Narrow Results

In this paper we investigate whether a CEV model can account for the observed variation in the at-the-money implied volatility as a function of the level of the at-the-money forward rate. We also determine which exponent β in the CEV process for the swap rate best accounts for the observed behaviour of the implied volatilities.

An arbitrage-free CEV economy driven by Brownian motion in independent, continuous random time is introduced. European options are priced by the no-arbitrage principle as conditional averages of their classical CEV values over the CEV-modified random time to maturity. A novel representation of the classical CEV price is used to investigate the asymptotics of the average implied volatility. It is shown that the average implied volatility of the at-the-money call option is lower and of deep out-of-the-money call options, under appropriate sufficient conditions, greater than the implied CEV volatilities. Unlike in the classical CEV model, the shape of the out-of-the-money tail can be both downward and upward sloping depending on the tails of random time. The model is implemented in limit lognormal time. Its multiscaling law is shown to imply a term structure of implied volatility that is qualitatively more sensitive to changes in the time to maturity than is the classical CEV model.