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## 5.2 Exposure Management 

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 

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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 

• 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) 

• 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). 

In two dimensions this optimization can easily be done graphically. 

## 6 Conclusion 

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, 

