Are probabilities of certain events (e.g. amount of rainfall over a period, probability of a Fed rate hike) inferred from derivatives on non-tradeable random variables (e.g. Weather Futures, Fed Funds Futures) stated in the risk-neutral measure (with the money market numeraire) or real-world measure?
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2 Answers
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St Louis Fed published in 2006 a very nice paper: What Are the Odds? Option-Based Forecasts of FOMC Target Changes by William Emmons, Aeimit Lakdawala, and Christopher Neely, which discusses futures on fed funds, options on such futures, implied risk-netural probabilities, and how they differ from objective real-world probabilities.
answered Apr 26, 2020 at 19:03
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Excellent question to grasp the concepts. Basically prices NEVER indicate anything about real-world probabilities. Prices are formed by the interaction of supply and demand from economic agents who might not know (or even care) of the underlying’s future behaviour.
Risk-neutral measures’ existence is a consequence of the FTAP, based on possible replication strategies for contingent claims. Either you can risklessly replicate the payments for such an asset, and the risk-neutral measure used to price it is unique because the no-arbitrage price, which is the value of the replicating portfolio, is unique. If you cannot replicate the claim, there is a range of NA prices (e.g. for a call option, it is $\left[ \left[PV\left(F_T - K\right)\right]^+, PV(F_T) \right]$, $F_T$ being the underlying’s forward price and $PV$ the present value) and infinitely many risk-neutral measures. For no reason should any one of them be particularly informative about real-world probabilities.
For non-traded assets, the “market” (investment banks offering tradeable prices) will use historic data to infer the behaviour of the underlying’s price, and allow for a good “margin of error”. E.g., if the historical standard deviation of temperatures is 15%, they might price using models with standard deviation ranging between 20% and 22%. But the future standard deviation could well be 30% due to climate instability, they wouldn’t know!
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$\begingroup$ I'm trying to reconcile this answer with my understanding that, while "drift" parameters change (roughly due to the risk premium) when I change from risk neutral to real-world measures, volatility / "diffusion" does not (per the diffusion invariance principle). Does this mean I can actually infer real-world implied volatilities from option prices? $\endgroup$– MikeRandAug 14, 2020 at 16:46
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$\begingroup$ "real-world implied volatilities" : what do you mean? By definition, an "implied" parameter is extracted from market information, typically prices. My point is that when working on a pricing model you want to assign positive Arrow-Debreu prices (Q-probabilities) to possible (P-)events ; but the real-world possibility of something is always subjectively estimated! The equivalent measure assumption is to avoid arbitrage (which can be doable or not in practice), by the FTAP. $\endgroup$– siou0107Aug 14, 2020 at 18:41