0.35 0.4 0.45 0.5 0.55 0.6 0.65 0 0.5 1 1.5 2 2.5 3 volatility where V$_{i}$ is the value based on our subjective proba- bility distribution. We suggest to maximize the objective function ξ ( n$_{1}$, . . . , n$_{N}$ ) = Π − Π m subject to constraints on selected risk and possibly exposure measures ( e.g. no short positions). Two examples of possible risk con- straints are In two dimensions this optimization can easily be done graphically. We criticised the stochastic risk-neutral pricing ap- proach for its lack of subjectivity and its mistreat- ment of uncertainty as risk. In particular, risk- neutral pricing is founded on the questionable argu- ment that any drift is worth sacrificing to eliminate whatever uncertainty there is. In this paper an alternative approach was pre- sented which acknowledges that information and ex- pectations are subjective. Market participants should use Bayesian probabilistic reasoning to rationally ex- press their beliefs in terms of probabilities. Observed market prices can be translated into im- plied probability distributions, for which futures im- ply expected returns. Given no significant extra information it is reason- able for market participants to adopt this expecta- tion. We showed that if a market participant is ad- ditionally sure about the standard deviation then he could rationally (due to a maximum entropy argu- ment) use Black-Scholes pricing to subjectively value his position. If he is not sure about the standard deviation he would value his position according to a skew. We commented on possible applications to deriva- tives exposure and risk management. Furthermore, 13 Here we make the simplifying assumption that V$_{i}$ does not depend on the n$_{j}$ . This is true for most investments, but not in general. Consider the value of bottled water to you in the desert, where you are dehydrated. The first bottle will be considerably more worth to you then the 1,000’s bottle. Figure 7: The resulting implied probability skew given the probability distribution of figure 5 for the second moment. • if there is a negative final portfolio value, its ex- pected absolute value is less than z (expected value conditional on that there is a loss). • the probability of a loss (including initial costs) must be smaller than y (this measure is only de- pendent on the relative number of contracts) 6 Conclusion Let us assume we have a set of derivatives (on the same underlying) we can invest in. Prices of these are determined by the market, and hence are compat- ible with the implied probability distribution p$_{m}$ ( x ). Our valuations and associated risks though are de- termined by our subjective probability distribution p ( x | I ). Consider a portfolio consisting of n$_{i}$ contracts of instrument i . The current portfolio market value is where V m i is the current market value of instrument i . The value of the portfolio to us is 13 Π m = N ∑ i =1 n$_{i}$V m i , (3) Π = N ∑ i =1 n$_{i}$V$_{i}$, (4) 5.2 Exposure Management 8