# Risk-neutral vs. physical measures: Real-world example

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.

• Who is this "prominent quant"? What he says does not make much sense to me... – Alex C Aug 20 '17 at 3:05
• Andrew Lesniewski. – arni Aug 20 '17 at 5:23
• @AlexC wow! The fact that $P \neq Q$ is a pretty basic fact of financial math. Let's say between 60% and 90 % of people have fire insurance on their house, how common is having a fire based on past data? Roughly 380.000 per year in the US, i.e. 0.3% of the 125Million houses. Which means that in 30 years you have roughly a probability of 10%. So why do so many people indeed buy fire insurance? Because insurance pays off exactly in the most painful of the events: when there is a fire! So in this case high $Q$ but low $P$! – fni Aug 20 '17 at 9:23
• your title is probably wrong: you probably meant "real-world vs. risk-neutral measure". Moreover, they are already a lot of similar questions around, like this one: quant.stackexchange.com/questions/9253 – lehalle Aug 20 '17 at 11:13

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!

• Can you elaborate on this in terms of the problem given, i.e., stocks up and strong earnings, but consumers being pessimistic and households cutting back. – arni Aug 20 '17 at 10:17
• Suppose it is more likely to have a recession, but people are more used to go through recessions. It is true that $P(recession)$ is high, i.e. high probability of recessions, but if $M(recession)$ went down more than proportionally, e.g. people are less risk averse, then $Q(recession)=P(recession)\times M(recession)$ actually went down, meaning that the price of the asset went up. To see this, think about an asset that pays $1$ in boom and $0$ in recession, then $$Price_t=(1-Q(recession))\times 1 + Q(recession)\times 0=(1-Q(recession))$$, hence when $Q(recession)$ goes down, price goes up. – fni Aug 20 '17 at 12:26
• Very good explanation. – Daneel Olivaw Nov 13 '17 at 12:53

First of all, you need to understand risk-neutral measures are not meant to make predictions of future prices, but they are meant to allow hedging (ie risk replication). Historical measures on their side, are meant to predict future prices.

I used the plural because they are many risk-neutral and historical measures. In both cases you choose one measure when you choose an underlying model.

• Risk neutral measure is explaining the map of current market prices: under a specific model (for instance martingale, or semi-martingale) you like all available financial products asking their prices to be consistant with respect to the associated claims.

• Historical measure uses past prices to estimate future ones. As usual in prediction, it needs to stationarize the data, and the usual way is to use prices returns. But some methods can get rid of stationarization locally. In any case you will need conditionings (ie month of the year, news, day of the week, etc). The way to disentangle and recompose basic elements to obtain the prices tomorrow (or newt week), is your model.

That being said, why is it expected to have different pricing for risk-neutral (RN) and historical (HIST) valuation?

1. First of all, from the viewpoint of the the HIST measure, prices today can be cheap or expensive, it is not meant to be the case for the RN measure.

• This shows that HIST can give you arbitrage opportunities for hedge funds: buy now if it is cheap, you will earn money selling your position soon.
• And RN gives you (rare) arbitrage opportunities for banks: you identify one bank is underpricing a derivative product with respect to all other market prices. Usually this mispricing is small, hence you need the leverage of a bank to earn a decent amount of money. And the gain will be instantaneous.
2. You now see that you need to be able to take risk to take profit of HIST arbitrage opportunities (because you will have to hold a portfolio waiting for the price to come back) and you need leverage to take profit of RN opportunities but your reward is riskless.

3. Last but not least, thanks to Bachelier's replication principle (ie Black-Scholes like framework), if you use a versatile enough RN measure and you replicate your risk frequently enough (to not suffer from gamma-driven lost). I.e. you will not loose (too much) money to cancel any risk. Of course you need the market to be complete (ie having enough tradable instruments to span the Euclidian space generated by your risk).

• I understand talking about risk-neutral probabilities when pricing/hedging derivatives consistently with the underlying, i.e. doing replication and not really trying to predict anything. However, when it comes to pricing something in absolute terms, e.g. thinking about the price of the underlying asset (something like S&P500) I feel like it is odd to talk about risk-neutral probabilities. – arni Aug 21 '17 at 8:28
• Think about a binomial tree with $S=94$ going to $S=110$ w.p. $p_u=0.3$ and $S=90$ w.p. $p_d=0.7$. We have $94<96=0.3*110+0.7*90$ due to risk aversion, and the risk-neutral measure places a higher probability on the down-state: $94=0.2*110+0.8*90$. If we decrease $p_u$ to 0.1 and increase $p_d$ to 0.9 (i.e. pessimistic view), but still $S=94$, then $94>92=0.1*110+0.9*90$ and the risk-neutral measure places a higher probability on the up-state, and it is as if investors have become risk-seeking. – arni Aug 21 '17 at 8:57
• It seems your question is now "why the price under risk-neutral measure could be non martingale?" @arni ; you may have a look at my answer to this question: quant.stackexchange.com/questions/29881/… – lehalle Aug 22 '17 at 6:38

This was born as a comment, but is too long, so I'll post it as an answer.

Excuse me, @AlexC, but the guy interviewed is right. Even when pricing options with a trivial binomial tree you notice that the probabilities of the different outcomes are re-weighted when you switch from the physical measure to the risk neutral one. Then, we can reason a little bit: if we consider the stock price as the present value of future cash flows - definition stolen from a corporate finance textbook, we need it before doing fancy mathematics - it is immediate to get how a stock price already incorporates historic prediction - i.e. the reaction of the market to particular events - and to even notice that this kind of analysis doesn't differ at all with respect to the predictions based on market sentiment - that is, the attitude of the market participants. Both are two different ways of doing basically the same thing.

Imagine we're in a classic, linear asset pricing framework. As @fnic explains, information can arrive either about future cash flows or discount rates. Newspaper articles typically are written as if changes in prices reflect changes in future cash flows, but this need not be the case. The stock market may also go up because the price of risk has declined.

If stock prices change, you don't know whether:

• Future cash flows changed
• Discount rates changed

Markets can go up while future cash flows remain unchanged if discount rates go down.

Also be aware that people say all kinds of bizarre, internally inconsistent or nonsensical stuff on CNBC or in newspaper articles.

### Classic SDF framework

The current price of a security is given by:

$$V_t = \sum_\omega P(\omega) M_{t+1}(\omega) X_{t+1}(\omega)$$

### Example:

And let's say the probability measure $P$, stochastic discount factor $M_{t+1}$ and payoff $X_{t+1}$ are defined as follows:

$$\begin{array}{cccc} \text{State} & \text{Probability: }P & \text{SDF: }M_{t+1} & \text{Payoff: }X_{t+1}& \\ \omega_1 & \frac{1}{4} & \frac{1}{1.1} & 110 \\ \omega_2 & \frac{1}{4} & \frac{1}{1.1} & 100 \\ \omega_3 & \frac{1}{4} & 1& 110 \\ \omega_4 & \frac{1}{4} & 1& 100\end{array}$$

• If all the states are possible, the current price of the payoff is $V_t \approx 100.23$.
• If only the states $\{w_3, w_4 \}$ with the high SDF value are possible, the price would be 105.

$$\operatorname{E}[ M_{t+1} X_{t+1} \mid \omega \in \{\omega_3, \omega_4\}] = .5 \cdot 1 \cdot 110 + .5 \cdot 1 \cdot 100 = 105$$

• If only the states $\{w_1, w_3 \}$ with the high cashflow are possible, the price would be 105.

$$\operatorname{E}[ M_{t+1} X_{t+1} \mid \omega \in \{\omega_1, \omega_3\}] = .5 \cdot \frac{1}{1.1} \cdot 110 + .5 \cdot 1 \cdot 110 = 105$$

Thus if we observe that price increases from 100.23 to 105, it could be due to either information about cashflows $X_{t+1}$ or the stochastic discount factor $M_{t+1}$.

In my view, this comment from the newspaper is related to expectations . If we consider the risk neutral measures as the financial market expectations we can understand the rally in the financial market (there is an expectation of a better conditions in the near future). This could be different from the historical measure (such as data showing a consumer pull back spending).

However, I thought risk neutral measures only apply for pricing financial derivatives products...I really doubt that we can extend that concept to the stock market. Can anyone clarify this please?

Perhaps it was just the way it was quoted but I have a serious problem with the statements by a prominent quant.

First, prediction via risk-neutral DGP makes no sense at all. Nobody (except those who dogmatically hold to this view in spite of losing money) in their right mind would predict using the Q-measure. Short rates would zip up to forwards, carry would not exist, options would have no value.

One predicts with P. Historic data gives more insight into evolutions of asset prices than do forward curves. In fact, one does use forwards in best predictions, but not at all in a method that is consistent with Q measure evolutions.

Price and risk-manage with Q measure. Predict with P. Of course, evolve by Q measure to see the discrepancies with P to determine market prices of risk. What is overvalued and what is undervalued.

Meucci has a reasonable tiny writeup on Q vs P measures. Each has its place, but don't start predicting with Q. 'P' Versus 'Q': Differences and Commonalities between the Two Areas of Quantitative Finance