Cauchy Theorem and Derivative Pricing in Simple Terms

In quantitative finance, one of the main challenges is valuing conditional derivative instruments, such as options whose price depend on specific conditions like future market prices, volatility or other criteria.

A key aspect of evaluating these instruments correctly is the convergence of price functions. Convergence here means that the sequence of a derivative’s values "stabilizes" toward a precise value as calculations are refined. As we approach the maturity date or the conditional event, the value of the derivative should converge to a unique price.

To ensure this convergence, mathematical notions of compact and closed sets are applied within the complex number space (ℂ). A compact set is closed and bounded, meaning sequences of values do not diverge to infinity but remain within a defined space. This ensures the convergence of price values in financial models.

Analytic functions play a crucial role in quantitative finance as they model derivative prices in a continuous and smooth way. A function is analytic when it can be represented as a power series (an infinite sum of terms a_n(z - z₀)^n, with a_n as coefficients).

Analytic functions are highly regular: infinitely differentiable and stable, which simplifies the valuation of derivatives.

The behavior of analytic functions, and hence pricing models, is guaranteed by the convergence of their power series on compact sets. Stability of values depends on the series’ controlled convergence.

The Cauchy integral formula is a key result in complex analysis for calculating the value of an analytic function using its values on a closed contour (*). If a function f is analytic in a domain containing a closed contour γ (gamma) and a point inside z₀, then f(z₀) can be computed through an integral along γ.

The formula allows one to calculate the coefficients a_n of an analytic function without differentiating it, which can be challenging for financial functions.

This implies that a function's power series, and its approximate value at any point, can be derived by integration. This greatly simplifies analysis, especially for derivatives with prices depending on many parameters.

An intuitive way to understand Cauchy’s formula is to see the integral along γ as a "weighted average" of the function’s values on the contour.

This is powerful: the value of the function at z₀ can be found by averaging its values along γ. Instead of knowing the function everywhere inside the domain, it's enough to know its values on the "boundary" (γ).

Thus, Cauchy’s formula turns a "global" problem (the function’s value over an entire disk) into a "local" one (the values only on the contour), easing the convergence of power series.

In finance, this means that knowing prices or market conditions over an interval may be sufficient to infer the prices or values of a conditional derivative within that boundary.

(*) In complex analysis, a contour is a closed and oriented curve in the complex plane (ℂ). Mathematically, it is a continuous line, often parameterized by a function γ(t) where t is a parameter varying within a certain interval [a, b], with the condition that γ(a) = γ(b) to ensure the curve is closed. This curve is traversed in a given direction, which defines its orientation.