Portfolio with stochastic arbitrage return and its e ect on option pricing

The purpose of this paper is to explore the role that random arbitrage opportunities playin pricing financial derivatives. We use a non-equilibrium model to set up a stochasticportfolio and for the random arbitrage return, we choose a stationary ergodic random processrapidly varying in time. We exploit the fact that option price and random arbitragereturns change on different time scales which allows us to develop an asymptotic pricingtheory involving the central limit theorem for random processes. We restrict ourselves tofinding pricing bands for options rather than exact prices. The resulting pricing bandsare shown to be independent of the detailed statistical characteristics of the arbitragereturn. It is well-known that the classical BlackScholes formula is consistent with quotedoptions prices if different volatilities are used for different option strikes and maturities.It is well-known that arbitrage opportunities always exist in the real world. Of course,arbitragers ensure that the prices of securities do not get out of line with their equilibriumvalues, and therefore virtual arbitrage is always short-lived. In this paper, we follow anapproach where option pricing with stochastic volatility is considered. We exploit the factthat option price and random arbitrage return change on different time scales allowing usto develop an asymptotic pricing theory by using the central limit theorem for randomprocesses. The approach yields pricing bands that are independent of the detailed statisticalcharacteristics of the random arbitrage return.

Keywords:

Black Scholes , Arbitrage , Portfolio , Option pricing , financial markets