Multiple Risk-Neutral Measures in Incomplete Derivative Models

Summary

The document explains why calibrating one arbitrage-free model does not always settle derivative valuation uniquely in practice. For model-dependent products, especially exotic derivatives, banks may assess values under multiple models and vary free parameters. Those choices can imply different risk-neutral densities and therefore different theoretical prices, even when the models are consistent with observed market data.

The answer frames this as a model-risk exercise: vanilla options may still be priced correctly across the alternatives, while more complex products can expose differences between model assumptions. This gives a practical reason to study incomplete-market pricing approaches and multiple candidate measures. The response is brief and does not specify a particular calibration procedure, selection criterion, or numerical example, so it establishes the motivation for comparing models rather than prescribing how to choose among them.

Key ideas

• Different model specifications or parameter choices can imply different risk-neutral densities.
• Banks may compare model-dependent products under several models as part of model-risk evaluation.
• Exotic derivatives can have different theoretical values across calibrated models.
• Vanilla prices may match market observations even when model-implied exotic values differ.
• The document motivates comparison but does not give a rule for selecting a preferred measure.

Tags

# In which scenario would we end up with more than one $\mathbb{Q}$ after calibrating an incomplete model?


# In which scenario would we end up with more than one $\mathbb{Q}$ after calibrating an incomplete model?












Reading the literature I see that quite an effort is made to price derivatives in an incomplete setting. I see stuff like efficient hedging, indifference pricing, choosing $\mathbb{Q}$ by considering some metric to $\mathbb{P}$ etc. However I cannot think of a situation where on would use this. I imagine the derivatives pricing pipeline as follows:

- Choose an arbitrage free model in its risk neutral form

- Calibrate the free parameters to market price

- Price your derivative of interest

So after step 2 we have exactly one risk neutral measure $\mathbb{Q}$. In which scenario would we have to deal with multiple rn measures?

## Answer by Kermittfrog (score 3)

https://quant.stackexchange.com/a/67844

From a model risk perspective, banks are required to re-evaluate their model-dependent products (think: exotics) using more than one model and - if there are free parameters - by varying those parameters. This will result in multiple theoretical values, and hence with multiple risk neutral densities. Note that, usually, all vanillas will be priced correctly.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.