From measure changes to time changes in asset pricing

Consequently, starting with a general multidimensional stochastic process S defined on a probability space (Ω, F,Ft, P) and representing the prices of primitive securities, the no-arbitrage assumption allows, for any chosen numéraire, to obtain a martingale representation for S under a probability measure QS equivalent to P. This route will be particularly beneficiary for the pricing of complex contingent claims. Alternatively, changing the clock, i.e., changing the filtration (Ft), we can recover the Brownian motion and normality of returns. In all cases martingales appear as the central representation of asset prices, either through a measure change or through a time change.