  We propose a probabilistic framework for pricing derivatives, which
acknowledges that information and beliefs are subjective. Market prices can be
translated into implied probabilities. In particular, futures imply returns for
these implied probability distributions. We argue that volatility is not risk,
but uncertainty. Non-normal distributions combine the risk in the left tail
with the opportunities in the right tail -- unifying the "risk premium" with
the possible loss. Risk and reward must be part of the same picture and
expected returns must include possible losses due to risks. We reinterpret the
Black-Scholes pricing formulas as prices for maximum-entropy probability
distributions, illuminating their importance from a new angle. Using these
ideas we show how derivatives can be priced under "uncertain uncertainty" and
how this creates a skew for the implied volatilities. We argue that the current
standard approach based on stochastic modelling and risk-neutral pricing fails
to account for subjectivity in markets and mistreats uncertainty as risk.
Furthermore, it is founded on a questionable argument -- that uncertainty is
eliminated at all cost.
