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What is the Taylor Series?
The Taylor series is a mathematical representation of a function as an infinite sum of terms. It allows us to approximate a function using its derivatives at a particular point. The series is named after the mathematician Brook Taylor, who introduced it in the 18th century.
The Taylor series expansion of a function f(x) around a point a is given by:
f(x) = f(a) + f'(a)(x-a)/1! + f''(a)(x-a)²/2! + f'''(a)(x-a)³/3! + ...
Each term in the series corresponds to a derivative of the function evaluated at the point a, multiplied by powers of the difference between x and a, divided by the factorial of the order of the derivative. The terms capture the local behavior of the function at the point a. The series is often truncated after a certain number of terms to create an approximation of the function. The more terms included, the closer the approximation is to the original function.
The Taylor series expansion is valuable in calculus and mathematical analysis. It provides a way to represent functions that may be difficult to work with directly. It allows for the estimation of function values beyond the range of known values and aids in the understanding of the behavior of functions near a particular point. In quantitative finance, the Taylor series is used in various ways to approximate and analyze financial functions and models, such as:
Option Pricing Models: The Taylor series expansion is employed to approximate option pricing models, such as the Black-Scholes model. By expanding the model's equations using the Taylor series, it is possible to derive simpler approximations or closed-form solutions for pricing options. These approximations can help in quickly estimating option prices and Greeks (sensitivity measures) without relying on complex numerical methods.
Numerical Methods: The Taylor series is utilized in numerical methods to approximate financial derivatives, such as option sensitivities (e.g., delta, gamma, vega). By approximating the derivative using the Taylor series expansion, numerical techniques like finite difference methods can be employed to calculate sensitivities accurately and efficiently.
Risk Management Models: The Taylor series is incorporated into risk management models, such as risk factor models or stress testing frameworks. By expanding the models using the Taylor series, the impact of changes in risk factors on portfolio risk can be analyzed. This enables the assessment of potential losses under different scenarios or shocks.
It's important to note that the Taylor series approximations are most effective for small deviations from the expansion point and may introduce errors as the deviation increases. Careful consideration and validation are necessary when employing Taylor series approximations in quantitative finance to ensure their accuracy and reliability.
The Taylor series is covered in more detail in module 1 of the CQF program.