Finiteness of variance is irrelevant in the practice of quantitative finance

Outside the Platonic world of financial models, assuming the underlying distribution is a scalable "power law", we are unable to find a consequential difference between finite and infinite variance models –a central distinction emphasized in the econophysics literature and the financial economics tradition. While distributions with power law tail exponents α>2 are held to be amenable to Gaussian tools, owing to their "finite variance", we fail to understand the difference in the application with other power laws (1<α<2) held to belong to the Pareto-Lévy-Mandelbrot stable regime. The problem invalidates derivatives theory (dynamic hedging arguments) and portfolio construction based on mean-variance. This paper discusses methods to deal with the implications of the point in a real world setting. 1. The Four Problems of Practice This note outlines problems viewed solely from the vantage point of practitioners of quantitative finance and derivatives hedging, and the uneasy intersection of theories and practice; it aims at asking questions and finding robust and practical methods around the theoretical difficulties. Indeed, practitioners face theoretical problems and distinctions that are not visibly relevant in the course of their activities; furthermore some central practical problems appear to have been neglected by theory. Models are Platonic: going from theory to practice appears to be a direction that is arduous to travel. In fact, the problem may be even worse: seen from a derivatives practitioner’s vantage point, theory may be just fitting (albeit with considerable delay), rather than influence, practice. This article is organized around a class of such problems, those related to the effect of power laws and scalable distributions on practice. We start from the basis that we have no evidence against Mandelbrot’s theory that financial and commodity markets returns obey power law distributions [Mandelbrot, 1963, 1997], (though of unknown parameters). We do not even have an argument to reject it. We therefore need to find ways to effectively deal with the consequences. We find ourselves at the intersection of two lines of research from which to find guidance: orthodox financial theory and econophysics. Financial theory has been rather silent on power laws (while accepting some mild forms of "fat tails" though not integrating them or taking them to their logical consequences) –we will see that power laws (even with finite variance) are totally incompatible with the foundations of financial economics, both derivatives pricing and portfolio theory. As to the econophysics literature: by adopting power laws, but with artificial separation parameters, using α=2, it has remedied some of the deficits of financial economics but has not yet offered us help for our problems in practice. We will first identify four problems confronting practitioners related to the fat tailed distributions under treatment of common finance models, 1) problems arising from the use by financial theory, as a proxy for "fat tails", of milder forms of randomness too dependent on the Gaussian (non-power laws), 2) problems arising from the abstraction of the models and properties that only hold asymptotically, 3) problems related to the temporal independence of processes that lead to assume rapid convergence to the Gaussian basin, and, 4) problems related to the 1 There has been a family of econophysics papers that derive their principal differentiation from Mandelbrot (1963) on the distinction between Levy-Stable basins and other power laws (infinite v/s finite variance): Plerou et al (2001), Stanley et al (2001), Gabaix et al (2003a, 2003b), Gopikrishnan et al (1998, 1999, 2000) and others. Furthermore, Plerou et al (2002) derive the “cubic” tail exponent α=3 from order flow (a combination of α=3/2 for company and order size and nonlinear square root impact for order impact) –a causal argument that seems unconvincing to any professional, particularly when the paper ignore the share of equity variations that comes from jumps around corporate announcements –such as earnings or mergers. These discontinuities occur before the increase in volume.