Pricing European Options in Realistic Markets

Abstract

We investigate the relation between the fair price for European-style vanilla options and the distribution of short-term returns on the underlying asset ignoring transaction and other costs. We compute the risk-neutral probability density conditional on the total variance of the asset's returns when the option expires. If the asset's future price has finite expectation, the option's fair value satisfies a parabolic partial differential equation of the Black-Scholes type in which the variance of the asset's returns rather than a trading time is the evolution parameter. By immunizing the portfolio against large-scale price fluctuations of the asset, the valuation of options is extended to the realistic case\cite{St99} of assets whose short-term returns have finite variance but very large, or even infinite, higher moments. A dynamic Delta-hedged portfolio that is statically insured against exceptionally large fluctuations includes at least two different options on the asset. The fair value of an option in this case is determined by a universal drift function that is common to all options on the asset. This drift is interpreted as the premium for an investment exposed to risk due to exceptionally large variations of the asset's price. It affects the option valuation like an effective cost-of-carry for the underlying in the Black-Scholes world would. The derived pricing formula for options in realistic markets is arbitrage free by construction. A simple model with constant drift qualitatively reproduces the often observed volatility -skew and -term structure.