options - Implied probability density (Question 2 - Applications and Interpretation)

Using the second derivative of the Call-Option-Price one can try to recover the pricing density.

Formally: Assuming a constant interst rate $r$ and also not making any assumptions on the model used to evolve $S_t$

$C(t,S_t,K,r,T)=e^{-r(T-t)}\int_0^{\infty}(S_T-K)^+f(S_T|S_t)dS_T$

The density is then recovered via

$p(S_T|S_t)=e^{r(T-t)}\frac{\partial^2 C(t,S_t,K,r,T)}{\partial K^2}|_{K=S_T}$

As a follow-up to my last question:

1. 
   

   What are the applications of this recovered density ?

   
   


2. 
   

   How can we Interpret it ? (can it be considered the "real" probability density ? - seeing how it is used in a pricing context it should still be risk neutral)

   
   

Answer

1) A straigthforward application is to price any complex payoff at maturity using this.

By that I mean a payoff that is such that the price of the option is

$$P = e^{-r(T-t)}E[f(S_T)] $$

Which you can then calculate by integrating $f(S_T)$ w.r.t. to your density.

One of the challenges though is to have a proper marks and inter/extrapolation for the implied vols of the wings (i.e. far away from current forward) so that your density does not become negative.

Therefore another application for this density is that it is a good way to check for no arbitrage on a term option volatility curve.

2) Yes, it is just the probability which you can hedge against by using derivatives on the market. It's a bit like if a horse has 1 against 4 odds, that does not mean that there is a 20% chance that he will win, it just (sort of) means that if you have some product whose price needs to know that probability, the right number to put in the price is 20% because the hedging instrument you will use (i.e. better 1 against 4 on that horse, or against) will cost exactly that.