Derivative valuation theory is based on the formalism of abstract probability theory and random variables. However, when it is made part of the pricing tool that the 'quant' (quantitative analyst) develops and that the option trader uses, it becomes a pricing technology. The latter exceeds the theory and the formalism. Indeed, the contingent payoff (defining the derivative) is no longer the unproblematic random variable that we used to synthesize by dynamic replication, or whose mathematical expectation we used merely to evaluate, but it becomes a contingent claim. By this distinction we mean that the contingent claim crucially becomes traded independently of its underlying asset, and that its price is no longer identified with the result of a valuation. On the contrary, it becomes a market given and will now be used as an input to the pricing models, inverting them (implied volatility and calibration). One must recognize a necessity, not an accident, in this breach of the formal framework, even read in it the definition of the market now including the derivative instrument. Indeed, the trading of derivatives is the primary purpose of their pricing technology, and not a subsidiary usage. The question then poses itself of a possible formalization of this augmented market, or more simply, of the market. To that purpose we introduce the key notion of writing. |