How to account for virtual arbitrage in the standard derivative pricing

In this short note we show how virtual arbitrage opportunities can be modelled and included in the standard derivative pricing without changing the general framework. Whatever people say about the drawbacks of the Black-Scholes (BS) approach [1] to derivative pricing, it is a standard method and almost any pricing and hedging software in nancial institutions is based on it. Practitioners have got used to BSlike partial di erential equations, martingales and other related mathematical animals. Both analytical and numerical methods are well developed and it is hardly surprising that practitioners are rather reluctant to \buy" complicated new theories. That is why it is interesting to see how some limitations of BS analysis can be overcome in the same mathematical framework without disturbing the foundations. One way to improve BS is to use a more realistic price process instead of the geometrical random walk. The most popular alternatives are ARCH-GARCH models where the volatility of the return is assumed to be stochastic. Although they are a better approximation for the price process the description is far from perfect [2]. Another line of attack is the no-arbitrage constraint. Some empirical studies have demonstrated existence of the short lived arbitrage opportunities [4, 5]. In this paper we show how to generalize BS analysis to the case where the virtual arbitrage opportunities exist. In Ref [3] Ilinski and Stepanenko proposed BS-based model to account for virtual arbitrage. The model substitutes the constant interest rate r0 as a rate of return on a riskless portfolio by some stochastic process r0 + x(t). In the limit of fast enough E-mail: [email protected]