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There is a well known result that $\frac{\partial^2 c(K)}{\partial K^2}= e^{-rT}f(K)$, where $K$ is the strike and $f$ is the risk neutral density at time $T$. With the left calculated from market prices, this formula can be used to obtain the implied risk neutral density.

However, what happens if the market prices are such that this doesn't give a valid density. For example, the market may give you a $\frac{\partial^2 c(K)}{\partial K^2}$ such that $f(K)$ doesn't integrate to unity.

Are there any known conditions on the implied volatility or $c(K)$ such that $f(K)$ is a valid density?

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If you recall the derivation from Breeden and Litzenberger (1978), all you need (other than no-arbitrage and infinitely many call options) is the following

  1. $\max\{S_0e^{−qT} − Ke^{−rT} , 0\} \leq C(S_0,K,T) \leq S_0e^{−qT}$ for all strikes $K \geq0$,
  2. $\frac{\partial C(S_0,K,T)}{\partial K}\geq -e^{-rT}$ for all strikes $K \geq0$,
  3. $\lim\limits_{K\to\infty}\frac{\partial C(S_0,K,T)}{\partial K}=0$ and
  4. The call option price $C(S_0,K,T)$ as a function of the strike price $K$ is twice differentiable, monotone decreasing and convex.

Then, there exists a well-defined risk-neutral density function (i.e. positive and integrates to one) given by $$q(x) = e^{rT}\frac{\partial^2 C(S_0,K,T)}{\partial K^2}\bigg|_{K=x}.$$

Typically, one can interpolate the liquidly traded strikes where the option is ATM well. The probablem arises when you need option values with small and large strikes which are not well-traded. One possibility is to propose a parametric model for the tail behaviour. Taylor (Asset Price Dynamics, Volatility, and Prediction, 2005) has an entire chapter on distracting the risk-neutral density and is quite applied.

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    $\begingroup$ Got a reference for condition 3? I don't see it in Breeden and Litzenberger. $\endgroup$
    – LCE
    Sep 6, 2019 at 10:59
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    $\begingroup$ From Breeden and Litzenberger, we obtain $\mathbb{Q}\left[\left\{S_T\leq x\right\}\right]=1+e^{rT}\frac{\partial C(S_0,K,T)}{\partial K}\bigg|_{K=x}$. Firstly, $\mathbb{Q}$ needs to be mon increasing, i.e. $C(S_0,K,T)$ convex in $K$. Furthermore, $\mathbb{Q}$ needs to be in $[0,1]$, i.e. $\frac{\partial C}{\partial K}\in(-e^{-rT},0)$ for all strikes $K$. This follows from (2) and (3). $\endgroup$
    – Kevin
    Sep 6, 2019 at 18:15
  • $\begingroup$ Have a look at Proposition 1 at warwick.ac.uk/fac/soc/wbs/subjects/finance/research/… $\endgroup$
    – Kevin
    Sep 6, 2019 at 18:18

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