hence limit the applicability of many methods. Also, one could argue that the neglect of available infor- mation besides historic data is rather a short-coming than an advantage. A good overview of these issues is given in [1]. This paper is organized as follows. We start by ex- posing short-comings of the traditional approach in section 2. In section 3 we present our approach and discuss the interpretation of futures and risk. In sec- tion 4 we discuss the interpretation of Black-Scholes pricing in our context. In section 5 we present ap- plications of the probabilistic framework to pricing under an uncertain second moment (uncertain uncer- tainty), which naturally leads to an implied volatility skew, derivative exposure and risk management. risk. Consider for example the standard log-normal case with where X is a “normally distributed random vari- able” $^{4}$. It is clear that d S becomes ‘increasingly de- terministic’ as σ goes to zero — for σ = 0 it just expresses the deterministic drift. Hence we want to argue that d S is really a mixture of a certain and uncertain component. As one cannot eliminate the dependence of the portfolio value change on the dif- ferent components of d S separately, it is not possible to follow a hedging strategy which keeps the drift, but eliminates the randomness. Now, by eliminating d S one also discards the non- random drift term and the resulting hedging is gen- erally (except for a drift equal to the risk free rate) suboptimal. This is most easily illustrated in ex- treme (hypothetical) cases with huge (compared to the volatility) positive or negative drift. It appears as if it is often not understood what it really means to have a drift in the distribution. This is not just something observed in historical data, but something we claim to know about the possibilities in the future. If we use a log-normal distribution with a 1000% drift rate and 10% volatility then we also say that it is practically impossible to observe a draw- down over the next year (and even to see a return of less than 900%) — and we claim to know this. Yet risk-neutral pricing tells us to ignore the knowl- edge of the extreme drift and to hedge as if there is a 50/50 chance of a draw-dawn (below the risk-free rate). Having a short put-delta position will lead us with near certainty to buy back the asset at higher prices. Figure 1 illustrates this. The x -axis shows the un- derlying asset value, with x = 1 being the current spot price. The solid line in the upper graph is a log- normal distribution with an expected return equal to the risk-free rate (here 10%). The lower graph shows the corresponding Black-Scholes put-delta po- sition for a 110% strike. For this case it makes sense The standard approach to derivatives pricing based on stochastic models and risk-neutral measures (see appendix A for a short review) raises a number of issues. Firstly, it does not allow for subjectivity as it assumes the existence of an objective “real distri- bution”, and hence does not take into account that people have different information and believes $^{2}$. This effectively leads to the idea that there is a “correct” value for a derivative, independently of its current trading price — it would possibly be “mispriced”. We think that this is misleading. Every market participant will assign a different value according to his state of knowledge and beliefs. If there are sell- ers and buyers with different subjective valuations, it might result in trades. These in turn might re- sult in observed market prices. One should note that the market price then does not necessarily represent the subjective valuation of neither the buyer nor the seller $^{3}$. Secondly, even if we assume this awkward “under- lying real distribution”, we find it difficult to fol- low the risk-neutral hedging argument, as the elim- inated term does not only contain uncertainty and $^{4}$It is questionable whether the concept of a random variable can be consistently defined [1]. One can argue that “random- ness” is a result of insufficient information about the state of the system. A random variable results then from uncontrolled (and hence unknown) initial conditions. 2 A good pricing theory should be able to value an asset even for an inside-trader. $^{3}$See appendix E. 2 d S = µS d t + σS d X, (1) 2 The Traditional Approach and its Problems