  We analyse derivative securities whose value is NOT a deterministic function
of an underlying which means presence of a basis risk at any time. The key
object of our analysis is conditional probability distribution at a given
underlying value and moment of time. We consider time evolution of this
probability distribution for an arbitrary hedging strategy (dynamically
changing position in the underlying asset). We assume log-brownian walk of the
underlying and use convolution formula to relate conditional probability
distribution at any two successive time moments. It leads to the simple PDE on
the probability measure parametrized by a hedging strategy. For delta-like
distributions and risk-neutral hedging this equation reduces to the
Black-Scholes one. We further analyse the PDE and derive formulae for hedging
strategies targeting various objectives, such as minimizing variance or
optimizing quantile position.
