What It Means for the Real-World and Risk-Neutral Measures to Coincide

# What is the interpretation if the real world measure $\mathbb P$ is equal to the martingale measure $\mathbb Q$


# What is the interpretation if the real world measure $\mathbb P$ is equal to the martingale measure $\mathbb Q$












Out of interest, is there anything noteworthy about a market when its real world measure $\mathbb P$ is actually also its martingale measure. In other words the real world measure $\mathbb P$ is equal to the martingale measure $\mathbb Q$. Particularly the case when the market is complete would be interesting as we would thus obtain the real world measure is the only martingale measure (given no-arbitrage).

## Answer by Jan Stuller (score 4, accepted)

https://quant.stackexchange.com/a/60942

## Drifts under $\mathbb{Q}$ and $\mathbb{P}$

Some good answers already. Let me just repeat for clarity: under the risk neutral measure $\mathbb{Q}$, the drift of all assets has to equal to the rate at which the Numeraire appreciates, i.e. typically this is the risk-free rate $r$ of the money market.

The reason for this is the "no-arbitrage" argument:

- suppose there is an asset that costs $S_{t_0}$ money today

- you could borrow $S_{t_0}$ money today and buy one unit of the asset

- you could sell the asset at some future date $t_1$ for price $S_{t_1}$ (this price is unknown at time $t_0$)

- at $t_1$ you will need to return the borrowed money with interest accruing at the numeraire rate, i.e. you need to return $S_{t_0}e^{r(t_1-t_0)}$

Suppose someone wants to buy a forward contract on the asset $S$ at time $t_0$ that expires at time $t_1$. What should the price of the forward contract be? It has to be $S_{t_0}e^{r(t_1-t_0)}$, otherwise there would be an arbitrage.

If we model the price process of the asset $S(t)$ via some stochastic differential equation, that equation will have a stochastic part and a drift part. Usually we choose the stochastic part to be a diffusion of the type $W_t$, which is zero in expectation. So the drift determines the expected future value of the asset $S_t$. That's why the drift under the measure $\mathbb{Q}$ has to equal to $e^{rt}$ (if we assume continous compounding), otherwise the expected value of the asset $\mathbb{E}^{\mathbb{Q}}[S_{t_1}]$ would not be equal to $S_{t_0}e^{r(t_1-t_0)}$ and the forward contract would be miss-priced under $\mathbb{Q}$.

So in conclusion, the risk-neutral measure $\mathbb{Q}$ is a mathematical tool used only for pricing of Assets under no-arbitrage conditions: when we take an expectation under the risk-neutral measure $\mathbb{Q}$ of a future distribution of an asset price, the expectation under $\mathbb{Q}$ is not meant to reflect the future "average" value or "value that the market on average expects"; instead, the expectation just gives the no-arbitrage price and has no probabilistic meaning in the "likelihood of outcomes" sense.

## How does $\mathbb{P}$ work in practice

The real world measure $\mathbb{P}$ is meant to reflect future distribution of asset prices in terms what market participants actually believe (i.e. believe in the "likelihood of outcomes" sense). It's difficult to come up with one unique measure $\mathbb{P}$ even for one asset, because every market participant will have his or hew own Bayesian view about future outcomes.

But here is a practical example of $\mathbb{P}$: if we look at historical default rates of corporate bonds in the USA, we see that these are lower than the average yield-premium of these bonds (i.e. yield premium above the "risk-free" rate of US Treasury bonds): this tells us that the price at which investors are willing to purchase these corporate bonds is such that the yield (i.e. return) on these bonds is (over long time periods) higher not only than the ultra-safe US Treasury bonds, but also high enough so that the investors get return even if (sometimes) some of the corporate bond issuers default. This tells us that under the real world measure $\mathbb{P}$, the market as a whole is able to judge the real future expected break-even return on these corporate bonds and price them in such a way that the actual realized return is higher than break-even (i.e. the drift under $\mathbb{P}$ is higher than $e^{rt+dt}$, where $d$ stands for annual default frequency).

## Examples of $\mathbb{P}=\mathbb{Q}$

So in conclusion, if $\mathbb{P}$ = $\mathbb{Q}$, then the drifts of the stochastic processes we choose to model risky assets would be the same under both measures and this would mean that:

(i) Investors in risky assets do not demand risk premium above the risk-free asset (that is totally unrealistic: i.e. why would investors buy junk corporate bonds at the same yield as US Treasuries or the German Bunds? Or why would investors purchase a risky growth stock with no revenue but a promise of a (stochastic) revenue in the future, and only expect to get the same return as holding safe US Treasuries or German Bunds?)

(ii) It would also mean that risky derivative market-makers are happy to sell derivatives such as CDS or Options with the same risk-premium as the risk-free asset: again, totally unrealistic, because the real-world risk taken by the CDS issuer is real and they are willing to write CDSs precisely only because they get compensated for taking on this risk; i.e. they can get a higher return than just sitting at home whilst being invested in some risk-free asset.

(iii) Another interpretation of $\mathbb{Q}=\mathbb{P}$ would be that all real-world risk would disappear and all assets would be risk-less: this could be the case in some fictional, non-stochastic universe (different to the one in which we live in)