3.4.1 The Volatility Surface from Zero

Basic Prerequisites: Black–Scholes–Merton (or none - this page is written to stand alone).

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1. Intuition & Practical Objective

Start with the only question that matters: what is an implied volatility? An option has a price $C_{\text{mkt}}$ that you can observe. BSM gives you a formula that turns a vol $\sigma$ into a price. Invert the formula - find the $\sigma$ that makes the formula reproduce the market price - and that number is the implied volatility. It is quote-convention, not a forecast.

Now the empirical fact that the whole pillar is built on: if you do this for every strike and every maturity, you do not get one number - you get a surface. For equity index options the surface slopes down in strike (out-of-the-money puts trade at higher implied vol than out-of-the-money calls). This is the skew. For currency options it is U-shaped (the smile). It flattens as maturity grows.

Three “aha”s:

1. Implied vol is the market’s price of the option, re-labelled. Quoting in vol units is just a monotone relabelling of the price: one number per strike, but comparable across strikes and maturities in a way raw prices are not. Vega ($\partial C/\partial \sigma >0$) makes the map price↔vol one-to-one and invertible.

2. A flat vol is a statement about the distribution. BSM assumes $\ln S_T$ is normal with one variance. A skewed implied-vol surface says the risk-neutral distribution of $\ln S_T$ has a fat left tail and a thin right tail (equity). The slope of the smile is the third moment; its curvature is the fourth. So the surface is not decoration - it is the market’s option-implied density (Breeden–Litzenberger: $\varphi (K,T)=e^{rT}{\partial }^{2}C/\partial K^2$).

3. Why is equity skew downward-sloping? Three durable explanations (Hull ch 20 §20.3): (i) leverage - when the stock falls, the firm’s debt/equity ratio rises, so the equity’s volatility rises; (ii) volatility feedback / clustering - a drop raises future expected volatility, and high vol discounts the stock; (iii) crashophobia - post-1987, traders pay up for crash protection. Also, for single stocks, high credit spreads create skew directly through default risk (Gatheral ch 6): a call priced with the risky rate $r+\lambda$ has a downward-sloping BSM implied vol.

The practical objective: internalize that the constant-$\sigma$ model is a quote translator, and read the surface as the primitive object that every real model must reproduce and move correctly.

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2. Mathematical Ground Truth & Derivations

2.1 Implied volatility is a root of a monotone function

Given $C_{\text{mkt}}$, define

$$f(\sigma )=C_{BS}(S,K,T,r,\sigma )-C_{\text{mkt}}.$$

Since $C_{BS}$ is strictly increasing in $\sigma$ with derivative (vega)

$$\nu =\frac{\partial C_{BS}}{\partial \sigma }=Sn(d_1)\sqrt{T}>0,$$

$f$ has a unique root for any price inside the no-arbitrage bounds (Hull 20.1; Haug §2.3). Newton–Raphson converges quadratically:

$${\sigma }_{n+1}={\sigma }_n-\frac{C_{BS}(S,K,T,r,{\sigma }_n)-C_{\text{mkt}}}{\nu ({\sigma }_n)}={\sigma }_n-\frac{C_{BS}({\sigma }_n)-C_{\text{mkt}}}{Sn(d_1)\sqrt{T}}.$$

2.2 Who cares about the smile? - the risk-neutral density

The smile is not cosmetic; it encodes the distribution. Breeden–Litzenberger (Hull app. 20A; Gatheral §1):

$$\varphi (K,T)=e^{rT}\frac{{\partial }^{2}C(K,T)}{\partial K^2}.$$

A skewed surface (down in strike) means the implied $\varphi$ is heavier on the low-$K$ side: a fat left tail. A symmetric smile (FX) means heavier tails both sides: excess kurtosis. If ${\partial }^{2}C/\partial K^2<0$ somewhere, the implied density is negative - a butterfly arbitrage, and the surface is impossible.

2.3 Why one vol cannot fit the smile

BSM forces one $\sigma$ for all $K$. The market’s prices across strikes are only consistent with different vols. Let the “market” quote a downward skew (an equity index): $28\%$ at $K=80$, $20\%$ at-the-money, $15\%$ at $K=120$. Priced by a single $20\%$ BSM vol, the $K=80$ call is too cheap by the amount below - the model cannot represent the skew, because the smile is the distribution, and one number is one distribution.

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3. Computational Implementation - inverting the smile

A “market” quote set generated from a downward skew, inverted strike-by-strike with Newton–Raphson on vega. Stdlib only.

The inversion recovers the skew exactly (28/24/20/17/15%) - confirming that the surface is nothing more than the market’s prices expressed in vol units. The last line is the point: at $K=80$ a single $20\%$ BSM vol underprices by $1.66$ because it cannot bend to fit the smile.

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4. Failure Modes & First-Principles Breakdowns

1. Treating implied vol as a forecast. Implied vol is a price under $\mathbb{Q}$ (risk-neutral), not the realized vol under $\mathbb{P}$. It embeds a variance risk premium; equity implied vols exceed subsequent realized vols on average. Never substitute one for the other.
2. Inversion breaks down where vega $\to 0$. Deep in/out-of-the-money and very short-dated options have $\nu \approx 0$; Newton then steps explosively or returns garbage. Guard with the no-arbitrage price bounds, widen to bisection, or invert in total-variance/$\ln K$ coordinates.
3. Reading the smile on the wrong axis. Skew is measured in log-moneyness or delta (Hull §20.5, “50-delta” quotes), not raw strike; a smile that looks flat vs $K$ can be steep vs $\ln (K/F_T)$ once you strip the forward and the maturity. The natural axis for the surface is $y=\ln (K/F_T)$ with total variance $w={\sigma }_{BS}^{2}T$.
4. Assuming one market’s smile is another’s. FX is a symmetric smile (both tails heavy); equities are a skew; commodities and rates have their own shapes. The “shape is model-generic” result (Gatheral ch 7) is about fitted SV models, not about copying one asset’s smile to another.

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5. References

• Hull, Options, Futures, and Other Derivatives
• Gatheral, The Volatility Surface
• Haug, The Complete Guide to Option Pricing Formulas

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6. Connected Graph Bridges

• Base: Stochastic Calculus & Itô · Black–Scholes–Merton · 03 · Pricing Formulas
• Continue: 02 · Implied vs Local Vol · Index Hub
• Sibling: Implied Volatility Surface & Smiles