Stability of calibration procedures: fractals in the Black-Scholes model

TL;DR

This paper investigates the effects of allowing complex volatility in the Black-Scholes model, revealing complex roots, essential singularities, and fractal structures in implied volatility calculations.

Contribution

It introduces the concept of complex volatility in Black-Scholes, analyzing the resulting complex roots, singularities, and fractal behavior in implied volatility.

Findings

Complex roots of implied volatility exist near the real axis.

Essential singularities occur at zero and infinity in the complex plane.

Newton-Raphson method exhibits chaotic fractal patterns when computing complex implied volatility.

Abstract

Usually, in the Black-Scholes pricing theory the volatility is a positive real parameter. Here we explore what happens if it is allowed to be a complex number. The function for pricing a European option with a complex volatility has essential singularities at zero and infinity. The singularity at zero reflects the put-call parity. Solving for the implied volatility that reproduces a given market price yields not only a real root, but also infinitely many complex roots in a neighbourhood of the origin. The Newton-Raphson calculation of the complex implied volatility has a chaotic nature described by fractals.