct and Inverse Incrementation).⁸ While earlier mathematicians like James Gregory and even some from the Kerala School of Astronomy and Mathematics in India had explored similar series expansions, Taylor was the first to publish a general method for constructing these series for any function for which they exist.⁷,⁶ His work laid a fundamental groundwork in calculus and mathematical analysis, though the full significance of Taylor's theorem was not universally recognized until later in the 18th century when mathematicians like Joseph-Louis Lagrange championed its importance.

Key Takeaways

The Taylor series is an infinite sum that approximates a function using its derivatives at a specific point.
It is a foundational tool in quantitative finance for simplifying complex models and performing sensitivity analysis.
Applications include option pricing, bond duration, and risk management.
The accuracy of a Taylor series approximation depends on the number of terms included and the distance from the expansion point.
A special case of the Taylor series, centered at zero, is known as the Maclaurin series.

Formula and Calculation

The Taylor series for a real or complex-valued function (f(x)) that is infinitely differentiable at a real or complex number (a) is given by the power series:

f(x) = f(a) + \frac{f'(a)}{1!}(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \dots + \frac{f^{(n)}(a)}{n!}(x-a)^n + \dots

This can be written compactly using sigma notation:

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n

Where:

(f(x)) is the function to be approximated.
(a) is the point around which the function is expanded (also known as the expansion point or center).
(f^{{(n)}(a)) is the (n)-th derivative of the function (f(x)) evaluated at the point (a). The derivative of order zero, (f}{(0)}(a)), is simply (f(a)).
(n!) denotes the factorial of (n) (i.e., (n \times (n-1) \times \dots \times 1)).
((x-a)^n) represents the (n)-th power of the difference between (x) and (a).

When the series is truncated after a finite number of terms, it forms a Taylor polynomial, which serves as an approximation of the original function. The more terms included, the more accurate the approximation, especially near the expansion point.

Interpreting the Taylor Series

The Taylor series provides a method to approximate the behavior of a function locally, around a specific point. The first few terms of the series often provide a sufficient approximation for many practical purposes, particularly when the variable (x) is close to the expansion point (a). The first term (f(a)) is the value of the function at the expansion point. The second term, involving (f'(a)), provides a linear approximation, effectively the tangent line to the function at (a). Subsequent terms, incorporating higher-order derivatives, capture the curvature and other nuances of the function's shape, refining the approximation. This ability to capture local behavior is crucial in fields like financial modeling for understanding how financial variables react to small changes.

Hypothetical Example

Consider the need to approximate the value of an investment portfolio (P(r)) as a function of the prevailing interest rate (r). Suppose we know the portfolio's current value at a reference rate (r_0 = 5%), (P(0.05) = $1,000,000). We also know its first derivative (related to bond duration), (P'(0.05) = -$5,000,000), and its second derivative (related to convexity), (P''(0.05) = $100,000,000).

To estimate the portfolio value if the interest rate changes slightly to (r = 0.051) (5.1%), we can use a second-order Taylor series approximation:

P(r) \approx P(r_0) + P'(r_0)(r-r_0) + \frac{P''(r_0)}{2!}(r-r_0)^2

Plugging in the values:
(r-r_0 = 0.051 - 0.05 = 0.001)

P(0.051) \approx \$1,000,000 + (-\$5,000,000)(0.001) + \frac{\$100,000,000}{2}(0.001)^2

P(0.051) \approx \$1,000,000 - \$5,000 + \$50,000 \times 0.000001

P(0.051) \approx \$1,000,000 - \$5,000 + \$50

P(0.051) \approx \$995,050

This hypothetical example illustrates how the Taylor series can be used to quickly estimate changes in a portfolio's valuation given small shifts in underlying parameters, without needing to re-calculate the entire portfolio from scratch.

Practical Applications

The Taylor series is extensively applied in financial engineering and quantitative analysis to approximate complex financial models and functions. Key applications include:

Option Pricing and "Greeks": The Taylor series is fundamental in deriving and understanding the "Greeks" (Delta, Gamma, Vega, Theta, Rho), which measure the sensitivity of an option's price to changes in underlying parameters like asset price, volatility, time, and interest rates. For instance, Delta is the first-order derivative (first term of the Taylor expansion) of an option price with respect to the underlying asset's price, while Gamma is the second-order derivative. This allows for rapid calculation and risk management in trading.⁵
Bond Duration and Convexity: As seen in the example, the Taylor series is used to approximate changes in bond prices due to interest rate fluctuations.⁴ The first-order approximation gives the bond's duration, a measure of interest rate sensitivity, while the second-order term introduces convexity, which accounts for the curvature of the price-yield relationship and improves the accuracy of the approximation.
Numerical Methods: In computational finance, the Taylor series is employed in various numerical methods to solve differential equations and approximate financial derivatives, enhancing computational efficiency for complex calculations.³
Risk Factor Models: It is incorporated into risk management models and stress testing frameworks to analyze the impact of changes in various risk factors on a portfolio's risk profile.²

Limitations and Criticisms

Despite its wide applicability, the Taylor series has important limitations, particularly in financial contexts where assumptions of smoothness and small changes may not always hold.

One significant criticism is related to its accuracy and convergence. A Taylor series provides a local approximation, meaning its accuracy degrades as one moves further away from the expansion point (a). For some functions, the series may not converge at all, or it may converge to a value that is not equal to the function's actual value outside a specific radius of convergence. In financial models, this implies that Taylor series approximations, particularly first and second-order ones (like those used for Delta-Gamma approximations), may be inaccurate for large changes in market variables or for functions with significant non-linearities.

A paper by the Federal Reserve Bank of New York highlighted that for a plausible range of parameter values, the Taylor series for the Black-Scholes option pricing formula can actually diverge.¹ It also noted that even when the series converges, very high-order approximations might be necessary to achieve acceptable levels of accuracy, which complicates practical implementation. This means that while useful for small movements, relying solely on low-order Taylor series approximations can lead to substantial errors in scenarios of significant market shifts or stress. Furthermore, financial markets often exhibit jump discontinuities and sudden changes that are not well-captured by smooth, differentiable functions assumed by the Taylor series.

Taylor Series vs. Maclaurin Series

The Maclaurin series is a special case of the Taylor series. Specifically, a Maclaurin series is a Taylor series expansion of a function around the point (a = 0).

Feature	Taylor Series	Maclaurin Series
Expansion Point	Any point (a) where the function is differentiable	Specifically at (a = 0)
Formula	( \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n )	( \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n )
Generality	More general, can be centered anywhere	A specific instance of the Taylor series

Confusion often arises because the Maclaurin series is frequently used due to its simplicity, especially when dealing with functions that are well-behaved around the origin. However, the choice of expansion point (a) in a general Taylor series is critical for achieving a good local approximation in other contexts.

FAQs

Why is the Taylor series important in finance?

The Taylor series is crucial in finance because it allows quantitative analysts and financial engineers to simplify complex financial models and functions into more manageable polynomial forms. This simplification aids in quickly estimating values, understanding sensitivities (like the "Greeks" in option pricing), and assessing risk management without computationally intensive full re-evaluations.

What are "derivatives" in the context of the Taylor series?

In the context of the Taylor series, "derivatives" refer to the mathematical concept of rates of change. The first derivative indicates the instantaneous rate of change (slope), the second derivative indicates the rate of change of the slope (curvature), and so on. These mathematical derivatives are crucial for constructing the series, as each term relies on a higher-order derivative of the function evaluated at a specific point.

Can a Taylor series perfectly represent any function?

No, a Taylor series does not perfectly represent every function. It provides a perfect representation only for "analytic" functions, which are functions that are equal to their Taylor series in some open interval. For other functions, the Taylor series acts as an approximation that is most accurate near the expansion point and may lose accuracy or even diverge further away. Its convergence depends on the specific function and the interval of interest.