When Implied Volatility Skew Has a Zero-Slope Strike

# Conditions for implied volatility to attain a minimum (or maximum) value


# Conditions for implied volatility to attain a minimum (or maximum) value












Suppose

$$ dS = \sigma S \left(\rho dW + \sqrt{1-\rho^2} dZ \right) $$

and

$$ d\sigma = a(\sigma,t) dt + b (\sigma,t) dW $$

with $dW dZ = 0$.

What are the conditions necessary such that the implied volatility skew of vanilla options attains a (global/local) minimum or maximum value for each maturity date $T$? In other words that the slope of the implied volatility skew is zero at some strike $K$ for each $T$, where $K = K(T)$, i.e. the strike could be different depending on $T$.

## Answer by fes (score 2)

https://quant.stackexchange.com/a/57185

Ok here is my shot at this. Let me know if I am missing something. For each $T$ and other relevant parameters implied volatility is defined via

$$C^{BS}(\sigma^{IV}(K),K)=e^{-rT}\mathbb{E}^{Q}[(S_T-K)^{+}]$$

Taking derivative wrt. to $K$ gives

$$\frac{\partial}{\partial \sigma}C^{BS}(\sigma^{IV}(K),K)\frac{d\sigma^{IV}(K)}{dK}+\frac{\partial}{\partial K}C^{BS}(\sigma^{IV}(K),K)=e^{-rT}\mathbb{E}[-I(S>K)]=-e^{-rT}\mathbb{P}(S>K)$$

Using the fact that BS vega is always positive $\frac{d\sigma^{IV}(K)}{dK}=0$ iff

$$\frac{\partial}{\partial K}C^{BS}(\sigma^{IV}(K),K)=-e^{-rT}\mathbb{P}(S>K)$$

Using a standard result for the derivative of BS price wrt. to $K$ gives

$$-e^{-rT}N(d_2)=-e^{-rT}\mathbb{P}(S>K)$$

or

$$N(d_2)=\mathbb{P}(S>K)$$

This means that we need to find a point such that the risk neutral probability that the option ends in the money is the same under BS and your model. Using Ito's lemma for $\log(S_t)$ we have

$$S_t=S_0\exp(\int_{0}^{t}\sigma_sdW_s-\frac{1}{2}\int_{0}^{t}\sigma_s^2ds)$$

Hence our condition is

$$N(d_2)=\mathbb{P}(S_0\exp(\int_{0}^{T}\sigma_tdW_t-\frac{1}{2}\int_{0}^{T}\sigma_t^2dt)>K)$$

or

$$N(d_2)=\mathbb{P}(\log(S_0)+\int_{0}^{T}\sigma_tdW_t-\frac{1}{2}\int_{0}^{T}\sigma_t^2dt>\log(K))$$

Hence a necessary and sufficient condition is that for each $T$ there is some $K(T)$ such that

$$N(d_2(K(T))=\mathbb{P}(\int_{0}^{T}\sigma_tdW_t-\frac{1}{2}\int_{0}^{T}\sigma_t^2dt>\log(K(T)/S_0))$$

## Answer by dm63 (score 0)

https://quant.stackexchange.com/a/51428

Intuitively, $b>0$ should be sufficient for the existence of a global minimum implied vol. The minimum should be more pronounced if $b$ is large , and the strike price with the minimum implied volatility should be close to ATM when $\rho $ is close to zero. I’m pretty sure these assertions could be proven from the Hagen approximation to the SABR model.