Beyond Black-Scholes Option Theory

This article presents a new option pricing principle that is more useful than the no-arbitrage principle, especially for incomplete markets. The focus here is on ideas behind mathematics— why the new theory is warranted, and how common sense dictates its construction. I. Complete and Incomplete Markets The real financial world is too complicated for anyone to understand it completely. Thus a simplified model world is needed for two purposes: (i) To help us understand the real world, in the sense that a real-life phenomenon is explained if it is expected in a model world. (ii) To help us make decisions in the real world, i.e., applying rules proven to be correct in the model world to the real world. There are two types of model world in option theory; one is the so-called complete markets and the other, incomplete markets. The difference is that in complete markets, as in the famous Black-Scholes framework, derivatives carry no risks because they can be completely eliminated by the delta hedging scheme; whereas in incomplete markets, derivatives have inherent risks because perfect hedges are impossible. An example of incomplete markets is where the underlying is not a tradable instrument, e.g., real options and weather derivatives. Even if the underlying is continuously tradable, the presence of transaction costs, stochastic volatilities or jumps etc. renders the model market incomplete. In short, any imperfection of a complete market model makes it incomplete, so there are a lot more incomplete markets than complete ones. Complete markets are total fiction; reality is better modeled by incomplete markets because of unavoidable risks. For example, to take on a larger position in the real world, you demand a better unit price to compensate for the extra risks. But this natural risk averse behavior contradicts predictions of any complete market model where there is a unique price regardless trading sizes. The Black-Scholes-Merton complete market option theory is complete, i.e., well understood. Pricing options by replication is widely written and taught. However, despite its importance, the current option theory for incomplete markets is very incomplete. I will explain in this article why this is the case, and how it can be completed. II. Position and Risk When you are making a risk related decision, it is your risk that really matters. Your risk comes from your position, without a position you have no risk. I view this statement as common sense.