Applications of Laurent and Taylor Series in Finance in Simple Terms

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.