Question

Demonstrate how the discounted asset price can be a martingale under an equivalent martingale measure Q.

Answer 1

In the mathematical modeling of discounted asset-price processes in frictionless financial markets, martingales play a central role. The main reason is the celebrated general version of the Fundamental Theorem of Asset Pricing (FTAP) , is the powerful tool of stochastic integration with respect to general predictable integrands, that martingales are exactly tailored for, played a crucial role. The FTAP connects the economical notion of No Free Lunch with Vanishing Risk (NFLVR) with the mathematical concept of existence of an Equivalent Martingale Measure (EMM), i.e., an auxiliary probability, equivalent to the original (in the sense that they have the same impossibility events), that makes the discounted asset-price processes have some kind of martingale property. For the above approach to work one has to utilize stochastic integration using general predictable integrands, which translates to allowing for continuous-time trading in the market. Even though continuous-time trading is of vast theoretical importance, in practice it is only an ideal approximation; the only feasible way of trading is via simple, i.e., combinations of buy-and-hold, strategies.

Recently, it has been argued that existence of an EMM is not necessary for viability of the market. Even in cases where classical arbitrage opportunities are present in the market, credit constraints will not allow for the arbitrage to be scaled to any desired degree. It is rather the existence of a strictly positive supermartingale deflator, a concept weaker than existence of an EMM, that allows for a consistent theory to be developed.