TheQuantBateman

01

INTUITION

Identify the state variables and the behavior they add.

Heston replaces constant volatility with a random, mean-reverting variance process. Negative price–variance correlation creates equity-like downside skew; vol-of-vol controls smile curvature; mean reversion governs how quickly the variance state forgets shocks.

01

v₀ anchors short-dated variance.

02

θ anchors the long-run variance level.

03

κ controls mean-reversion speed, σᵥ the variance-of-variance and ρ the leverage channel.

02

WHY MARKETS CARE

Ask which instruments can identify the dynamics.

One parameter set can generate a full surface and richer dynamics than Black–Scholes, making Heston a durable benchmark for exotics, calibration studies and model comparison.

INSTRUMENTS

vanilla option calibration grids

barriers and forward-starts

variance-sensitive exotics

QUOTE CONVENTION

Calibration inputs must first be normalized under the asset class’s actual quote and forward conventions. Heston parameters are not portable across inconsistent surfaces.

03

MATHEMATICS

Write the dynamics before interpreting parameters.

Formula · Full derivation

Affine characteristic function

\phi(u,\tau)=\exp(C(u,\tau)+D(u,\tau)v_t+iux_t)

The conditional log-price transform is exponential-affine in the current variance and log spot, reducing the pricing problem to Riccati equations.

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Full derivation

Why the Heston characteristic function is affine

Fourier pricing is practical because the log-price/variance transform has an exponential-affine form.

01
State the risk-neutral spot dynamics
Begin with the joint risk-neutral spot and variance state; the spot diffusion uses the same stochastic variance factor that enters the affine transform.
$dS_t=(r-q)S_tdt+\sqrt{v_t}S_tdW_t^S$
02
Transform the state
Use log spot x = log S so the diffusion generator is polynomial-affine in variance.
$x_t=\log S_t$
03
Propose an affine transform
Conditioned on the current state, assume the characteristic function is exponential-affine in log spot and variance.
$\phi(u,\tau)=\exp(C(u,\tau)+D(u,\tau)v_t+iux_t)$
04
Insert into the backward equation
Matching constant and variance coefficients produces coupled Riccati ordinary differential equations for C and D.
$\partial_\tau D=a(u)+b(u)D+cD^2$
05
Solve and integrate
The closed transform is inserted into a stable Fourier inversion or COS method to recover vanilla prices.
$C(K,T)=\mathcal F^{-1}[\phi(u,T)]$
Numerical checkImplementation must manage branch choices, integration domains and numerical cancellation.

Affine structure makes Heston computationally tractable; it does not make calibration uniquely identified or hedging dynamics correct by construction.

Inputs

vₜ: instantaneous variance
κ: mean-reversion speed
θ: long-run variance
σᵥ: vol-of-vol
ρ: Brownian correlation
φ(u,τ): conditional characteristic function

Assumptions and limits

Five parameters can be weakly identified on sparse grids.
A good vanilla fit does not guarantee exotic hedge performance.
Naive Euler variance discretization can become negative.

Formula · Full derivation

Variance process

dv_t=\kappa(\theta-v_t)dt+\sigma_v\sqrt{v_t}dW_t^v

CIR-style mean reversion supports a non-negative variance state under suitable schemes.

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Formula · Definition

Leverage correlation

d\langle W^S,W^v\rangle_t=\rho\,dt

Negative correlation links spot selloffs to variance shocks and generates downside skew.

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Formula · Short derivation

Feller condition

2\kappa\theta\ge\sigma_v^2

A sufficient condition for the continuous-time variance process to stay strictly positive; market calibrations can violate it, demanding careful numerics.

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05

MODEL / PRICING

Calibrate, compute, and challenge the dynamics.

METHOD

Use an analytical characteristic-function engine, COS/Fourier inversion, PDE or a validated Monte Carlo scheme depending on payoff and required sensitivities.

CALIBRATION

Minimize weighted premium or volatility residuals across a selected grid. Constrain parameters, use multiple initial guesses and report parameter stability alongside fit error.

MODEL COMPARISON

Static fit is not dynamics.

Question	Black–Scholes	Local volatility	Heston
Volatility state	One constant σ	σ(S,t) deterministic	vₜ stochastic
Fits today’s surface	No	Exactly, in ideal theory	Approximately by calibration
Forward dynamics	Flat smile	Spot-driven	Variance + correlation driven
Primary strength	Transparent baseline	Vanilla-consistent diffusion	Richer smile dynamics
Primary failure	No smile	Often unrealistic forward skew	Parameter and calibration instability
Compute	Low	Medium: PDE/MC	Medium–high: Fourier/PDE/MC
Hedge implication	Greeks at one σ	State-localized vol hedge	Variance and vol-of-vol risk

Implementation with current QuantLib

Current QuantLib exposes HestonProcess, HestonModel and several engines. The process/model/engine separation is valuable, but API use comes after convention alignment and numerical validation.

API authority: upstream QuantLib reference pinned in the source registry.

06PYTHON IMPLEMENTATIONOpen the implementation and checks.

ARCHITECTURE

Separate the Heston process from pricing engines.
Validate parameter domains and correlation.
Benchmark analytical, numerical and Monte Carlo prices.
Track fit error and parameter drift through calibration runs.

PYTHON 3 · NUMPY / SCIPY

Full-truncation Heston paths

Simulate a stable educational path set with deterministic randomness and non-negative variance in diffusion terms.

REUSABLE EXAMPLE

01from __future__ import annotations

02

03import numpy as np

04

05def heston_paths(*, paths: int = 20_000, steps: int = 252, seed: int = 7) -> tuple[np.ndarray, np.ndarray]:

06 rng = np.random.default_rng(seed)

07 dt = 1.0 / steps

08 spot = np.full(paths, 100.0)

09 variance = np.full(paths, 0.04)

10 kappa, theta, vol_of_vol, rho, rate = 2.0, 0.04, 0.45, -0.70, 0.03

11 for _ in range(steps):

12 z1 = rng.standard_normal(paths)

13 z2 = rho * z1 + np.sqrt(1.0 - rho**2) * rng.standard_normal(paths)

14 v_pos = np.maximum(variance, 0.0)

15 spot *= np.exp((rate - 0.5 * v_pos) * dt + np.sqrt(v_pos * dt) * z1)

16 variance += kappa * (theta - v_pos) * dt + vol_of_vol * np.sqrt(v_pos * dt) * z2

17 return spot, np.maximum(variance, 0.0)

18

19spot_t, variance_t = heston_paths()

20assert np.isfinite(spot_t).all() and np.all(spot_t > 0.0)

21assert np.isfinite(variance_t).all() and np.all(variance_t >= 0.0)

22print(round(float(spot_t.mean()), 4), round(float(variance_t.mean()), 6))

07

INTERACTIVE LAB

Shock one parameter and trace the full response.

HESTON VARIANCE STATE AND LEVERAGE CORRELATION

Heston dynamics lab

Move mean reversion, long-run variance, vol-of-vol and correlation; inspect variance persistence and smile response.

SYNTHETIC · CONTROLLED SCENARIOS

SCENARIO INTENSITY70%

ρ-0.75

Vol-of-vol0.45

Long-run vol20.00%

Variance / smile response by Maturity (years)

Negative rho: Selloffs lift variance and downside skew.

expected variance
downside skew response

Maturity (years): 0.0Y. expected variance: 0.0776. downside skew response: 0.0187.

Use Left/Right or Up/Down arrows to inspect values; Home and End jump to the bounds.

08

FRONT OFFICE

Where the model meets the book.

ON THE DESK

“Calibration error is visible. Parameter instability is often more expensive.”

VISIBLE INPUTS

clean vanilla surface

curves and forwards

calibration weights

parameter bounds

engine tolerances

CALIBRATION

Use liquid strikes, multiple starts and stable parameter transforms. Evaluate premium residuals, bid/offer coverage and day-over-day parameter movement.

RISK

surface-vega buckets

spot/variance correlation

vol-of-vol exposure

forward smile dynamics

calibration jump risk

DAILY WORKFLOW

freeze market snapshot
select instruments
calibrate
validate repricing
compare parameters
run exotic risk
approve or fall back

Production failure modes

local optimizer traps
characteristic-function branch errors
bad variance discretization
unstable finite-difference Greeks
parameters pinned at constraints

09MACRO CONNECTIONOpen the transmission channel.

MACRO CONNECTION

Leverage and variance regimes

Risk-off moves often combine falling spot, higher variance and stronger downside skew—the channel represented by negative spot–variance correlation.

01Risk-off shocktransmits

spot falls and protection demand rises

02Variance statetransmits

jumps above long-run mean

03Mean reversiontransmits

controls decay of the shock

04Option surfaceoutput

level and skew reprice

10COMMON PITFALLSOpen the failure checklist.

01

Reading calibrated parameters as directly observable economic constants.

02

Ignoring Feller violations in simulation choices.

03

Comparing fits without the same quote weights.

04

Using one calibration start and declaring uniqueness.

11SOURCES / FURTHER READINGOpen sources and continue the track.

research

Lectures 07 and 10 — stochastic volatility and Heston Monte Carlo

Research source for model progression and numerical experiments; text and code are independently implemented.

Source: Computational Finance Course
Author: L. A. Grzelak
Ref: main

OPEN ORIGINAL SOURCE ↗

implementation reference

Heston process, model, engines and test suite

Current implementation reference and validation architecture.

Source: QuantLib upstream
Author: QuantLib contributors
Ref: v1.42.1

OPEN ORIGINAL SOURCE ↗

Why the Heston characteristic function is affine

State the risk-neutral spot dynamics

Transform the state

Propose an affine transform

Insert into the backward equation

Solve and integrate

Leverage and variance regimes

Lectures 07 and 10 — stochastic volatility and Heston Monte Carlo

Heston process, model, engines and test suite