Yet, it seems that there might be some logical loophole here. Arbitrage pricing tells you a fair price, such that you should not trade below this price, or above this price. But it did not explain, why you should trade at the equilibrium price itself. If options are redundant, as they have risk, why bother trading them. If this was indeed the case, option markets would simply not exist. So strictly speaking, the Black-Scholes model is a model of a fake market. Of course today, everyone knows that options are risky and the Black-Scholes model is only an approximation to reality. But this is an approximation that seems to wash out risk. How can we model risky business, such as an option trading using models that have no risk. One of my favorite books on derivatives is the book, Traders, Guns, and Money by Satyajit Das. And he tells a story of a German bank that hired a professor from a leading university to quantify its risk. When after some months of calculations, the professor came back and reported that the bank had absolutely no risk. The head trader responded, "I certainly hope you are wrong, Herr Professor. If you are correct then, we can't be making any money". So what this hilarious story tells us, is that risk is a central object in finance. And the absence of risk is a severe deficiency of the original Black-Scholes model. Of course, we all know a long list of other problems with the Black-Scholes model. It doesn't match option pricing data. It doesn't match stock pricing data. Stock prices are not lognormal. It neglects transaction costs. It assumes complete markets and continues rehedging. Both assumptions in contradiction with reality and market practices. And finally, it loses risk and options, so as we can see there so many problems here that it's not even clear where to start. And one question that could be asked here is, what would be a minimum change to the Black-Scholes formula. That would be useful and meaningful yet would have the same or similar level of tractability as the Black-Scholes. Now, one possible way is to follow the mainstream approach in quantitative finals called the risk neutral approach. As follows from its very name, this approach ignores the problem of absence of risk in the Black-Scholes model. And instead, it focuses exclusively on matching market quotes, using some modifications of the original log-normal dynamics of the Black-Scholes model. This became a dominant mantra of most of work that the do. Before you apply such models to price and hedge and option, you have to calibrate it, which by itself can be a very time-consuming or tedious work. I would differentiate between two classes of such risk neutral models. There are parametric models and there are non-parametric models. The first class includes stochastic volatility models, jump-diffusion models, and Levy models. The second class includes local volatility models, maximum entropy models, and non-parametric Bayesian models. And I have in fact spent a large part of my professional life in finance, developing various maximum entropy models and other non-parametric models. And I have been quite happy with them for a while, because they are so flexible. They almost always work wonderfully, and match data quite well. But then, I found the better approach that I will discuss later. It pointed to the source of the problems with these models, namely, that they are missing risk, and risk is kind of the most important thing in the whole game. Now, this approaches start to remind me of Ptolemy's epicycles. Very similar to risk-neutral models, both parametric and non-parametric. Ptolemy explains apparent imperfections in motions of planets by introducing hidden variables called Deferent and Epicycle. So that the absorbed planet motion becomes super position of two hidden processes. And Ptolemy had details, calculations, and fitted model parameters for more than 40 heavenly bodies that were known at that time. But how meaningful this approach is? There is a remarkable story of a cult that was created during the second World War. Japanese and then the allies have built military bases and air field on small islands in the Pacific. Food for soldiers was delivered there once a week, from the air by planes dropping boxes with cargo. Soldiers were friendly with the native people who did not know much of civilization before that. So after the allies left, they started worship a deity called John Frum, who they expected back at some point with cargo for them. And they built wooden rifles, straw airplanes in their worshiping of John Frum. So I thought that this would be a good analogy to what we do in quantitative finance, RPG, CPUs, and GPUs. How different are we from these gentlemen, whose there wooden rifles and saw aeroplanes, if exclude risk in options from our models. What exactly are we after in this approach. Someone famous, they said that, a string fury, about string theory that it's not even wrong. And we can say essentially, the same thing about this risk-neutral models of quantitative finals. Does the longest date, do not include risk? These models are not even wrong. They're just about something else. It's just a branch of math called mathematical finance.