# What the implied distribution really is?

From volatility surfaces we have a implied distribution of $S_T$. This distribution is the real world distribution or this is a risk neutral distribution?

In this related question How to derive the implied probability distribution from B-S volatilities?, it is shown how to infer the implied probability density of the future prices of a risky asset from a continuum of call prices written on that asset (Breeden-Litzenberger identity).

The developments, which I invite you to read, basically rely on the fact that the call price writes $$C=e^{-rT} \int_0^\infty (S-K)^+ p(S) dS$$ or equivalently $$C = e^{-rT} \Bbb{E} \left[ (S_T-K)^+ \right] \tag{1}$$ under some measure where $S_T$ is a random variable with probability density function: $$d\Bbb{P}(S_T \leq S)/dS = p(S)$$

Now, the fundamental theorem of asset pricing tells us that equation $(1)$ holds under the so-called risk-neutral measure (a measure equivalent to the real-world measure but under which the $t$-value of any self-financing strategy is a martingale when expressed with respect to the risk-free money market account numéraire).

Consequently, the implied density $p(S)$ you compute by evaluating $$p(S) = e^{rT} \frac{\partial^2 C}{\partial K^2}(K=S)$$

is indeed a risk-neutral pdf (because it relies on the risk-neutral expression of the call price $(1)$.)

• So the expected value of the impled distribution of $S_T$ is always $S_0 e^{rT}$. Correct?
– John
Dec 14 '16 at 11:37
• Omitting any other carry effects (e.g. dividends), yes. More accurately the expectation is equal to the forward price. Dec 14 '16 at 12:28

I'm not a probabilist but I tend to think real-world distribution in financial world doesn't exist (or at least is not a proper term). None of the financial events are really repeatable or IID.

And to your question, it should be risk-neutral density function.

• That's not true. Of course that the real world measure exists, this has nothing to do with any statistical property such as IIDness. The time series that you work with every day are precisely realisations of stochastic processes under the real world measure. In addition, everything which can be expressed as an expectation under $\Bbb{Q}$ can also be expressed as an expectation under $\Bbb{P}$ since $\Bbb{E}^\Bbb{Q}_t [ X_T ] = \Bbb{E}^P_t \left[ X_T \left. \frac{d\Bbb{Q}}{d\Bbb{P}} \right\vert_{\mathcal{F}_T} \right]$. Dec 14 '16 at 9:19
• So, yes, the real world measure exists. However, it is not relevant for option pricing purpose. To convince yourself go back to the binomial pricing framework (or BS continuous time framework) and see how real world probabilities disappear from the pricing equation due an absence of arbitrage opportunities + static (resp. dynamic) replication argument. Dec 14 '16 at 9:21
• @Will Gu Without the the real world measure, we haven't anything.
– user16651
Dec 14 '16 at 10:55
• @Quantuple Is there a way o getting the real distribution of $S_T$?
– John
Dec 14 '16 at 12:10
• This would require some working modelling assumptions, since after all you are looking at modelling some future state of the economy based on your current knowledge. See the many existing and related SE questions. Dec 14 '16 at 12:30