Pricing European Options in Realistic Markets

We investigate the relation between the fair price for European-style vanilla options and the probability of short-term returns on the underlying asset in the absence of transaction costs. 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 the absence of arbitrage opportunities. However, the evolution in general is in the variance v of the asset’s returns rather than in trading time. The variance of the asset’s returns when the European option expires is the only uncertainty in this case. By immunizing the portfolio against large-scale price fluctuations of the asset, the valuation of options is extended to the realistic case 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 of the asset’s returns includes at least two different options. The fair value of an option in this case is determined by a drift function α(x, v) 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 changes in the asset’s returns. It affects the option valuation like a cost-of-carry for the underlying would. The derived pricing formula for options in realistic markets is arbitrage free by construction. A simple model with constant drift α > 0 qualitatively reproduces the often observed volatility -skew and -term structure.