Why Risk-Neutral Option Densities Do Not Determine Real-World Probabilities

# How to infer real world measure from risk neutral measure


# How to infer real world measure from risk neutral measure












Assume we have inferred risk neutral density of stock price at time T from option prices. Assume we have obtained a parameterized density p(S). How can we infer real world measure? I know about Girsanov's Theorem but I am not sure whether I can use it for this.

## Answer by Quantuple (score 4)

https://quant.stackexchange.com/a/37583

Let $\Bbb{P}$ denote the physical measure and $\Bbb{Q}$ the risk-neutral one.

First of all, it's important to realise that while $\Bbb{P}$ exists but is not tractable (it is the measure under which we observe realisations of the various market quantities), $\Bbb{Q}$ is a pure mathematical construct that cannot be observed in the real world but can be tied to $\Bbb{P}$ under some assumptions.

These assumptions are: picking a valuation model and postulating the absence of arbitrage opportunities (= fair valuation). Without this, there is no way to relate the 2 measures.

This caveat can be further understood by noting that (using a mathematical concept known as change of measure) \begin{align} \Bbb{P}[S_T \leq K] &= \Bbb{E}^\Bbb{P} \left[ \Bbb{1}\left\{ S_T \leq K \right\} \right] \\ &= \Bbb{E}^\Bbb{Q} \left[ \Bbb{1}\left\{ S_T \leq K \right\} \left. \frac{d\Bbb{P}}{d\Bbb{Q}} \right\vert_{\mathcal{F}_T} \right] \end{align} where the expectation on the RHS is the one you should compute under the risk-neutral measure $\Bbb{Q}$ to end up on the CDF under the real-world measure $\Bbb{P}$.

Now although the probability density function of $S_T$ under $\Bbb{Q}$ $$ q(T, S) = \frac{d\Bbb{Q}[S_T < S]}{dS} $$ can be computed in a model-free way from vanilla option prices (Breeden-Litzenberger identity), the Radon-Nikodym derivative $$ \left. \frac{d\Bbb{P}}{d\Bbb{Q}} \right\vert_{\mathcal{F}_T} $$ remains model-specific. Hence without a model, you are stuck.

Actually, even with a model, estimating the parameters appearing in the Radon-Nikodym derivatives (market prices of risk) can prove quite tricky if not impossible, see discussion here.

Some people worked on a model-free methods for moving from $\Bbb{P}$ to $\Bbb{Q}$ by e.g. minimising the Kullback-Leibler divergence (or relative entropy) between the two densities subject to some relevant constraints (see work of Derman and Zou) . I haven't found that to work very well in practice though.