Phase 15 - Lesson 15.4

Stochastic Volatility and the Heston Model

Giving volatility its own random driver - and getting realistic smile dynamics in return.

⏱ 55 min● Advanced🔗 Prereqs: 15.3

Builds on: 15.3’s deterministic local vol produced wrong forward-smile dynamics.
Leads to: 15.5 assembles these into the full surface and its arbitrage constraints.

Learning Objectives

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Write the Heston model and identify the role of each parameter (\(\kappa,\theta,\xi,\rho,v_0\)).
Explain how a second Brownian driver and spot–vol correlation generate a persistent skew.
State the Feller condition and why it matters for positivity of the variance process.
Describe how Heston is priced via its characteristic function and Fourier inversion.
Simulate the Heston SDEs and price an option by Monte Carlo, contrasting with local vol.

Key Vocabulary

Stochastic volatility: A model in which volatility itself follows a random process driven by its own Brownian motion.

Heston model: Variance follows a square-root (CIR) mean-reverting diffusion, correlated with the spot’s Brownian.

Mean reversion \(\kappa\): The speed at which variance is pulled back toward its long-run level \(\theta\).

Vol of vol \(\xi\): The volatility of the variance process; controls the convexity (curvature) of the smile.

Correlation \(\rho\): Correlation between the spot and variance shocks; its sign sets the skew direction.

Feller condition: \(2\kappa\theta\ge\xi^2\): guarantees the variance process stays strictly positive.

Characteristic function: \(\E[e^{iu\ln S_T}]\); known in closed form for Heston, enabling Fourier pricing.

Intuition & Motivation

Intuition

Local vol made volatility a function of the spot; stochastic vol makes it a process with its own noise. In Heston the instantaneous variance \(v_t\) mean-reverts (calm and turbulent regimes) and is correlated with the stock’s shocks. That correlation is the key: when \(\rho\lt 0\), falling prices coincide with rising vol, which fattens the left tail and produces a persistent equity skew that does not flatten unrealistically as the spot moves. You buy two things over local vol - realistic forward-smile dynamics and a genuine vega/vol-of-vol risk - at the cost of an incomplete market (volatility risk is not perfectly hedgeable) and more parameters.

The Heston model

Under the risk-neutral measure the spot and its instantaneous variance \(v_t\) solve

\[dS_t=rS_t\,dt+\sqrt{v_t}\,S_t\,dW_t^{S},\qquad dv_t=\kappa(\theta-v_t)\,dt+\xi\sqrt{v_t}\,dW_t^{v},\] (15.9)

\[d\langle W^S,W^v\rangle_t=\rho\,dt.\] (15.10)

The variance is a Cox–Ingersoll–Ross square-root process: it mean-reverts to \(\theta\) at speed \(\kappa\), with volatility \(\xi\). The \(\sqrt{v_t}\) diffusion keeps variance non-negative, and under the Feller condition

\[2\kappa\theta\ge\xi^2\] (15.11)

it stays strictly positive (never touches zero). Each parameter shapes the smile in an interpretable way:

Parameter	Name	Effect on the smile
\(v_0\)	initial variance	Overall ATM level (short end)
\(\theta\)	long-run variance	ATM level at long maturities
\(\kappa\)	mean-reversion speed	Term structure: how fast smile approaches \(\sqrt\theta\)
\(\xi\)	vol of vol	Convexity / curvature (smile ‘wings’)
\(\rho\)	spot–vol correlation	Skew: \(\rho\lt 0\) tilts to a down-sloping equity skew

Key Idea

Correlation \(\rho\) does what a deterministic local-vol slope cannot: it makes the skew a genuine feature of the dynamics, so the model’s forward smile stays realistic. That is the whole reason to pay for the extra factor.

Pricing by the characteristic function

Heston has no closed-form call price, but the characteristic function of \(x_T=\ln S_T\) is known explicitly:

\[\phi_T(u)=\E\big[e^{iu x_T}\big]=\exp\!\big(C(u,T)+D(u,T)v_0+iu x_0\big),\] (15.12)

where \(C\) and \(D\) solve Riccati ODEs in closed form. The call price then follows from a single numerical Fourier integral (Heston’s formula, or the Carr–Madan FFT). This makes calibration to a whole surface fast - the reason Heston is an industry workhorse.

Worked Example - How each parameter moves the smile

Raise \(v_0\): the entire short-dated smile shifts up (more ATM vol now).

Raise the vol of vol \(\xi\): the smile gains convexity - wings lift relative to ATM (fatter both tails).

Make \(\rho\) more negative: the smile tilts to a steeper down-sloping skew (rich OTM puts) - the equity signature.

Raise \(\kappa\): variance reverts faster, so long-dated ATM vol pins closer to \(\sqrt\theta\) and the term structure steepens toward it.

Check \(2\kappa\theta\ge\xi^2\): if violated, simulated variance can hit zero and full-truncation schemes are needed.

Interactive: simulate Heston and price by Monte Carlo

Discretize (15.9) with correlated normals and full-truncation on the variance, then average the discounted payoff.

Common Mistakes to Avoid

Letting the discretized variance go negative: without full truncation (or a scheme like QE) \(\sqrt{v}\) becomes complex and the sim blows up.
Ignoring the Feller condition when \(\xi\) is large: near-zero variance needs careful handling.
Assuming volatility risk is hedgeable: stochastic vol makes the market incomplete; a variance risk premium appears in the pricing measure.
Reusing local-vol intuition for exotics: Heston’s forward smile differs, which is exactly why you switched models.

Quant Practitioner Tips

Calibrate via the characteristic function/FFT, not Monte Carlo - it is orders of magnitude faster for a full surface.
\(\rho\) controls skew and \(\xi\) controls convexity: use them to fit the smile’s slope and curvature respectively.
For accurate simulation use the Andersen QE scheme; plain Euler needs many steps and still biases the wings.
Blend with local vol (local-stochastic vol) when you must fit vanillas exactly and keep realistic dynamics.

Knowledge Check

Q1 Medium

In the Heston model the parameter that primarily controls the direction (sign) of the volatility skew is:

The mean-reversion speed \(\kappa\)

The long-run variance \(\theta\)

The spot–variance correlation \(\rho\)

The initial variance \(v_0\)

Q2 Medium

The Feller condition \(2\kappa\theta\ge\xi^2\) ensures that:

The smile is symmetric

The variance process stays strictly positive and never hits zero

The model has a closed-form call price

Volatility risk is hedgeable

Q3 Hard

The main advantage of Heston over pure local volatility is that it:

Fits today’s vanilla surface more exactly

Produces realistic forward-smile dynamics via a genuine second (correlated) volatility factor

Requires fewer parameters

Makes the market complete

Practical Exercise

(a) Write the Heston SDEs and label each parameter. (b) You calibrate to an equity index and obtain \(\rho=-0.7\), \(\xi=0.5\), \(\kappa=1.2\), \(\theta=0.04\). Interpret the sign of \(\rho\) and check the Feller condition for \(\kappa\theta\). (c) Explain why, unlike Black–Scholes, Heston implies an incomplete market and where that shows up in pricing.

▶ Show full solution

(a) \(dS=rS\,dt+\sqrt v\,S\,dW^S\), \(dv=\kappa(\theta-v)dt+\xi\sqrt v\,dW^v\), \(d\langle W^S,W^v\rangle=\rho\,dt\): \(v_0\) initial variance, \(\theta\) long-run variance, \(\kappa\) reversion speed, \(\xi\) vol of vol, \(\rho\) spot–vol correlation.

(b) \(\rho=-0.7\lt 0\) means volatility rises when the index falls - the leverage effect - producing a down-sloping skew with rich OTM puts. Feller: \(2\kappa\theta=2(1.2)(0.04)=0.096\) versus \(\xi^2=0.25\). Since \(0.096\lt 0.25\), the Feller condition is violated: variance can reach zero, so simulation needs full-truncation/QE handling and the wing fit should be checked.

(c) Volatility is driven by \(W^v\), which is not a traded asset, so you cannot perfectly hedge vol risk with spot and cash alone - the market is incomplete. Consequently the risk-neutral measure is not unique; a variance (volatility) risk premium enters, appearing as an adjustment to the drift of \(v\) under \(\Q\). Prices depend on that premium, unlike the unique Black–Scholes price.

After the reveal, answer for yourself: If Feller is violated in calibration, would you re-calibrate with a penalty, or accept it and fix the simulation scheme?

Lesson Summary

Heston gives volatility its own mean-reverting square-root process, correlated with the spot. The correlation \(\rho\) generates a persistent, realistic skew and the vol-of-vol \(\xi\) its convexity; the Feller condition \(2\kappa\theta\ge\xi^2\) keeps variance positive. Pricing uses the closed-form characteristic function and Fourier inversion, making surface calibration fast. Unlike local vol it delivers realistic forward-smile dynamics, at the cost of an incomplete market and a volatility risk premium.

Formula Sheet Additions

Heston dynamics

\[dS=rS\,dt+\sqrt v\,S\,dW^S,\quad dv=\kappa(\theta-v)dt+\xi\sqrt v\,dW^v,\quad d\langle W^S,W^v\rangle=\rho\,dt\]

Mean-reverting CIR variance correlated with the spot; \(\rho\) sets skew, \(\xi\) convexity.

Feller condition

\[2\kappa\theta\ge\xi^2\]

Keeps the variance process strictly positive.

Error Log Checklist

Did I apply full truncation so variance never goes negative in simulation?
Did I check the Feller condition against my calibrated \(\xi\)?
Did I interpret \(\rho\) (skew) and \(\xi\) (convexity) correctly?
Did I acknowledge market incompleteness / the variance risk premium?

Retrieval Practice

Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.

▶ Show retrieval prompts & answers

Q: How does Heston generate a persistent equity skew that local vol cannot?
A: Through a negative correlation \(\rho\) between spot and variance shocks: vol rises as the spot falls, fattening the left tail as a dynamic (forward-consistent) feature.

Q: State the Feller condition and what it guarantees.
A: \(2\kappa\theta\ge\xi^2\); it guarantees the CIR variance process stays strictly positive (never reaches zero).

Q: Why is the Heston market incomplete, and how does that affect pricing?
A: Volatility is driven by a non-traded Brownian, so vol risk can’t be perfectly hedged; the risk-neutral measure is non-unique and a variance risk premium enters the pricing.

Flashcards

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Heston model

CIR variance \(dv=\kappa(\theta-v)dt+\xi\sqrt v\,dW^v\), correlation \(\rho\) with spot; stochastic vol.

Feller condition

\(2\kappa\theta\ge\xi^2\) keeps variance strictly positive.

\(\rho\) vs \(\xi\)

\(\rho\) sets skew direction; \(\xi\) (vol of vol) sets smile convexity.

Completion Checklist

I can explain the core ideas in my own words
I worked the derivations/examples by hand
I completed the interactive workbench(es)
I passed the knowledge check

Confidence / mastery rating

Personal notes

Source References

This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.

The Volatility Surface (Jim Gatheral, 2006) foundational - Ch. 2–3 - Ch. 2–3: the Heston model, its characteristic function, and the parameter–smile map.
Stochastic Calculus for Finance II (Steven Shreve, 2004) foundational - Ch. 6–8 - Ch. 6–8: multi-factor diffusions, the CIR/square-root process, and incomplete-market pricing.
Methods of Mathematical Finance (Karatzas & Shreve, 1998) foundational - Ch. 1–2 - Ch. 1–2: incomplete markets, non-unique martingale measures, and risk premia.

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