Time Dynamics of Probability Measure and Hedging of Derivatives

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.