Abstract:
This paper describes a two-factor model for a diversified index that attempts to explain
both the leverage effect and the implied volatility skews that are characteristic of index options.
Our formulation is based on an analysis of the growth optimal portfolio and a corresponding random
market activity time where the discounted growth optimal portfolio is expressed as a rime transformed
squared Bessel process of dimension four. It turns out that forthis index model an equivalent risk neutral
martingale measure does not exist because the corresponding Radon-Nilcodym derivative process is a
strict local martingale. However. a consistent pricing and hedging framework is established by using
the benchmark approach. The proposed model. which includes a random initial condition for market
activity, generates implied volatility surfaces for European call and put options that are typically
observed in real markets. The paper also examines the price differences of binary options for the
proposed model and their Bleck-Scholes counterparts.
