Why Option Calibration Focuses on Risk-Neutral Dynamics

Summary

The note explains why option models are often calibrated directly under a risk-neutral measure using observed option prices, without first specifying a corresponding physical-measure process. The calibrated model is used to extrapolate prices beyond the observed instruments and to describe dynamics relevant to pricing and hedging.

The proposed practical check is whether the model fits observed option and underlying dynamics well enough for dynamic hedging to behave reliably. Poor dynamics can create systematic hedging biases and indicate a weak model. The response argues that practitioners often give limited attention to real-world dynamics because those dynamics are difficult to know and improving their description may not improve option pricing or hedging. This is a pragmatic modeling rationale, not a proof that an equivalent physical measure exists for every chosen risk-neutral model.

Key ideas

• Risk-neutral calibration uses option prices to build models for pricing and extrapolation.
• Observed option and underlying dynamics help assess whether a calibrated model supports dynamic hedging.
• Systematic hedging bias is evidence that the model dynamics may be inadequate.
• The rationale does not establish equivalence between the calibrated measure and a physical measure.

Tags

# Why calibration in $Q$ against option prices without showing that $Q$ is equivalent to $P$?


# Why calibration in $Q$ against option prices without showing that $Q$ is equivalent to $P$?












In practice, I have seen articles and financial textbooks on calibration of processes directly under the risk neutral world without showing that the measure is equivalent to a physical measure $P$. They seem to make an assumption that in the physical world, the market is arbitrage free, and that there exists an equivalent measure $P$, whatever the process defined on $Q$ looks like in $P$. Is there a reason for this and why people don't bother with even checking that there exists an equivalent measure $P$?

## Answer by Mark Joshi (score 2, accepted)

https://quant.stackexchange.com/a/35999

Fundamentally, option pricing is an extrapolation exercise. Fitting a q-measure model to the observed option prices gives a way of performing the extrapolation.

If q-measure model gives reasonable dynamics to the option prices observed and the underlying then the process of dynamic hedging with the options will work. If it doesn't, there will be systematic biases and the model will be poor.

Practitioners don't worry about the real-world process because it's unknowable and modelling it better rarely helps with the modelling and hedging.

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