# implied volatility and strike price

Assume for simplicity that the expiration time of an option is $1$ the initial stock price is $1$ and there is no dividend yield and the risk free return is $0$.

How is it possible to show that the following holds for the implied volatility : $$\sigma_{\operatorname{imp}}^2 = \operatorname{O}(\log K)$$

Where $K$ is the strike price of the option.

• Does the Roger Lee reference mentioned in the first sentences of this paper google.be/url?sa=t&source=web&rct=j&url=https://… help? Aug 12, 2016 at 15:34
• You need a jump-free distribution for the underlying I guess? Aug 12, 2016 at 16:07

Notations:

• $$T,K$$ are the maturity and the strike of a vanilla call of price $$C(T, K)$$.

• $$(S_t)_{t\in [0,T]}$$ the price process of the underlying.

• $$\mathbb{Q}$$ is the risk neutral measure.

• $$x(T,K) = \log(K/S_0) - rT$$ is the log-moneyness.

Fix a maturiy $$T^* > 0$$ and define :

$$C(K) := C^{BS}(T^*, K, \sigma_{imp}(K))$$

S.t the R.H.S is the Black-Schiles price of a vanilla call with strike $$K$$ and maturity $$T^*$$

If $$\mathbb{E}_{\mathbb{Q}}\left(S_{T^*}^2\right) < \infty$$, one can easily show that :

$$\forall \ K > 0 : \quad C(K) \leq e^{-rT^*} \frac{\mathbb{E}_{\mathbb{Q}}\left(S_{T^*}^2\right)}{4K}$$

We will prove the following inequality : $$\exists \ \kappa > 0, \ \forall \ x > \kappa : \sigma_{imp}^2(x(K)) \leq 2\frac{x(K)}{T^*}\quad \quad (*)$$

First, notice that by using the bound on the call price we have : $$\lim_{x\to +\infty} C(K(x)) = 0 \quad \quad (1)$$

Knowing that $$\sigma \mapsto C^{BS}(T^*, K, \sigma)$$ is continuous and increasing, it suffices to show that : $$C^{BS}(T^*, K, \sigma_{imp}(x)) \leq C^{BS}(T^*, K, \sqrt{2\frac{x(K)}{T^*}})$$

Now, using l’hôpital’s rule we can prove that :

$$\lim_{x\to +\infty} C^{BS}(T^*, K, \sqrt{2\frac{x(K)}{T^*}}) = \frac{S_0}{2}$$ So : $$\exists \ A > 0, \ \forall \ x>A: \quad C^{BS}(T^*, K, \sqrt{2\frac{x(K)}{T^*}}) - \frac{S_0}{2} \geq - \frac{S_0}{4}$$ So that : $$\exists \ A > 0, \ \forall \ x>A: \quad C^{BS}(T^*, K, \sqrt{2\frac{x(K)}{T^*}}) \geq \frac{S_0}{4}$$

By (1), the definition of the limit gives: $$\exists \ B > 0, \ \forall \ x>B: \quad C(K(x)) \leq \frac{S_0}{4}$$ Puting $$\kappa := \max(A,B)$$ completes the proof.

Now expressing (*) in the strike forme rather than log-moneyness proves that : $$\sigma_{imp}^2(K)= \underset{K\to +\infty}{\mathcal{O}}(\log(K))$$