A Subjective and Probabilistic Approach to Derivatives U. Kirchner ICAP PO Box 1210, Houghton, 2041, South Africa ulrich.kirchner@icap.co.za November 1, 2018 We propose a probabilistic framework for pricing derivatives, which acknowledges that information and beliefs are subjective. Market prices can be trans- lated into implied probabilities. In particular, futures imply returns for these implied probability distribu- tions. We argue that volatility is not risk, but uncer- tainty. Non-normal distributions combine the risk in the left tail with the opportunities in the right tail — unifying the “risk premium” with the possible loss. Risk and reward must be part of the same picture and expected returns must include possible losses due to risks. We reinterpret the Black-Scholes pricing formulas as prices for maximum-entropy probability distribu- tions, illuminating their importance from a new an- gle. Using these ideas we show how derivatives can be priced under “uncertain uncertainty” and how this creates a skew for the implied volatilities. We argue that the current standard approach based on stochastic modelling and risk-neutral pric- ing fails to account for subjectivity in markets and mistreats uncertainty as risk. Furthermore, it is founded on a questionable argument — that uncer- tainty is eliminated at all cost. There are two ways to justify the Black-Scholes equa- tion. On the one hand there is the standard stochas- One could best describe our approach as a proba- bilistic one embracing subjectivity and Bayesian in- ference, while the traditional approach is based on statistical/stochastic ideas and methods. These su- perficially appear as if they are objective in nature as they do not use prior distributions and do not ex- plicitly involve subjective information. This, how- ever, hides the fact that particular priors (usually as- sumptions about normality) are already assumed and We want to offer an alternative framework in- spired by the second approach, which, in essence, ac- cepts the subjectivity of information/beliefs with a Bayesian interpretation of probabilities [1] and rein- terprets the meaning of asset prices, futures, and risk. It yields the Black-Scholes pricing equations (but without the usual assumptions like no trading costs) as maximum-entropy solutions for known first two moments, but the concept extends beyond this special case. We remove hedging arguments and re- place them with probabilistic reasoning $^{1}$. The second approach is considerably simpler, neater, and a direct consequence of probabilistic rea- soning. However, it cannot explain why the expected return should be equal to the risk-free rate — and presumably that is the reason why this derivation is hardly ever presented. tic approach with dynamic hedging. On the other hand one can simply calculate the expected pay-off for the log-normal probability distribution with an expected return equal to the risk-free rate. arXiv:1001.1616v1 [q-fin.PR] 11 Jan 2010 Abstract 1 Introduction $^{1}$Given the solution hedging prescriptions can then be re- constructed. 1