we considered the case of uncertain standard devia- tion. For a log-normal probability distribution for the standard deviation we illustrated how a “volatility- smile” was implied. While based on simple probabilistic principles, our approach is quite a deviation from what are current standard methods in finance (and particularly in fi- nancial mathematics) and we believe that it offers an interesting different perspective on financial markets. All well known textbooks present the standard ap- proach based on stochastic differential equations, for example [4, 5]. Risk-neutral pricing has its origins in the stochastic approach to derivatives. Hence it is based on the idea that there is a “real” distribution describing the underlying return process (the “random variable”). In the Black-Scholes framework the “random pro- cess” of the underlying asset price is described by the infinitesimal log-normal random price movements where S is the asset price, µ is the drift, σ is the volatility, and d X is a normally distributed random variable. This is the only uncertain/random contri- bution to the price evolution. Hence it is argued that a portfolio is “risk-free” if this term is elimi- nated through a suitable (dynamic) hedging strategy, in which case the portfolio should earn cash returns. Assuming this is done one arrives at the risk-neutral pricing formulas, which do not depend on the drift rate µ . This concept is extended to other distributions, for which the risk-neutral measure is linked to the “real- world” measure through a measure transformation, which sets the expected return to the risk-free rate. This then leads to the notion that there are two dis- tributions — the “real” distribution underlying the process and the “risk-neutral” distribution according to which derivatives should be priced. It is important to understand that the “real” distribution is thought As K → ∞ this gives the first moment m$_{1}$ of x (ex- pected value) as Similarly one finds with z as above and the inverse x = exp(ˆ σz + ν ) (note the usual completing of the square in the exponent) where N ( K ) is the cumulative normal distribution function Substituting z = ln( x ) − ν ˆ σ we find Let ν and ˆ σ 2 be the first moment and second central moment of the log-price probability distribution for some time t from now, i.e. the distribution expressing our beliefs about future market levels. Having given no other information our best guess is the maximum- entropy distribution for the log-price given the first two moments as constraints. This is a Gaussian dis- tribution for the log-price, or the log-normal distri- bution for the price, which is given by to be necessary, because otherwise there would be no risk-premium. Hence the dogma that futures tell us nothing about the “real” distribution. A Origins of risk-neutral Pric- ing B Deriving Black-Scholes from the log-normal Distribution p ( x ) = 1 x ˆ $_{σ}$√$_{2}$$_{π}$ { exp − [ln( x ) − ν ] 2 2ˆ σ 2 } . (6) d S = µS d t + σS d X, (5) ∫ K 0 p ( x )d x = 1 √$_{2}$$_{π}$ ∫ z ( K ) −∞ e − z $^{2}$/ 2 d z = N ([ln( K ) − ν ] / ˆ σ ) , (7) N ( x ) def = 1 2 π ∫ x −∞ d z e − $_{z}$2 2 . (8) ∫ K 0 xp ( x )d x = ∫ z ( K ) −∞ exp(ˆ σz + ν ) √$_{2}$$_{π}$ e − z $^{2}$/ $^{2}$d z = e ν +ˆ $_{σ}$2 2 1 √$_{2}$$_{π}$ ∫ z ( K ) −∞ e − ( z − ˆ σ ) 2 2 d z = e ν +ˆ $_{σ}$2 2 N ( ln( K ) − ν ˆ σ − ˆ σ ) . (9) m$_{1}$ = e ν +ˆ σ $^{2}$/ $^{2}$. (10) 9