# Convert implied probability into real probability

A risk-neutral world is one where all investors are indifferent to risk and don’t require any extra risk premium for the risk they bear. In this world, all assets (irrespective of their risk) will earn the risk-free rate. Investors’ risk appetite and true/real world probabilities of a given event both play a role in the determination of the risk-neutral probabilities. Since in the real world investors are risk averse, they are more concerned about bad outcomes (for example a market drop), so the associated risk-neutral probabilities are higher than the real ones. Similarly, the implied probabilities associated to good outcomes is lower than the real ones.

So I am understanding that in general, for example on stocks, we would always find that implied probabilities of negative moves are higher than what we would really get if we could compute real world probabilities (and viceversa for positive moves). Then I read:

In practice, to obtain the implied probability density function we can follow these steps:

1. Calculate option prices P at various strikes K by using strikes with a distance ΔK extremely small. If not all strikes are available in the market, use interpolation to find the missing ones.
2. Calculate the difference between consecutive prices ΔP
3. Calculate the ratio between ΔP and ΔK (this can be seen as the first derivative of the price with respect to strike)
4. Calculate the difference between consecutive ΔP/ΔK (as calculated in previous step)
5. Use the difference from previous step and divide by ΔK

At this point, by plotting the results from step 5 against the strikes, we can see the probability distribution as implied by the prices we used.

So it seems pretty straighforward to get the implied probabilities once I have the option prices. But I have the following question. Suppose that I am interested in the real world probabilities of a market drop and have calculated the implied probability distribution following the steps from the article, how do I know how I should correct these probabilities to adjust for the fact that these are risk-neutral so usually inflated for negative moves in the market as compared to what the corresponding real world probabilities would be?

In other words, is there a generaland parctical way (a trader's rule of thumb) to have an idea of what a given implied probability would correspond to in the real world? The article mentions that we need to take risk-aversion into account but what is the risk-aversion of the market that would allow me to convert implied into real probability?

• There is something called the Ross Recovery Theorem which supposedly recovers real world distribution from options prices. But honestly I don't really understand it. However, you might. Here is the paper: nber.org/system/files/working_papers/w17323/w17323.pdf
– user34971
Jul 27, 2022 at 21:22
• Recovery is a pretty cool new research avenue as @FridoRolloos says. However let me add that the current empirical evidence is mixed at best. Jul 27, 2022 at 22:25
• Please note that if you could back out real probabilities from risk-neutral probabilities, you'd have a money printing machine. So from a philosophical point of view, it should not be possible to reliably back out one from the other. Aug 1, 2022 at 12:13
• @Kermittfrog can you please elaborate a bit more as to why you would have a money printing machine if you manage to know what are the market-expected real world probabilities of a given move? After all, those are still only probability and there is no certaninty that the market will go in one direction or the other. Aug 1, 2022 at 18:51
• If you can obtain real world probabilities from option prices, you can position yourself in options whose price ( derived from risk neutral probability) is below the real-world-probability-weighted payoff. Of course, this is a gamble - But you can make use of the law of large numbers. Does that make sense? Aug 2, 2022 at 8:03