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We know the numeraires for the forward measure, the risk-neutral measure, etc. What is the numeraire for the real world measure $\mathbb{P}$?

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Let $N_t$ be the numeraire for the real world measure $P$. Let $r$ be the risk free rate and $P^*$ be the risk neutral measure. Then from the standard change of measure results one must have $ \frac{dP^*}{dP} = \frac{e^{rt} N_0}{N_t} $ so that we get the general result $$ \boxed{N_t = N_0 e^{rt} \frac{dP}{dP^*}} $$

Now in the particular 1 dimensional Black & Scholes - GBM case we have for the stock price under $P$ $$ \frac{dS_t}{S_t} = \alpha dt + \sigma dW_t $$ and again standard results tell us that $\frac{dP^*}{dP} = e^{-\frac{\lambda ^2}{2} t - \lambda dW_t}$ (where $\lambda = \frac{\alpha - r}{\sigma}$ is the sharpe ratio) and we get $N_t = N_0 e^{(r+\frac{\lambda ^2}{2})t + \lambda W_t}$. Now if we differentiate that formula we obtain $$ \frac{dN_t}{N_t} = \lambda dW_t + (r + \lambda ^2) dt= \frac{\lambda }{\sigma}\frac{dS_t}{S_t} + r \left(1 - \frac{\lambda }{\sigma}\right) dt $$ and we see that the numeraire $N_t$ is the value of a portfolio that invests on every instant a proportion $\lambda/\sigma $ of its value in stock and $1 - \lambda/\sigma $ of its value in the risk free asset.

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  • $\begingroup$ I agree if the market consists only of the risk free asset $r$, and the stock $S$. However, in the general case, the numeraire shouldn't depend on a stock, but on the market (CAPM), or its factors (APT) instead. $\endgroup$ – byouness May 16 '18 at 10:00
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To build on Antoine's answer (which covers the case where the market consists only of a stock $S$ and a risk free asset $r$). In the general case, if the real world measure $\mathbb{P}$ numéraire depends on a stock $S$, then this means each asset will have its own real world measure... which is clearly not the case. Here, one needs to resort to a framework such as CAPM or APT.

If one assumes a CAPM (or more generally an APT framework), the risk premium (resp. premia if many factors) should be defined with respect to the undiversifiable risk, i.e. the market portfolio (resp. APT factors). The rationale here is that the idiosyncractic risk (a.k.a. diversifiable risk) of stocks can be diversified away, so there is no risk premium to remunarate against assuming this risk.

In this case:

  • The real-world numéraire in this market is the so-called market numéraire $M_t$. It is the process starting at $M_0 = 1$ and with volatility $\lambda_t$ the risk premium (vector of risk premia if we have many APT factors): $$\frac{dM_t}{M_t} = \lambda_t dW^\mathbb{P}_t$$

  • Under this numéraire, the expected return of each financial asset $S$ in this market is given by: $$\mu^S_t = r_t + \langle \frac{dM_t}{M_t}, \frac{dS_t}{S_t} \rangle_t $$ Equivalently, if we denove the instantaneous covariance as $\sigma^{S,M}_t$, we can write (CAPM or APT style): $$\mu^S_t - r_t = \frac{\sigma^{S,M}_t}{\sigma^{M,M}_t} (\mu^M_t - r_t)$$

  • The claims can be priced using the real-world measure and this numéraire like this:

$$\text{Price}_t = \mathbb{E}^\mathbb{P}\left[\text{Payoff}_T \frac{M_t}{M_T} | \mathcal{F}_t \right]$$

This is what is done in insurance for example, where the claims are valued as the expectation of the discounted terminal values under the real-world measure.

  • A change of measure from $\mathbb{P}$ to the risk-neutral proba $\mathbb{Q}$ removes the risk premium from the drift term, leaving only the risk free rate $r_t$, and vice versa, a change from $\mathbb{Q}$ to $\mathbb{P}$ adds the risk premium to the drift term.

To sum up:

  • Each market is fully characterized by its so-called market numéraire, which depends on the risk premia.
  • Under this numéraire, the drift of a given asset is given by the risk free rate and instantaneous covariance between the market numéaire and the asset;
  • and claims can be priced by taking the expectation under the real-world measure.
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