TheQuantBateman

01

INTUITION

Build the state before the equation.

Local volatility replaces one constant σ with a deterministic function σloc(S,t). Given a smooth arbitrage-consistent vanilla surface, Dupire identifies a diffusion that matches all European marginal distributions in ideal theory.

01

Exact vanilla fit is a static statement.

02

Differentiation amplifies quote noise.

03

Forward smile dynamics can be less realistic than the initial fit.

02

WHY MARKETS CARE

The product exists before the model.

Local volatility is a baseline for barrier and path-dependent pricing and a building block for local-stochastic volatility models.

INSTRUMENTS

barriers

digitals

autocallables

local-stochastic-volatility hybrids

QUOTE CONVENTION

Surface inputs must share forward, discount, dividend, expiry and strike conventions. Local variance is annualized decimal variance.

03

MATHEMATICS

Notation, units and exact claims.

σloc(S,t): local volatilityC(K,T): call surfaceD(0,T): discount factor∂KKC: density term

QUANT NOTESYNTHETIC · EDUCATIONAL

Local-vol diffusion

Volatility is deterministic conditional on current spot and time.

dS_t=(r-q)S_tdt+\sigma_{loc}(S_t,t)S_tdW_t

Read the equation together with its financial domain and convention.

QUANT NOTESYNTHETIC · EDUCATIONAL

Dupire formula

The call surface determines local variance where derivatives and density are well behaved.

\sigma_{loc}^2(K,T)=\frac{\partial_TC+(r-q)K\partial_KC+qC}{\tfrac12K^2\partial_{KK}C}

Read the equation together with its financial domain and convention.

QUANT NOTESYNTHETIC · EDUCATIONAL

Density requirement

A small or negative denominator makes inversion unstable or invalid.

\partial_{KK}C(K,T)>0

Read the equation together with its financial domain and convention.

04

DERIVATION

Do not jump to the final expression.

DERIVATION

Inverting the forward equation

Match the Fokker–Planck evolution of the local-vol diffusion to strike derivatives of call prices.

01
Start with the diffusion
Specify risk-neutral drift and a state-time diffusion coefficient.
$dS=(r-q)Sdt+\sigma_{loc}(S,t)SdW$
02
Write density evolution
The forward Kolmogorov equation evolves terminal density under the diffusion.
03
Connect density to calls
Use Breeden–Litzenberger to replace terminal density by the second strike derivative of call prices.
$\partial_{KK}C=Df(K,T)$
04
Differentiate through maturity
Differentiate the discounted payoff expectation in T and substitute the density evolution.
05
Isolate local variance
Rearrange the forward PDE to obtain Dupire’s numerator over the convexity denominator.
NUMERICAL CHECKSmooth prices or total variance before differentiating; raw finite differences are unstable.

Dupire converts a full static surface into one diffusion, but the inversion is only as reliable as surface cleaning, smoothing and boundary treatment.

05

MODEL / PRICING

Fit, compute, then challenge the assumptions.

METHOD

Build an arbitrage-controlled call-price surface, compute stable derivatives, floor diagnostics rather than values, and solve PDEs with consistent boundaries.

CALIBRATION

The model is implied from the vanilla surface rather than optimized to it; smoothing hyperparameters are the effective calibration choices.

LIMITATIONS

Noise amplification in derivatives.

Unrealistic forward smile dynamics can mis-hedge exotics.

Boundary extrapolation materially changes local vol.

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 separates market structures, processes, instruments, engines and calibration helpers. Use those abstractions only after the lesson’s conventions, domains and numerical checks are explicit.

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

06

PYTHON LAB

Theory → implementation → checks.

PYTHON 3 · NUMPY / SCIPY

Dupire denominator diagnostic

Detect unstable density denominators before computing local variance.

REUSABLE EXAMPLE

01from __future__ import annotations

02

03import numpy as np

04

05def convexity(strikes: np.ndarray, calls: np.ndarray) -> np.ndarray:

06 if strikes.shape != calls.shape or strikes.size < 5:

07 raise ValueError("aligned grid with at least five nodes required")

08 h = np.diff(strikes)

09 if not np.allclose(h, h[0]):

10 raise ValueError("teaching implementation requires uniform strikes")

11 return (calls[:-2] - 2.0 * calls[1:-1] + calls[2:]) / h[0]**2

12

13k = np.arange(80.0, 125.0, 5.0)

14c = np.array([22.0, 18.0, 14.5, 11.5, 9.0, 7.0, 5.5, 4.5, 3.8])

15density_term = convexity(k, c)

16assert np.isfinite(density_term).all()

17assert np.all(density_term > 0.0)

18print(float(density_term.min()))

07

INTERACTIVE LAB

Move the state. Challenge the equation.

SURFACE DERIVATIVES AND DUPIRE STABILITY

Dupire stability lab

Change smoothing and grid spacing; inspect call convexity, local variance and failure regions.

SYNTHETIC · CONTROLLED SCENARIOS

SCENARIO INTENSITY70%

ATM local vol21.96%

Max local vol27.40%

Noise stateSmooth

08

FRONT OFFICE

Where the model meets the book.

ON THE DESK

“The calibration is exact because the interpolation made the difficult decisions first.”

VISIBLE INPUTS

clean call surface

curves and forwards

smoothing rule

grid

boundaries

CALIBRATION

The model is implied from the vanilla surface rather than optimized to it; smoothing hyperparameters are the effective calibration choices.

RISK

local-vol vega

barrier sensitivity

surface derivative risk

grid risk

DAILY WORKFLOW

clean prices
enforce shape
differentiate
diagnose denominator
solve PDE

Production failure modes

negative density
noisy time derivative
wing explosion
boundary artifacts

09

MACRO CONNECTION

Map the transmission channel.

MACRO CONNECTION

Spot moves through a state-dependent surface

A local-vol model maps today’s skew into spot-dependent future diffusion, linking selloffs to higher instantaneous volatility without an independent variance shock.

Current skew

defines state dependence

Spot selloff

moves into higher local vol

Path distribution

changes barrier probabilities

Exotic hedge

inherits local dynamics

10

COMMON PITFALLS

Most failures begin outside the formula.

01

Differentiating raw implied vols.

02

Calling exact vanilla fit a validation of dynamics.

03

Flooring negative local variance silently.

04

Ignoring extrapolation in path-dependent pricing.

11

SOURCES / FURTHER READING

Attribution with implementation authority.

researchBSD-3-Clause

Volatility, Monte Carlo and stochastic-volatility lectures

Research map for the mathematical progression and numerical experiments; prose, examples and code are original.

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

OPEN ORIGINAL SOURCE ↗LICENSE ↗

implementation referenceQuantLib permissive license

Current volatility structures, processes, calibration helpers and tests

Implementation reference for production abstractions and validation patterns.

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

OPEN ORIGINAL SOURCE ↗LICENSE ↗

Local-vol diffusion

Dupire formula

Density requirement

Inverting the forward equation

Start with the diffusion

Write density evolution

Connect density to calls

Differentiate through maturity

Isolate local variance

Volatility, Monte Carlo and stochastic-volatility lectures

Current volatility structures, processes, calibration helpers and tests