Bounding Large-Strike Implied Volatility with a Finite Second Moment
Summary
This argument derives an asymptotic upper bound on Black–Scholes implied variance for calls with increasingly large strikes. It assumes a fixed maturity and that the underlying's terminal price has a finite second moment under the risk-neutral measure. A moment-based bound on call prices then implies that call prices approach zero as strike rises.
The proof compares those diminishing market call prices with Black–Scholes call prices evaluated at a trial volatility whose squared value grows proportionally to log-moneyness. The latter prices approach a positive fraction of the spot price, so eventually they exceed the market prices. Since the Black–Scholes call price increases with volatility, implied volatility must be below the trial level for sufficiently large strikes, yielding the stated order bound in log strike. The result is an upper bound, not an exact asymptotic formula or a description of the volatility smile at ordinary strikes. It also relies on the finite-moment assumption and the fixed-maturity setup.
Key ideas
- A finite risk-neutral second moment gives an upper bound on call prices that decays as strike increases.
- At fixed maturity, the call price approaches zero in the large-strike limit under that moment condition.
- Black–Scholes call prices increase with volatility, allowing a trial volatility to bound implied volatility.
- A trial squared volatility proportional to log-moneyness produces call prices with a positive limiting value.
- The conclusion is an asymptotic upper bound and depends on the stated moment and maturity assumptions.
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Full text
# implied volatility and strike price
# 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.
## Answer by Greyearl (score 1)
https://quant.stackexchange.com/a/70898
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 : $$ \sigma_{imp}^2(K)= \underset{K\to +\infty}{\mathcal{O}}(\log(K))$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.