Modern quantitative finance is just like the fortress of Aristotelian philosophy, except that the content arises from Brownian motion theory introduced in 1900 by Louis Bachelier in his thesis but the real foundations of this Aristotelian orthodoxy comes from early 1970s Black-Scholes-Merton option pricing models. Theory comes from the Arbitrage pricing, and some of the major papers establish the framework of equivalent martingale measures. It’s pretty powerful, this infrastructure, but flawed as SCIENCE. It’s a purely mathematical and theoretical edifice, self-contained and seems much more solid than empirical world’s ugly deviations from it. harrison-kreps-1979 HarrisonKreps78 harrison-pliska cox-ross-alternative

The powerful theory here, like all powerful theories, never match empirical reality perfectly. I first learned this material in 1995, and was overwhelmed at the time, at my first job at Lehman Brothers, and I worked on and off in Wall Street quant jobs for years with serious doubts about this enterprise. I worked with serious raw scientific data at a biotech on signal processing of mass spectrometric data around 2003-5 and studied a great deal of Paul Feyerabend’s philosophy of science during this time confronting mass spectrometric data, especially the problem of lineshapes in mass spectra. In continuous-time finance, the theory is more or less all built on models with Brownian motion driving log-prices, and it was clear for decades that this was not appropriate until early 2000s saw the publication of books with more accurate jump-diffusion and Levy process models, of which I had been recently fitting — following Wim Schoutens’ work — the Meixner distributions in the past few days to data. But all this finance theory and the quantitative improvements look at finance in my opinion too much through the eyes of option pricing and management from the viewpoint of the investors and traders and not through the eyes of the billions of people for whom, whether they have investment in the global capital markets or not, face severe problems from volatility storms, turbulence in the global financial markets. The change in viewpoint does not diminish the actual accomplishments of the pioneers of what I am calling the Aristotelian Orthodoxy of Quantitative Finance, but does provide a new perspective: option pricing that is accurate is valuable, but a view towards accuracy in option pricing produces a severe bias regarding the fundamental measurable quantity from the larger perspective which is volatility. Univariate simple modeling assumptions lead to convenience in option pricing, but it introduces volatility as a tunable parameter whether constant or following some sort of Ornstein-Uhlenbeck process for example. But if one looks at volatility of massive numbers of instruments and calculate the covariance matrix, one sees that it makes little sense to consider the volatility of an individual instrument as an independent object at all. Volatility is a global object that is a function of the entire market, this is very clear as it is not interpretable at all for individual stocks or instruments meaningfully. Second, the quantitative source of turbulence in global financial markets is most likely that the underlying stochastic process probability density follows a NONLINEAR diffusion as this is the source of turbulence widely known in other scientific applications of diffusions — there is an analogue of the Reynold’s number phase transition to turbulence of Navier-Stokes equations to the purely diffusion with nonlinearity of the KPP equation. Recall that Navier-Stokes has the form and the KPP has the form while the heat equation (whose sample paths are Brownian motions) has the form . In both the nonlinear modifications of the heat equation, there is turbulence whatever are the precise conditions governing turbulent regimes which are delicate matters. The problem with this Aristotelian Orthodoxy of Quantitative Finance is that all the technicalities of this edifice obscures the fact that TURBULENCE cannot possibly occur in any of these linear diffusion models, while it is observed in the global markets. This is the second issue that I am concerned with besides volatility being a global object — nonlinear diffusions seems to be the source of turbulence but it is not convenient at the moment in established orthodoxy. Now if one gives up need for neat, cute, and clever closed form solutions and rely on numerical methods fully, then there may be a way out: we can numerically estimate appropriate probability densities and use something like the Esscher transform to calculate equivalent martingale measures and push through some of the orthodoxy with nonlinear univariate diffusions with volatility being estimated in some global manner.

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