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?