# 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 $P$?

• The argument goes the other way around. The only real measure is $\Bbb{P}$: the financial time series we all observe are realisations of stochastic processes under $\Bbb{P}$. Yet, obivously, $\Bbb{P}$ remains unknown in practice. By making the theoretical assumptions of (i) no arbitrage (ii) complete market, we can show that there exists a unique probability measure $\Bbb{Q}$, equivalent to $\Bbb{P}$ (in the mathematical sense), under which the discounted value of self-financing portfolios are martingales, which allows us to price instruments by taking expectations under $\Bbb{Q}$. – Quantuple Sep 11 '17 at 12:23
• If there is either arbitrage or the market is incomplete, then this result does not hold (either there is no such measure, or it is not unique). You can be sure that a lot of people bothered demonstrating this. – Quantuple Sep 11 '17 at 12:25
• While I agree with @Quantuple's first comment, the second I believe is misleading...the market may not be complete, but by observing the option prices that do exist one can observe the "Q" measure that the market has chosen. – user9403 Sep 11 '17 at 12:46
• @user9403, sorry if this is misleading. I agree that by calibrating to option prices, we will agree on the risk-neutral dynamics of the underlying. But, if the market is incomplete, that does not mean we will necessarily agree on how to translate that back to the $\Bbb{P}$-world. Take stochastic volatility à la Heston for instance. True we'll have the same set of 5 parameters under $\Bbb{Q}$, but depending on our view of the market price of volatility risk, we'll potentially have different RN derivatives for the change of measure to express this dynamics back under $\Bbb{P}$ (...) – Quantuple Sep 11 '17 at 13:50
• Well it's as harsh as any modelling assumption I suppose: "all models are wrong, some models are useful". The real test at the end of the day is to hedge with your model and see if you manage to defend the margin that you charged at inception. – Quantuple Sep 11 '17 at 14:21

• Thank you for your answer. Before accepting it I have one small question. Why do practitioners want the process to be a martingale in the $Q$-measure if they do not bother with that it is equivalent to the real world measure? Couldn't they just choose any kind of process and fit it to the option data if they do not bother with the real world at all? If they can not confirm equivalence of measures, the use of fundamental theorem of asset pricing fails anyway, no? – noidea Sep 12 '17 at 9:03