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I am working on a density forecasting project using options. Using the Breeden-Litzenberger formula it is possible to find the implied density at maturity under the risk neutral probability of an underlying:

$$ C\left(0, S, K, T\right)=E_{Q}\left[e^{-r T }\left(S_T-K\right)_+\right] $$

$$ \frac{\partial^2 C\left(0, S, K, T\right)}{\partial^2 K}=e^{-r T} f_{S_T}(K) $$

I was wondering if it was possible (and if so how) to write this probability density under the historical probability, i.e. to do a change of measure.

I specify that we do not make any hypothesis on the dynamics of the underlying and thus that the theorem of Girsanov does not help us.

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    $\begingroup$ If no make further assumptions at all, it's pretty difficult to make many statements about the real-world distribution. Google ``recovery theory’’ which aims to extract $\mathbb{P}$-densities from market data. It's empirical success is debatable. $\endgroup$
    – Kevin
    Oct 21, 2022 at 7:17
  • $\begingroup$ Hi Kevin, thanks for your response. I will look into the theory of recovery. The only problem with doing assumptions is that I don't want to make an assumption that involves the distribution of the underlying asset at maturity. For example, if we take the Black-Scholes assumptions, the underlying asset at maturity will follow a log-normal distribution. $\endgroup$
    – Petra Di Mario
    Oct 21, 2022 at 11:04
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    $\begingroup$ If you make no assumptions about the distribution of the underlying, the parametrisation of the prices of risk or the stochastic discount factor but still want to estimate the entire distribution of real-world returns, then you're asking a very difficult question. Almost Nobel Prize worthy :) $\endgroup$
    – Kevin
    Oct 21, 2022 at 12:22

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