Arbitrage‐Free Pricing

This chapter addresses the issue of using the dynamics to price contingent claims that is derivative contracts. Probability in finance does not have much to do with beliefs or experiments, but much more with the concepts of arbitrage and of replicating portfolios. In an economy in which the risk is priced into the assets, the assumption that one does not care about such premium in order to price derivatives means that one has accounted for risk in another way, that is by changing the probability assigned to the scenarios. This change of measure technique is the main instrument of work for pricers. The stochastic processes described previously should be changed to martingale if they are to be used for pricing financial products or strategies whose payoffs are linked to the dynamics of a risky asset – derivatives. Not only should they be martingale processes, but they must also be such as to deliver consistent prices for the option contracts traded in the market. It is known as calibration problem.