Chapter 11 — Local and Stochastic Volatility

Guiding question

Black–Scholes assumes constant volatility. Markets produce an implied volatility surface that varies by both strike and maturity — the "volatility smile." What extensions of GBM reproduce this surface, and what are the trade-offs between local and stochastic volatility models?

The empirical failure of Black–Scholes is specific: the implied volatility $\sigma_{\rm imp}(K,T)$ extracted from market option prices is not flat. Out-of-the-money puts trade at higher implied vol than at-the-money options (the skew), and implied vol varies with maturity (the term structure). Two classes of models address this:

1. Local volatility (Dupire): $\sigma$ depends on $S_t$ and $t$, not on a separate random factor.
2. Stochastic volatility (Heston, SABR): $\sigma_t$ is driven by its own SDE.

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11.1 The implied volatility surface

Given market prices $C_{\rm mkt}(K,T)$ for all strikes $K$ and maturities $T$, the implied volatility $\sigma_{\rm imp}(K,T)$ is the unique $\sigma$ in Black–Scholes that matches each price:

$$C_{\rm BS}(S_0,K,T,r,\sigma_{\rm imp}(K,T)) = C_{\rm mkt}(K,T).$$

The surface $\sigma_{\rm imp}(K,T)$ is the market's consensus view of future realized volatility, embedded in option prices. Key features:

• Skew: $\partial_K\sigma_{\rm imp} < 0$ for equities (puts more expensive than calls at same distance from ATM).
• Smile: $\sigma_{\rm imp}$ is U-shaped in $K$ for many asset classes (FX, rates).
• Term structure: $\sigma_{\rm imp}$ changes with $T$.

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11.2 Local volatility: Dupire's formula

Assume the stock follows:

$$dS_t = r S_t\,dt + \sigma_{\rm loc}(t,S_t)\,S_t\,dW_t.$$

The local volatility function $\sigma_{\rm loc}(t,S)$ is a deterministic function of time and spot. This model is complete — every option payoff is replicable.

Dupire's equation

Given a complete set of European call prices $C(K,T)$ that are free of arbitrage, the unique local volatility that reproduces them is:

$$\sigma_{\rm loc}^2(T,K) = \frac{\partial_T C + rK\partial_K C}{\tfrac12 K^2\partial_{KK}C}.$$

Derivation from the forward Kolmogorov equation

The call price satisfies the Dupire forward PDE (in terms of $K$ and $T$, not $S$ and $t$):

$$\partial_T C = \tfrac12\sigma_{\rm loc}^2(T,K)K^2\partial_{KK}C - rK\partial_K C.$$

Solving for $\sigma_{\rm loc}^2$ gives the formula above. This is Kolmogorov's forward equation written for the European call as a function of its boundary data $(K,T)$.

Local vol as conditional expectation of realized variance

The local variance $\sigma_{\rm loc}^2(t,K)$ equals the expected instantaneous variance of the stock, conditional on $S_t = K$:

$$\sigma_{\rm loc}^2(t,K) = \mathbb E[\sigma_t^2\mid S_t = K].$$

This holds in any model where $\sigma_t$ is a function of $(t,S_t)$ or is stochastic. Local vol is the "smile-consistent" variance — it is the exact variance needed to match option prices, but says nothing about how variance moves in the future.

Limitation of local vol. Local volatility models fit the current surface exactly, but their dynamics are unrealistic: forward implied vols are "sticky delta" in the wrong direction (empirically, vol tends to rise when $S$ falls more than local vol predicts). They also produce dynamics where the ATM vol smile flattens too fast as $T$ increases.

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11.3 Stochastic volatility: the Heston model

The Heston model adds a mean-reverting variance process:

$$dS_t = rS_t\,dt + \sqrt{V_t}\,S_t\,dW_t^S,$$ $$dV_t = \kappa(\theta - V_t)\,dt + \xi\sqrt{V_t}\,dW_t^V,$$ $$d[W^S, W^V]_t = \rho\,dt.$$

Here $V_t$ is the instantaneous variance (so $\sqrt{V_t}$ is the instantaneous volatility), $\theta$ is the long-run variance, $\kappa$ is the mean-reversion speed, $\xi$ is the vol-of-vol, and $\rho$ is the correlation.

Heston stochastic volatility model

κ=2.0θ=0.040ξ=0.50ρ=-0.70

Heston: vol is stochastic, mean-reverting. ρ < 0 (leverage effect) creates the implied volatility skew: OTM puts are expensive, OTM calls are cheap. Increasing ξ (vol-of-vol) steepens the smile. Feller condition 2κθ > ξ² keeps vol strictly positive.

Feller condition: $2\kappa\theta > \xi^2$ ensures $V_t > 0$ a.s. (Chapter 5).

Risk-neutral measure. The Heston model has one stock and two Brownian motions. The market is incomplete: there is a one-parameter family of risk-neutral measures indexed by the market price of variance risk $\lambda$. Typically one sets $\lambda = 0$ (zero variance risk premium) or calibrates $\lambda$ from the term structure of implied vol.

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11.4 The Heston characteristic function

The key tractability of the Heston model: the log-stock has an affine conditional characteristic function. Define $\phi(u;t,T,S,V) = \mathbb E^{\mathbb Q}[e^{iu\log S_T}\mid S_t,V_t]$. By affine theory:

$$\phi(u) = \exp\!\bigl(C(u,\tau) + D(u,\tau)V_t + iu\log S_t\bigr),$$

where $\tau = T-t$ and $(C,D)$ solve a system of Riccati ODEs.

Heston call price via Fourier inversion

The European call price is:

$$C = S_0 P_1 - Ke^{-rT}P_2,$$

where

$$P_j = \frac12 + \frac{1}{\pi}\int_0^\infty\text{Re}\!\left[\frac{e^{-iu\log K}\phi_j(u)}{iu}\right]du$$

and $\phi_j$ are related to $\phi$ by a measure-change. This is a one-dimensional numerical integral — fast to evaluate.

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11.5 The implied volatility smile from Heston

The Heston model produces a characteristic smile:

Parameter	Effect on smile
$\rho < 0$	Left skew (equity-like): high vol when $S$ falls
$\rho > 0$	Right skew (FX-like): high vol when $S$ rises
$\xi$ large	Wider smile (more curvature)
$\kappa$ large	Flatter long-term smile (fast mean-reversion)
$\theta$ large	Higher overall vol level

Leverage effect and ρ < 0

Empirically, equity implied vol rises sharply when the market falls. This "leverage effect" means $\rho < 0$: negative correlation between price moves and volatility moves. In the Heston model, $\rho$ is the correlation between $dW^S$ and $dW^V$. A fall in $S$ (driven by $dW^S < 0$) coincides with a rise in $V$ (driven by $dW^V > 0$ when $\rho < 0$), reproducing the observed negative equity skew.

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11.6 SABR model

The SABR (Stochastic Alpha Beta Rho) model is widely used in interest rates:

$$dF_t = \alpha_t F_t^\beta\,dW_t, \qquad d\alpha_t = \nu\alpha_t\,d\hat W_t, \qquad d[W,\hat W] = \rho\,dt.$$

For $\beta = 1$: SABR with lognormal backbone. For $\beta = 0$: normal SABR.

The Hagan–Kumar–Lesniewski–Woodward (HKLW) approximation gives an explicit formula for implied vol in terms of $(F, K, T, \alpha, \beta, \nu, \rho)$:

$$\sigma_{\rm imp}(K) \approx \frac{\alpha}{\cdots}\left[1 + \cdots\right].$$

The formula is used to interpolate the vol surface from calibrated SABR parameters.

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11.7 Local stochastic volatility (LSV)

A practical compromise between local vol (exact fit) and stochastic vol (realistic dynamics):

$$dS_t = rS_t\,dt + \sigma_{\rm loc}(t,S_t)\sqrt{V_t}\,S_t\,dW_t^S.$$

The function $\sigma_{\rm loc}$ is chosen so the model exactly reproduces the market surface, while $V_t$ provides stochastic dynamics. The calibration condition is:

$$\sigma_{\rm loc}^2(t,K) = \frac{\sigma_{\rm Dupire}^2(t,K)}{\mathbb E[\sqrt{V_t}\mid S_t = K]}.$$

This requires a Fokker–Planck PDE for the joint density of $(S_t,V_t)$.

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11.8 Problems

Problem 1: Dupire in BS

Show that for constant $\sigma$ (Black–Scholes), Dupire's formula gives $\sigma_{\rm loc}^2(T,K) = \sigma^2$. (Hint: use the BS call formula and compute the derivatives.)

Problem 2: Heston and the Feller condition

For Heston with $\kappa=1$, $\theta=0.04$, $\xi=0.3$: check the Feller condition $2\kappa\theta > \xi^2$. What happens numerically when you violate it?

Problem 3: Smile curvature

In the Heston model, the at-the-money implied vol is approximately $\sqrt{\theta}$. The smile curvature (second derivative with respect to $K$ at ATM) is driven by $\xi$. Show informally why larger $\xi$ creates a wider smile.

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11.9 Selected solutions

Solution 1

For Black–Scholes, $C(K,T) = S_0 N(d_1) - Ke^{-rT}N(d_2)$ with constant $\sigma$. Compute:

$\partial_T C =$ (Theta in terms of $\tau$) $= S_0 N'(d_1)\sigma/(2\sqrt{T}) - rKe^{-rT}N(d_2)$.

$\partial_K C = -e^{-rT}N(d_2)$.

$\partial_{KK} C = e^{-rT}N'(d_2)/(K\sigma\sqrt{T})$.

After substitution into Dupire's formula $\sigma_{\rm loc}^2 = (\partial_T C + rK\partial_K C)/(\frac12 K^2\partial_{KK}C)$, one can verify algebraically that $\sigma_{\rm loc}^2 = \sigma^2$. The key identity is $S_0 N'(d_1) = Ke^{-rT}N'(d_2)$.

Solution 2

Feller condition: $2\times 1\times 0.04 = 0.08 > 0.09 = 0.3^2$. The condition is NOT satisfied. When violated, $V_t$ touches zero and the simulation may become unstable or require reflection/truncation.

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11.10 What this chapter gives you

Model	Key property	Tractability
Local vol (Dupire)	Exact fit, deterministic $\sigma(t,S)$	Dupire PDE, closed-form $\sigma_{\rm loc}$
Heston	Stochastic $V_t$, affine structure	Fourier inversion, semi-closed form
SABR	Free-lognormal stochastic vol	HKLW explicit approximation
LSV	Exact fit + stochastic dynamics	Fokker–Planck calibration

The next chapter applies these tools to the interest rate market, where the "underlying" is the entire yield curve.