Understanding the implied volatility surface for options on a diversified index

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Journal Article
Asia-Pacific Financial markets, 2005, 11 (1), pp. 55 - 77
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This paper describes a two-factor model for a diveersified 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 analhsis of the growth optimal portfolio and a corresponding random market activity time where the discounted growth optimal portfolio is expressed as a time transformed square Bessel process of dimension four. It turns our that for this index model an equivalent risk neutral martingale measure does not exist because the corresponding Radon-Nikodym derivative process is a strict local martingale. However, a consistent pricing and hedging framework is established by using the benchmark approach. The prposed model, which includes a random initial condition for market activity, generates implied colatility 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 th epropsed model and their Black-Scholes counterparts.
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