Derivatives Models on Models

11

Negative Volatility and the Survival of the Western Financial Markets

Knut K. Aase

This chapter discusses situations where certain parameters are given values that are outside their natural ranges. One case is obtained when plugging in a negative value for the volatility parameter σ in the Black and Scholes formula. This leads to seemingly “new” results.

A different setting is considered related to the developments in time of biological populations. Here deterministic models lead to chaotically fluctuating population sizes, which came as a surprise to workers with population data.

It is argued that the origins for the seemingly new and original results may be related.

1 Introduction

The motivation for this chapter is the entertaining chapter by Haug, “A Look in The Antimatter Mirror”, including the comic strip “The Collector” in “From Russia with vol”. The Collector had a nightmare, dreaming that the volatility had gone negative. An internet search for negative volatility came up with a single link – to Albert Shiryaev at the Steklov Mathematical Institute, Moscow, Russia. After kicking in Shiryaev's door, The Collector reveals his mission – to terminate the Russian secret weapon known as negative volatility. The Collector feared that this weapon might destroy the western capitalist system, and in particular ruin his own option strategy, buying cheap options for very low volatility. But Shiryaev reassures The Collector that negative volatility is “just a mathematical concept”. ...