3.5.1 Advanced Volatility from Zero

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

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1. Intuition & Practical Objective

In Black–Scholes the volatility is a number you look up. That is not a simplification you can get away with, because the market tells you it is false: invert the Black–Scholes formula on every listed option and you get a surface, not a number. The only honest reading of that surface is that the variance of the underlying is itself random.

This page builds the why with no prior stochastic-volatility background. The objective is one idea: a stochastic-volatility model fixes the smile because it makes the risk-neutral distribution a mixture - and a mixture is exactly what produces fat tails, a smile, and a skew. Everything in the rest of this folder is a choice about which process drives that mixture.

Three “aha”s:

1. Fat tails come from mixing. If the variance $v$ were known, $\ln S_T$ would be normal. Let $v$ be random with distribution $\pi (v)$. Then the density of $\ln S_T$ is the mixture $\int \mathcal{N}(x;\cdot ,v)\pi (v)dv$ - a scale mixture of normals. Its kurtosis exceeds $3$ whenever $\pi$ is non-degenerate. The smile is not an anomaly to be explained away; it is the generic consequence of random variance.

2. The skew needs correlation, not just randomness. A symmetric variance mixture produces a symmetric U-shaped smile (both tails fat - this is what FX options look like). Equity options show a downward skew: the left tail is fatter than the right. For that you need the two Brownian motions to be correlated: $⟨d\ln S,dv⟩=\rho \eta vdt$ with $\rho <0$. Prices fall because volatility rises - the leverage effect. One parameter, $\rho$, turns a smile into a skew.

3. Variance’s own “volatility” is the missing knob. Black–Scholes has one number; an SV model has a distribution of variance. Its dispersion, the vol-of-vol $\eta$, sets the curvature of the smile (how much the wings lift). This is why $\eta$ is the parameter that matters for exotic products: it controls convexity in vol, which is precisely what cliquets and forward-skew products pay for.

The practical objective: understand that a stochastic-volatility model is a distributional device, and that its parameters are not “a better volatility” - they are the shape of the variance distribution and its coupling to the spot.

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2. Mathematical Ground Truth & Derivations

2.1 Mixture of lognormals, in one line

Take $X_T=\ln (S_T/F_T)$. With random variance $v$ (independent of the Brownian motion driving $S$),

$$p(x)={\int }_0^{\infty }\mathcal{N}(x;-\frac{1}{2}Tv,Tv)\pi (v)dv,\mathcal{N}(x;m,s^2)=\frac{1}{s\sqrt{2\pi }}{e}^{-\frac{(x-m{)}^2}{2s^2}}.$$

Its cumulants are the cumulants of the variance-average, and the third cumulant (skewness) is exactly zero unless $v$ is correlated with $S$ - the mathematical statement of aha #2. The variance-mixture route is the classic “Taylor/SV” explanation of the smile (Hull ch 20 §20.3 gives the leverage + volatility-feedback + crashophobia trio for equities).

2.2 Why a single $\sigma$ cannot reproduce a mixture’s prices

Black–Scholes assigns one density. The mixture assigns another. If the two disagree in the wings, no choice of $\sigma$ can repair it: the model has one degree of freedom and the mismatch is a function of $K$. Concretely, with a two-point variance mixture,

$$C(K,T)=p\text{BSM}(F,K,T,{\sigma }_1)+(1-p)\text{BSM}(F,K,T,{\sigma }_2),$$

which is an exact closed form (no simulation needed) - the cleanest possible demonstration. Its ATM implied vol is below $\sqrt{\mathbb{E}[v]}$ (Jensen: option prices are convex in vol), and its wings rise.

2.3 The Heston programme, previewed

Heston replaces “distributed variance” by a concrete, tractable process:

$$d{S}_t=\sqrt{v_t}{S}_{t}d{Z}_1,d{v}_t=-\lambda (v_t-\bar{v})dt+\eta \sqrt{v_t}d{Z}_2,d{Z}_{1}d{Z}_2=\rho dt.$$

Failure to produce a smile is not an option here: the variance mixture induced by the CIR process $v_t$ is the smile generator. The four parameters have one job each - $\bar{v}$ sets the level, $\lambda$ the term structure, $\eta$ the curvature, $\rho$ the skew (§02–§04). The rest of this folder is spent making that precise, computing it, and finding where it breaks.

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3. Computational Implementation - a two-state mixture is a smile

We price a two-state variance mixture in closed form and invert the Black–Scholes formula strike-by-strike. No simulation, no free parameters to fit: the smile appears because the density is a mixture. Stdlib only.

Three facts to read off the table:

1. The implied vol is not constant. It runs $24.97\%$ at $K=70$ down to $19.98\%$ ATM and back up to $23.60\%$ at $K=130$ - a smile, produced by nothing but the randomness of the variance.
2. ATM implied vol $\ne \sqrt{\mathbb{E}[v]}$. $19.98\%$ against $22.36\%$. Option prices are convex in $\sigma$; mixing two pure-volatility worlds and then quoting the price back in one volatility systematically under-states the dispersion. This gap is the reason variance swaps and ATMF implied vols are different market parameters.
3. The mixture is symmetric - both wings lift. To tilt it down (equity skew) you must correlate the variance with the spot return. That single observation is why every equity SV model carries a $\rho \in [-1,0)$.

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4. Failure Modes & First-Principles Breakdowns

1. “A mixture is a model.” A two-point mixture reproduces a smile but has no dynamics: it says nothing about how the variance moves, so it cannot price a forward-start option, a cliquet, or a variance swap. Statics ≠ dynamics - the recurring theme of this folder (04 · SV Dynamics).
2. Confusing implied vol with expected vol. As the table shows, ${\sigma }_{BS}^{ATM}=19.98\%$ while $\sqrt{\mathbb{E}[v]}=22.36\%$. Convexity (Jensen) means implied variance is not the expected variance, and the two series (implied ATMF vs variance-swap) are distinct market quotes (Bergomi ch 5).
3. Forgetting that the risk-neutral mixture is not the real-world one. Everything above lives under $\mathbb{Q}$. The market’s ${\pi }^{\mathbb{Q}}(v)$ carries a variance risk premium and is not the histogram of realized variance. Calibrating dynamics to historical time series is a category error (Hull ch 23 is about $\mathbb{P}$, not $\mathbb{Q}$).
4. Assuming more parameters ⇒ better. The smile’s shape is nearly model-generic (Gatheral §7.8): once a model has fat tails via a variance mixture and a negative $\rho$, it will fit today’s slice. What differs is the dynamics, which today’s slice cannot see. Fitting harder is not the answer; testing dynamics is.

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5. References

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

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6. Connected Graph Bridges

• Base: Stochastic Calculus & Itô’s Lemma · Black–Scholes–Merton · BSM · 03 Pricing Formulas
• Continue: 02 · The Heston Model · Index Hub
• Sibling: VS · 01 From Zero · Heston & SABR (flat note)