Choosing a Pricing Measure for Derivatives in Incomplete Markets
Summary
The document explains why derivative pricing in an incomplete market does not determine a unique equivalent martingale measure. Multiple such measures can imply different arbitrage-free prices, so replication alone cannot select a single price when some risks cannot be hedged. The answer frames the choice of measure as linked to how an investor or market participant handles partial hedging and residual risk.
It outlines several approaches to hedging unreplicable payoffs, including superhedging, utility maximization, and mean-variance hedging. These approaches use different objectives or measures of risk and can therefore produce different hedge strategies and associated costs. A price may also include a premium for bearing residual risk; the excerpt notes that competition can reduce this premium, particularly for vanilla options. It does not give a calibration procedure or prescribe one measure for all markets. Calibration to observed option prices is posed in the question, but the response instead emphasizes the role of hedging choices, risk preferences, and market competition in selecting or constraining a pricing measure.
Key ideas
- Incomplete markets admit multiple equivalent martingale measures and a range of arbitrage-free prices.
- When perfect replication is unavailable, pricing depends on how unhedgeable risk is handled.
- Superhedging, utility maximization, and mean-variance hedging are distinct approaches to partial hedging.
- Different risk criteria can produce different hedge costs and valuations.
- The excerpt does not establish a universal measure-selection or calibration rule.
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Full text
# How to choose the martingale measure in incomplete markets # How to choose the martingale measure in incomplete markets 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? ## Answer by ir7 (score 4, accepted) https://quant.stackexchange.com/a/57396 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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