Chapter 5

Local Volatility and Implied Volatility

5.1 INTRODUCTION

It is well known that a single Black–Scholes model of the form

(5.1)

is inadequate to describe the prices of traded options accurately, for two reasons. Firstly, no term structure can be generated by a simple Black–Scholes model, though this can be easily added by imposing a term structure upon the drift and volatility terms

(5.2)

where μt and σt are deterministic. Note that the volatility term σt is an instantaneous volatility, and the implied volatility for an option with time to maturity T will be a root-mean-square quantity of the form

(5.3)

by additivity of variance.

Secondly, even with term structure taken into account, a different instantaneous volatility will in general be required for options with different strike K.

Dupire (1993) attempted to answer the question – is it possible to construct a state-dependent instantaneous volatility that, when fed into a one-dimensional diffusion of the form

(5.4)

recovers ...