Towards non-equilibrium option pricing theory

A recently proposed model (by Ilinski et al.) for the dynamics of intermediate deviations from equilibrium of financial markets ( “virtual” arbitrage returns) is incorporated within an equilibrium (arbitrage-free) pricing method for derivatives on securities (e.g. stocks) using an equivalence to option pricing theory with stochastic interest rates. Making the arbitrage return a component of a fictitious short-term interest rate (while the real risk-free rate is assumed to be constant) and thus treating it as another source of uncertainty (besides the security price) in a virtual world, the influence of intermediate arbitrage returns on derivatives in the real world can be recovered by performing an average over the (non-observable) arbitrage return at the time of pricing. Using a famous result by Merton, exact pricing formulas for European call and put options under the influence of virtual arbitrage returns (or intermediate deviations from economic equilibrium) are derived where only the final integration over initial arbitrage returns needs to be performed numerically. This result, which has not been given previously and is at variance with results stated by Ilinski et al., is complemented by a discussion of the hedging strategy associated to a derivative, which replicates the final payoff but turns out to be not self-financing. Numerical examples are given which underline the fact that an additional positive risk premium (with respect to the Black-Scholes values) is found reflecting extra hedging costs due to intermediate deviations from economic equilibrium.