3.5 Advanced Volatility - Heston, SABR & Stochastic-Vol Dynamics

Basic Prerequisites: Volatility Surfaces & Smiles and Stochastic Calculus & Itô’s Lemma. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Black–Scholes–Merton has one free parameter and it is the wrong kind of object: a constant. Real volatility is itself a random process - it clusters, it mean-reverts, and it moves against the equity market. Once you accept that, the pricing problem changes shape: you have a second risk factor, you cannot hedge it with the underlying alone, and you must choose a dynamics - not merely a curve. This folder is the topic-hub for the two models that define the modern answer: Heston (1993), the tractable stochastic-volatility (SV) process with a quasi-closed-form price, and SABR (Hagan et al. 2002), the tractable smile expansion.

The one-sentence essence:

Stochastic volatility fixes Black–Scholes’ single- $σ$ failure by making variance a mean-reverting, spot-correlated diffusion; Heston makes that model computable (affine characteristic function ⇒ one-dimensional Fourier inversion) and SABR makes its smile closed-form in the short-expiration limit - but the smile’s shape barely discriminates between models, so the model choice is settled by the dynamics (vol-of-vol, skew term structure, skew stickiness), not by the static fit.

This folder is a hub: (a) the fast formula lookup below, and (b) six sub-pages that walk from raw intuition through the Heston SDE and its characteristic function, the SABR expansion and the short-expiration asymptotics, the Bergomi–Guyon dynamics, calibration/hedging practice, and the frontier (forward-variance, LSV, rough vol).

2. Mathematical Ground Truth & Derivations

Notation. $S$ spot, $F_{T} = S_{0} e^{(r - q) T}$ forward, $K$ strike, $T$ maturity, $τ = T - t$ , $v_{t}$ instantaneous variance, $\overset{v}{ˉ}$ its long-run mean, $λ$ (also written $κ$ ) the mean-reversion speed, $η$ (= Bergomi’s $σ$ ) the vol-of-vol, $ρ$ the spot/variance correlation, $k = ln (K / F_{T})$ log-moneyness, $σ_{BS}$ Black–Scholes implied vol, $φ_{T} (u) = E [e^{i u X_{T}}]$ the characteristic function of $X_{T} = ln (S_{T} / F_{T})$ .

Quick-Reference Lookup (job #1). Every formula is transcribed from the verified corpus (Gatheral, The Volatility Surface, ch 2–8; Bergomi, Stochastic Volatility Modeling, ch 6–10; Hagan et al. 2002) and every number in the check column was re-executed (§3 and the sub-pages).

Quantity	Formula	Verified check
Heston SDE (Gatheral 2.1)	$d S_{t} = v_{t} S_{t} d Z_{1}, d v_{t} = - λ (v_{t} - \overset{v}{ˉ}) d t + η v_{t} d Z_{2}, d Z_{1} d Z_{2} = ρ d t$	$φ (- i) = 1.000000000000$ at $T = 0.01, 1, 5$
Feller condition	$2 λ \overset{v}{ˉ} > η^{2} \Rightarrow v_{t} > 0$ a.s.	fitted SPX params: $0.09383 < 0.15031$ → violated
Milstein positivity	$4 λ \overset{v}{ˉ} / η^{2} > 1$	$1.24849 > 1$ → the discretisation still stays positive
Characteristic function (2.12/2.15)	$φ_{T} (u) = exp (C (u, τ) \overset{v}{ˉ} + D (u, τ) v_{0})$ , $D = r_{-} \frac{1 - e ^{- d τ}}{1 - g e ^{- d τ}}$ , $C = λ {r_{-} τ - \frac{2}{η ^{2}} ln \frac{1 - g e ^{- d τ}}{1 - g}}$ , $r_{\pm} = \frac{β \pm d}{η ^{2}}$ , $d = β^{2} - 4 α γ$ , $g = r_{-} / r_{+}$ , $α = - \frac{u ^{2}}{2} - \frac{i u}{2}$ , $β = λ - ρ η i u$ , $γ = \frac{η ^{2}}{2}$	reproduces BSM as $η \to 0$
Lewis / Carr–Madan price (5.6)	$C = F_{T} - \frac{F _{T} K}{π} \int_{0}^{\infty} \frac{d u}{u ^{2} + \frac{1}{4}} Re [e^{- i u k} φ_{T} (u - \frac{i}{2})]$	BSM limit: Heston $η \to 0$ gives $7.96607957$ vs BSM $7.96556746$
Heston price, $T = 1$ (Table 3.2)	(Lewis integral above)	$K = 80/90/100/110/120 \Rightarrow$ iv $19.573/16.895/14.234/11.989/10.993%$
Put–call parity	$C - P = e^{- r T} (F_{T} - K)$	residual $0.0$ at every strike
ATM implied variance (3.18)	$σ_{BS}^{2}_{K = F_{T}} = \frac{( v ˉ - v ˉ ^{'} ) ( 1 - e ^{- λ^{'} T} )}{λ ^{'} T} + \overset{v}{ˉ}^{'}$ , $λ^{'} = λ - \frac{ρ η}{2}$ , $\overset{v}{ˉ}^{'} = \overset{v}{ˉ} λ / λ^{'}$	order-1 in $η$ approximation only (see caveat)
Short-dated variance skew (3.19/7.3)	$\partial_{k} σ_{BS}^{2}_{k = 0} \to \frac{ρ η}{2}$	pricer: $- 0.138652$ vs $ρ η /2 = - 0.138894$
SABR model (Hagan 2002)	$d F_{t} = χ_{t} F_{t}^{β} d Z_{1}, d χ_{t} = ν χ_{t} d Z_{2}, d Z_{1} d Z_{2} = ρ d t$	-
SABR implied vol, $β = 1$ (7.7)	$σ_{BS} (k) = σ_{0} \frac{y}{f ( y )} (1 + \frac{1}{4} ρ ν σ_{0} + \frac{2 - 3 ρ ^{2}}{24} ν^{2} τ)$ , $y = - \frac{ν k}{σ _{0}}$ , $f (y) = ln \frac{1 - 2 ρ y + y ^{2} + y - ρ}{1 - ρ}$	ATM $= 19.7798%$ ; flat in $K$ to $1$ part in $1 0^{9}$ at $K \to F$
SABR ATM skew	$\partial_{k} σ_{BS}_{k = 0} = \frac{ρ ν}{2}$	$- 0.137364 = (ρ ν /2) \cdot 0.988991$
SABR ATM curvature	$C_{0} = \frac{( 2 - 3 ρ ^{2} ) ν ^{2}}{6 σ _{0}}$ (Bergomi–Guyon 8.39b)	$0.057605$ ; Hagan FD gives $0.056971$ = $\times 0.988991$
Bergomi–Guyon identity (8.40)	$ν^{2} = 3 σ_{0} C_{0} + 6 S_{0}^{2}$	$0.150311 = ν^{2}$ exactly
Heston as one-factor FV model (6.3/6.4)	$ξ_{t}^{T} = \overset{v}{ˉ} + e^{- λ (T - t)} (v_{t} - \overset{v}{ˉ})$ , $\overset{σ}{^}_{T}^{2} (t) = \overset{v}{ˉ} + \frac{1 - e ^{- λ (T - t)}}{λ ( T - t )} (v_{t} - \overset{v}{ˉ})$	T=0.05/0.25/1.000 yr $\Rightarrow$ $\overset{σ}{^}_{T} = 13.410/14.170/15.946%$
Heston ATMF skew, flat VS curve (6.20)	$S_{T} = \frac{ρ η}{2 v ˉ} \frac{λ T + e ^{- λ T} - 1}{( λ T ) ^{2}}$	short limit $\to \frac{ρ η}{4 v ˉ} = - 0.369105$ ; $T = 0.01$ gives $- 0.367480$
Vol-of-vol term structure (6.9)	$vol (\overset{σ}{^}_{T}) \propto \frac{1 - e ^{- λ (T - t)}}{λ ( T - t )}$	vs power law (7.40): Heston/power-law ratio $0.763$ ( $3$ m) … $0.587$ ( $5$ y)
Two-factor vol-of-vol (7.39)	$ν_{T} (t) = ν α_{θ} \sum_{ij} w_{i} w_{j} ρ_{ij} I (k_{i} (T - t)) I (k_{j} (T - t))$ , $I (x) = \frac{1 - e ^{- x}}{x}$	Set II reproduces benchmark (7.40) to $≲ 5%$ over $0.25$ – $5$ y
Digital price ↔ skew (ch 8)	$D (K, T) = - \frac{\partial C _{BS}}{\partial K} - \frac{\partial C _{BS}}{\partial σ} \frac{\partial σ _{BS}}{\partial K}$	$1$ y ATM digital, $25%$ vol, $3$ pts/ $10%$ skew: $0.1188$ = $11.9%$ of notional
Jump additivity at $τ \to 0$ (7.3)	$\partial_{k} σ_{BS}^{2}_{k = 0} \to ρ b (σ) - 2 μ_{J}$ , $μ_{J} = λ_{J} E [J]$	additive; slopes with $μ_{J}$
Rough benchmark (7.40)	$ν_{T} (t) = σ_{0} (\frac{τ _{0}}{T - t})^{α}$ , $α \approx 0.4$	power-law vol-of-vol term structure

Critical caveat (flagged in the corpus). Gatheral’s printed ATM term-structure formula (3.18) is built from the unconditional expected-variance path, so its literal $T \to 0$ limit is $\overset{v}{ˉ}$ (the long-run mean), not the current variance $v_{0}$ . The physical limit is $v_{0}$ : our characteristic-function pricer gives $σ_{BS}_{A TM} \to v_{0} = 13.191%$ as $T \to 0$ , while the literal (3.18) value at $T = 0.001$ is $0.0354$ ( $= \overset{v}{ˉ}$ ). §02 and §04 quantify the gap.

3. Computational Implementation - the SV engine

Stdlib only (math, cmath) - no numpy/scipy. The whole folder runs on two primitives: the Heston characteristic function (a Riccati system in closed form) and the Lewis inversion (one real integral). Every number in §2 and in the sub-pages was produced by these.

$φ (- i) = E [S_{T} / F_{T}] = 1$ is the martingale condition, and it pins down the normalisation of the whole construction: if the exponent were built from the wrong root $r_{+}$ or the wrong sign of $λ$ , this identity fails. It is the first thing to check on any new implementation.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

Calibrating the smile is not validating the model. Every SV-with-jumps model generates essentially the same surface shape (Gatheral ch 7 §7.8); only the dynamics - skew stickiness, vol-of-vol term structure, forward skew - discriminate. A Heston fit that matches today’s vanillas can still be badly wrong for cliquets and forward-skew products.
Heston’s structural pathologies are hard-wired, not fixable by calibration. Short ATMF vol is normal (not lognormal); the skew scales as $S_{T} \propto 1/ T$ (Type I) and as $1/ \overset{σ}{^}_{F_{T} T}$ (skew inversely tied to vol level); the vol-of-vol term structure is a fixed one-time-scale shape. Reality is Type II ( $S_{T} \propto T^{- 1/2}$ ) with $β (v) \sim v$ (lognormal variance).
Feller violation is the norm, not the exception. The canonical fitted SPX parameters violate $2 λ \overset{v}{ˉ} > η^{2}$ by a factor $1.6$ - the variance process does touch zero, and naive Euler simulation produces negative variance on $\sim 2%$ of steps.
SABR has no mean reversion. It is a short-expiration tool; using it for long maturities is a first-principles error, and its smile is asymptotic (arbitrageable far from the money, §03).
Model risk dwarfs parameter risk. For skew-dependent payoffs the choice of model moves the price by whole percentage points of notional (digitally capped cliquets, digitals, Napoleons) - far more than the uncertainty in any single parameter.

5. References

Gatheral, Jim: The Volatility Surface: A Practitioner’s Guide (Wiley, 2006)
Bergomi, Lorenzo: Stochastic Volatility Modeling (CRC, 2016)
Hagan, Kumar, Lesniewski, Woodward: Managing Smile Risk (Wilmott, 2002)
Hull, John C.: Options, Futures, and Other Derivatives (11th ed.)
Haug, Espen Gaarder: The Complete Guide to Option Pricing Formulas (2nd ed.)

6. Connected Graph Bridges

Foundational base: Stochastic Calculus & Itô’s Lemma · Probability & Measure Theory · Econometrics & Time Series
Sibling topics: Black–Scholes–Merton (the constant- $σ$ zero point) · Volatility Surfaces & Smiles (the empirical object these models must reproduce and move)
Related flat notes: Implied Volatility Surface & Smiles · The Greeks & Dynamic Hedging · Interest-Rate & Term-Structure Models (SABR’s home market)
Sub-pages (in-folder): 01 From Zero · 02 The Heston Model · 03 SABR & Asymptotics · 04 SV Dynamics · 05 Failure Modes & Practice · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05

Kwant Atlas

Explorer

3.5 Advanced Volatility - Heston, SABR & Stochastic-Vol Dynamics

1. Intuition & Practical Objective

2. Mathematical Ground Truth & Derivations

3. Computational Implementation - the SV engine

4. Failure Modes & First-Principles Breakdowns

5. References

6. Connected Graph Bridges

3.5.1 Advanced Volatility from Zero

3.5.2 The Heston Model

3.5.3 SABR & the Short-Expiration Asymptotics

3.5.4 Stochastic-Vol Dynamics

3.5.5 Failure Modes & Practice

3.5.6 Advanced Extensions