Taken from a mid-July Wall Street Journal news story:
Surging optimism in financial markets hasn’t translated into a big pickup in economic growth. Stocks hit records Friday and big U.S. banks reported stronger-than-expected earnings. But new government data showed consumers pulled back spending at mid-year even as markets rallied. Households also grew less optimistic about the future and inflation on consumer purchases softened.
A prominent quant responds:
In the quant language: the physical measure and the risk neutral measure are different, and assign different probabilities to future events... There is really no logical contradiction between predictions based on history and those based on market sentiment.
Can someone elaborate a bit on what exactly this means, how it should be interpreted, and what it tells us about the future? I am familiar with risk-neutral pricing theory, but thinking about a real-world story like this one makes me quite confused.
Answer
You can price an asset paying $X_{t+1}$ in two ways: $$P_t=\frac{1}{R_f}\sum_{\omega} Q(\omega)X_{t+1}(\omega)$$ $$P_t=\sum_{\omega} P(\omega)M_{t+1}(\omega)X_{t+1}(\omega)$$ As you can see, the price is making a joint statement (i.e. you can recover $Q(\omega)$) regarding both the probability of an event $P(\omega)$ and how much people dislike that event, i.e. the discount factor $M_{t+1}(\omega)$. If I ask you to price an umbrella, not only the price reflects the likelihood that tomorrow is going to rain (i.e. $P(rain)$) but how much you dislike being under the rain without an umbrella (i.e., $M_{t+1}(rain)$). Therefore, when you observe the price of umbrellas going up, is it because it is more likely to rain or because people dislike more taking a shower? Unfortunately, so far there is no way to tell!
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