Stochastic arbitrage return and its implications for option pricing

The purpose of this work is to explore the role that arbitrage opportunities play in pricing financial derivatives. We use a non-equilibrium model to set up a stochastic portfolio, and for the random arbitrage return, we choose a stationary ergodic random process rapidly varying in time. We exploit the fact that option price and random arbitrage returns change on different time scales which allows us to develop an asymptotic pricing theory involving the central limit theorem for random processes. We restrict ourselves to finding pricing bands for options rather than exact prices. The resulting pricing bands are shown to be independent of the detailed statistical characteristics of the arbitrage return. We find that the volatility "smile" can also be explained in terms of random arbitrage opportunities.Comment: 14 pages, 3 fiqure

Arbitrage Opportunities and their Implications to Derivative Hedging

We explore the role that random arbitrage opportunities play in hedging financial derivatives. We extend the asymptotic pricing theory presented by Fedotov and Panayides [Stochastic arbitrage return and its implication for option pricing, Physica A 345 (2005), 207-217] for the case of hedging a derivative when arbitrage opportunities are present in the market. We restrict ourselves to finding hedging confidence intervals that can be adapted to the amount of arbitrage risk an investor will permit to be exposed to. The resulting hedging bands are independent of the detailed statistical characteristics of the arbitrage opportunities.

An Adaptive Method for Valuing an Option on Assets with Uncertainty in Volatility

We present an adaptive approach for valuing the European call option on assets with stochastic volatility. The essential feature of the method is a reduction of uncertainty in latent volatility due to a Bayesian learning procedure. Starting from a discrete-time stochastic volatility model, we derive a recurrence equation for the variance of the innovation term in latent volatility equation. This equation describes a reduction of uncertainty in volatility which is crucial for option pricing. To implement the idea of adaptive control, we use the risk-minimization procedure involving random volatility with uncertainty. By using stochastic dynamic programming and a Bayesian approach, we derive a recurrence equation for the risk inherent in writing the option. This equation allows us to find the fair price of the European call option. We illustrate numerically that the adaptive procedure leads to a decrease in option price.