In Chapter 14, we considered the option-pricing problem in ideal frictionless markets. Real markets are often efficient, but they are never ideal. In this chapter, we discuss how the complexity of modeling financial markets increases when we take into account aspects of real markets that are not formalized in the ideal model. These aspects are addressed in the literature as market microstructure [26] or market imperfections [127].

The terminology used in the economics literature suggests a clear parallel with similar scenarios observed in physical sciences. For example, it is much easier to construct a generalized description of the motion of a mechanical system in an idealized world without friction than in the real world. A similar situation is encountered when we compare equilibrium and non-equilibrium thermodynamics. In this chapter, we show that knowledge of the statistical properties of asset price dynamics is crucial for modeling real financial markets. We also address some of the theoretical and practical problems that arise when we take market imperfections into account.

Discontinuous stock returns

The existence of a portfolio containing both riskless and risky assets – replicating exactly the value of an option – is essential in determining the rational price of the option under the assumption that no arbitrage opportunities are present. Whether a portfolio is replicating or not depends on the statistical properties of the dynamics of the underlying asset. In the previous chapter, we saw that a replicating portfolio exists when the price of the underlying asset follows a geometric Brownian motion, but we also saw that this case cannot be generalized.