Laurent series are an extension of Taylor series and have significant applications in quantitative finance for analyzing functions around singularities. They allow for modeling asset prices,
valuing options, and solving differential equations.

A
series is an infinite sum of terms expressed as a_0 + a_1 + a_2 + …, which can either converge or diverge. For an analytic function, the Taylor series around a point z_0 is written
as:

f(z)
= a_0 + a_1 (z - z_0) + a_2 (z - z_0)^2 + …, where each coefficient a_n is related to the n-th derivative of the function evaluated at z_0.

Laurent
series generalize Taylor series by including negative powers:

f(z)
= … + a_{-2} (z - z_0)^-2 + a_{-1} (z - z_0)^-1 + a_0 + a_1 (z - z_0) + …

These
negative powers enable the modeling of singularities or points where the function becomes infinite or undefined, which Taylor series cannot handle. Laurent series thus break down a function into
a singular part (with negative powers) and a regular part (with positive powers).

For
a Taylor series, the region of convergence is a disk centered around the point z_0, where the series converges to the function it represents. There exists a radius R such that the series
converges for all points z satisfying |z - z_0| < R. This means that the Taylor series converges within a circle of radius R around z_0 but diverges outside this disk.

In
contrast, the Laurent series has a region of convergence called a "crown" (or annulus), defined by two concentric circles of radii R1 and R2 centered around the same point z_0. The series
converges if R1 < |z - z_0| < R2, forming a circular "band" around z_0.

Unlike
the Taylor series, the Laurent series can converge within an annulus that excludes z_0, making it useful for modeling singularities or poles. For example, a function with a simple pole at z_0,
such as f(z) = 1 / (z - z_0), will have a Laurent series dominated by (z - z_0)^-1, capturing the singularity caused by the zero in the denominator at z_0. This allows for better understanding
and modeling of complex behaviors, especially in finance, where markets may present significant irregularities.

The
negative powers of (z - z_0), like (z - z_0)^-1, (z - z_0)^-2, etc., become very large as z approaches z_0 because (z - z_0) becomes very small. These terms therefore tend toward infinity, which
is necessary to represent a singularity or pole.

While
the Taylor series converges within a disk around z_0, the Laurent series converges within a "crown", providing a better analytic representation when singularities are present.

**About the Author**

Florian Campuzan, a graduate of Sciences Po Paris (Economic and Financial section) with a degree in Economics (money and finance), is a CFA charterholder and AMF-certified professional. He began his career in private equity and venture capital as an investment manager at Natixis before transitioning to market finance as a proprietary trader. In the early 2010s, he founded Finance Tutoring, specializing in market and corporate finance training and consulting.

For over 12 years, Florian has led finance training, advised financial institutions and industrial groups on managing risks, and prepared candidates for AMF certification and the CFA exam. Passionate about quantitative finance and the application of mathematics, he focuses on making complex concepts intuitive, believing that mastering any topic starts with understanding its core intuition.

Écrire commentaire

0