Recovering Risk-Neutral Probability from Option Prices

Summary

The document addresses how implied probabilities relate to option prices when Black–Scholes implied volatility varies with strike and maturity. Its central result is that the second derivative of a call price with respect to strike, scaled by the discount factor adjustment, gives a probability density under the risk-neutral measure. This is the Breeden–Litzenberger relationship, cited through a reference to the original paper and a textbook appendix.

The relationship offers a way to infer a distribution from observed option prices rather than forcing all strikes and maturities into a single constant-volatility Black–Scholes specification. The document does not derive the formula or explain how to estimate derivatives from discrete, noisy market quotes, and it gives no empirical examples. Its brief answer identifies the key connection, leaving practical calibration and interpretation details open.

Key ideas

• The second strike derivative of call prices is linked to a risk-neutral probability density.
• The Breeden–Litzenberger relationship provides a way to infer distributions from option prices.
• Implied probabilities offer a distributional interpretation when implied volatility varies by strike and maturity.
• The document cites the result but does not cover numerical estimation from market quotes.

Tags

# Connection between implied volatily and implied probability


# Connection between implied volatily and implied probability












I am reading some lecture notes about Black-Scholes (BS) option pricing. Since the BS-formula is not supported by observed data because of the dependence of the implied volatility on the strik and time to maturity, three possible solutions are suggested:

- Stochastic Volatility Models

- Local Volatility Models

- Implied Probability.

The first two make intuitively sense to me but the third does not. So, in my notes is $\dfrac{\partial C(S,t,K,T)}{\partial K} = -e^{-r\tau} \{1 - Q(K)\} \\ \implies C(S,t,K,T) = e^{-r\tau} \int^{\infty}_{K} \overline{Q}(K) dK$

Could you please explain why these results are a solution to the implied volatility problem?

## Answer by jaamor (score 1, accepted)

https://quant.stackexchange.com/a/19078

Take a look at Hull's Appendix of the Volatility Smiles chapter. (Chapter 16 in my version). It gives a method to calculate the probability density function based on option prices:

$$ g(K) = e^{rT} \frac{\partial ^2 c}{\partial K^2} $$

This result comes from the Breeden Litzenberger 1978 paper.

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