  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.
