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:

  1. Stochastic Volatility Models
  2. Local Volatility Models
  3. 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?


1 Answer 1


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.

  • $\begingroup$ So, the idea is to retrieve the density of the real world option prices and use this density to price other options? $\endgroup$ Jul 29, 2015 at 12:49
  • $\begingroup$ That's not the idea. As you state in your question the BS model has some faulty assumptions, including constant volatility across strike prices. In practice, the market works around those issues by treating $\sigma $, the implied volatility as a "free" parameter that captures the real world deviations from the BS framework. The equation above outlines a way to back out the market implied risk neutral probability density function. It is an interesting exercise to try out the method described by Hull on a very liquid option chain of your choice. $\endgroup$
    – jaamor
    Jul 29, 2015 at 23:13

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