Let us assume now that the first moment m$_{1}$ is known to be equal to the present value x$_{0}$ times the risk-free growth factor, i.e. Substituting from above and solving for ν (the first moment of the log-price) gives and Δ the number of units of the underlying asset. The portfolio value is then given by Differentiating with respect to the underlying asset value S and demanding that the portfolio value is invariant gives For this value of ν let us define and hence For example, for the unknown variance case of section 5.1 this gives (assuming that p ( σ | I ) is independent of the spot level) and d$_{1}$ def = d$_{2}$ + ˆ σ . The value of an European put with strike K is the present value of the expected cash-flow. Hence where p ( σ | I ) is the probability distribution for the standard deviation and Δ $^{bs}$( r, σ ) is the Black-Scholes delta for given standard deviation and risk-free rate. where we used (7) and (9) to evaluate the integral. Given the second moment as an annualized variance ˆ σ 2 = σ $^{2}$t this agrees with the Black-Scholes pricing formula (for no dividends). The calculation for European call and binary put/call are very similar. One might believe that generally probabilities are about inferring a correct, but unknown value. For example, people might ask themselves where the mar- ket will be on a certain date, and then see whether they guessed the “correct” value. The problem here arises as it is implicitly assumed that the “correct value” exists objectively without being dependent on the actual betting game and our associated behaviour. Lets consider a specific example. Apparently there was a lottery system in Ireland where the “lucky num- bers” were assigned by a machine at the till. It so happened that the later winner was allowed by some- one else to go ahead and get his numbers first. Hence the winner claimed later that if that other person would have not let him go first, he would have not received the winning numbers. This certainly would be true if the numbers would have been drawn already at that point in time (and In this paper we emphasized that risk is not uncer- tainty. However, if one wants to eliminate uncer- tainty one can derive a “classic hedging prescription” from a given pricing equation. This is very similar to the usual procedure, except that the pricing has been found independently. Let us assume we want to create a portfolio of a derivative and its underlying asset such as to have it momentarily invariant under asset price changes. Let V ( S ) be our subjective value of the derivative D Real Uncertainties C Reconstructing Hedging Prescriptions 10 V$_{p}$ = e − rt ∫ K 0 d x ( K − x ) p ( x ) = e − rt ( KN ( − d$_{2}$ ) − e ν +ˆ σ $^{2}$/ $^{2}$N ( − d$_{1}$ ) = e − $^{rt}$KN ( − d$_{2}$ ) − x$_{0}$N ( − d$_{1}$ ) , (14) Δ = − ∫ ∞ 0 d σ p ( σ | I ) Δ $^{bs}$( r, σ ) , (18) Δ = − ∂V ∂S . (17) 0 = ∂ Π ∂S = ∂V ∂S + Δ , (16) − d$_{2}$ def = [ln( K ) − ν ] / ˆ σ = ln ( K x$_{0}$ ) − rt + ˆ σ $^{2}$/ 2 ˆ σ (13) ν = ln( x$_{0}$ ) + rt − ˆ σ $^{2}$/ 2 . (12) m$_{1}$ = x$_{0}$e $^{rt}$. (11) Π = V + Δ S. (15)