Completeness, Martingale Measures, and Derivative Price Intervals
Summary
The note contrasts pricing in complete and incomplete financial models. In a complete model, it describes a unique martingale measure and a derivative price given by the discounted expectation under that measure. In an incomplete model, multiple martingale measures can produce a range of discounted expectations, suggesting lower and upper arbitrage-free valuation bounds. The author then asks whether practitioners actually use such intervals and whether calibration makes incompleteness irrelevant in practice.
The proposed calibration maps market data to a risk-neutral measure drawn from the union of measures associated with a model family. Once a particular measure is selected, the formula yields one price, but that selection does not by itself show that the underlying model is complete or that alternative admissible measures are immaterial. The document poses questions rather than answering them and gives no market examples or empirical evidence. Its formulation also leaves unspecified the calibration criteria, admissibility constraints, and whether the displayed price interval includes its endpoints.
Key ideas
- Completeness is associated with a unique martingale measure and a unique arbitrage-free price in the note's setup.
- An incomplete model may admit multiple martingale measures and a range of discounted expected payoffs.
- Calibration can select one risk-neutral measure and produce a point estimate without eliminating model incompleteness.
- The document asks about practical price intervals but does not answer or substantiate the questions.
Tags
Full text
# Is completeness of a financial model relevant for derivatives pricing?
# Is completeness of a financial model relevant for derivatives pricing?
If a market model is complete then every derivative has a unique arbitrage free price. However we are not starting with a model but with a arbitrage free Model class $\mathcal{M}$ (E.g. the Blackscholes model family). Let $\Phi(m)$ be the set of Martingale measures attached to a specfic model $m \in \mathcal{M}$. Completeness (of a model family) then just means $|\Phi(m)|=1 \forall m \in \mathcal{M}$. We want to price a derivative $\Psi$. for a complete model we have $$\Pi(\Psi) = \mathbb{E}_{\Phi(m)}[\frac{\Psi}{B_T}]$$ For an incomplete model we have $$\Pi(\Psi) \in (\inf\limits_{Q \in \Phi(m)}\mathbb{E}_{Q}[\frac{\Psi}{B_T}],\sup\limits_{Q \in \Phi(m)}\mathbb{E}_{Q}[\frac{\Psi}{B_T}])$$
Therefore I thought that one would actually calculate in the incomplete case a price interval in practice and that this would differentiate complete from incomplete models, making the former easier and the latter more robust.
However Usually we calibrate the risk neutral distribution to market prices. Let $\mathcal{D}$ be the space of Market data. Let $\mathcal{M}_{rn} = \bigcup\limits_{m\in \mathcal{M}}\Phi(m)$. Calibration is then a map $c_{\mathcal{M}}: \mathcal{D} \to\mathcal{M}_{rn} $.
Therefore for any Model class $\mathcal{M}$ the price of a derivative would be calculated as $$\Pi(\Psi) = \mathbb{E}_{c_{\mathcal{M}}(D)}[\frac{\Psi}{B_T}]$$
Hence we arrive at a unique price and it doesn't really matter for the validity of this price that it comes from a model family consisting of incomplete models.
My questions would be:
- In practice, would one ever calculate the price intervall for an incomplete model? If yes, why?
- Am I correct that the topic of non-unique prices for incomplete models is a non-issue in practice?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.