Hey I know that when market is incomplete, then we have to choose an equivalent martingale measure (I heard about Escher Transform martingale measure, Mean correcting martingale measure, minimal entropy martingale measure). But in Bjork "Arbitrage Theory in Continuous Time" is written:

When dealing with derivative pricing in an incomplete market we thus have to fix a specific martingale measure Q, or equivalently a λ, and the question arises as to how this is to be done.

Question: Who chooses the martingale measure?

Answer: The market!

And I don't really understand it. So we do not have to look for an equivalent Martingale measure, but get it by calibrating the model (under the physical measure) to the current option prices?


The incompleteness property says that there are infinitely many martingale measures producing an interval of arbitrage-free prices. In reality one has to charge a reasonable price for partial hedging (not for total hedging) of the risks and bear some residual risk, which implies selecting an equivalent martingale measure (EMM) based on some 'optimality' concept.

I'll include Cont and Tankov view from 'Financial Modeling with Levy Processes' and 'Financial Modeling with Jump Processes'.

(Chapter 10 in the second reference) "In a complete market, there is only one arbitrage-free way to value an option: the value is defined as the cost of replicating it. In real markets, as well as in the models considered in this book, perfect hedges do not exist and options are not redundant: the notion of pricing by replication falls apart, not because continuous time trading is impossible in practice but because there are risks that one cannot hedge even by continuous time trading. Thus we are forced to reconsider hedging in the more realistic sense of approximating a target payoff with a trading strategy: one has to recognize that option hedging is a risky affair, specify a way to measure this risk and then try to minimize it. Different ways to measure risk thus lead to different approaches to hedging: superhedging, utility maximization and mean-variance hedging are among the approaches discussed in this chapter. Each of these hedging strategies has a cost, which can be computed in some cases. The value of the option will thus consist of two parts: the cost of the hedging strategy plus a risk premium, required by the option seller to cover her residual (unhedgeable) risk. We will deal here with the first component by studying various methods for hedging and their associated costs. Arbitrage pricing has nothing to say about the second component which depends on the preferences of investors and, in a competitive options market, this risk premium can be driven to zero, especially for vanilla options."

@river_rat mentions here (in the comments), in the context of Heston market price of volatility risk, that the extra EMM parameter could (should) be used "in the stability of the resulting hedge ratios (which is sadly usually of secondary concern)".


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