TheQuantBateman

01

INTUITION

Build the state before the equation.

A volatility surface stacks option-implied volatility across strike and expiry. Every visible point is a compact encoding of an option price under a stated convention; every gap between liquid points is a modelling decision.

01

A smile slice answers how downside, ATM and upside options differ at one expiry.

02

A term slice answers how the market distributes uncertainty and event risk through time.

03

The fitted surface marks vanilla books today. A dynamics model determines how it moves tomorrow.

02

WHY MARKETS CARE

The product exists before the model.

Desks need continuous values for marking, interpolation, scenario risk and calibration even though only a sparse set of options trades reliably.

INSTRUMENTS

listed or OTC vanilla grids

variance-sensitive products

barriers and digitals

forward-start and cliquet structures

QUOTE CONVENTION

This lesson uses spot moneyness K/S and ACT/365-like maturity for education. Production equity, FX and rates surfaces require their native forward, delta, premium, ATM and normal/lognormal conventions.

03

MATHEMATICS

Notation, units and exact claims.

k = log(K/F_T): log-forward moneynessw(k,T) = σ_imp²(k,T)T: total implied varianceC(K,T): call-price surfaceσ_loc(K,T): local volatility

QUANT NOTESYNTHETIC · EDUCATIONAL

Surface coordinate

Forward log-moneyness makes strike geometry comparable across maturities.

(k,T)\mapsto \sigma_{imp}(k,T),\qquad k=\log(K/F_T)

Read the equation together with its financial domain and convention.

QUANT NOTESYNTHETIC · EDUCATIONAL

Total variance

Many no-arbitrage and interpolation questions are more naturally expressed in total variance than volatility.

w(k,T)=\sigma_{imp}^2(k,T)T

Read the equation together with its financial domain and convention.

QUANT NOTESYNTHETIC · EDUCATIONAL

Risk-neutral density

Convex call prices imply a non-negative risk-neutral terminal density.

\partial_{KK}C(K,T)=D(0,T)\,f_{S_T}^{\mathbb Q}(K)\ge 0

Read the equation together with its financial domain and convention.

QUANT NOTESYNTHETIC · EDUCATIONAL

Dupire local variance

A sufficiently smooth arbitrage-consistent vanilla surface identifies one local-volatility diffusion.

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

Read the equation together with its financial domain and convention.

04

DERIVATION

Do not jump to the final expression.

DERIVATION

From a call-price surface to local volatility

Dupire’s result is an inversion of the forward equation. It shows why surface smoothness and convexity are numerical requirements, not styling preferences.

01
Assume risk-neutral local dynamics
Let instantaneous volatility depend on state and time while the drift remains risk-neutral.
$dS_t=(r-q)S_tdt+\sigma_{loc}(S_t,t)S_tdW_t$
02
Write the forward density equation
The transition density evolves under the Fokker–Planck equation associated with the diffusion.
$\partial_T p=-\partial_S((r-q)Sp)+\tfrac12\partial_{SS}(\sigma_{loc}^2S^2p)$
03
Differentiate option prices by strike
Breeden–Litzenberger links the strike curvature of call prices to the discounted risk-neutral density.
$\partial_{KK}C=D(0,T)p(K,T)$
04
Differentiate by maturity
Insert the forward density evolution into the maturity derivative of the call payoff integral and integrate by parts.
$C(K,T)=D(0,T)\int_K^{\infty}(S-K)p(S,T)dS$
05
Isolate local variance
Rearrange the resulting forward PDE. A non-positive denominator or noisy derivative makes the inferred local variance unstable.
$\sigma_{loc}^2=\frac{\partial_T C+(r-q)K\partial_K C+qC}{\tfrac12K^2\partial_{KK}C}$
NUMERICAL CHECKThis is a theoretical identity; production implementations require smooth arbitrage-aware interpolation and boundary handling.

A surface is simultaneously market data, an interpolation object and an input to dynamics. Those roles should never be silently conflated.

05

MODEL / PRICING

Fit, compute, then challenge the assumptions.

METHOD

Normalize option quotes, infer implied volatilities, choose a stable coordinate system, fit each slice, connect maturities and validate reconstructed premiums and static-arbitrage diagnostics.

CALIBRATION

The educational workbench exposes level, skew, curvature and term slope directly. Production calibration minimizes weighted premium or volatility errors subject to stability, liquidity and no-arbitrage constraints.

LIMITATIONS

Sparse wings force extrapolation choices.

A perfect static fit does not guarantee realistic forward dynamics.

Volatility-space interpolation can hide option-price arbitrage.

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 1.42 introduced PiecewiseBlackVarianceSurface for ragged grids and provides Black variance surfaces, smile sections and SABR structures. A library supplies tested representations; the user must still choose conventions, quotes, interpolation and extrapolation deliberately.

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

06

PYTHON LAB

Theory → implementation → checks.

PYTHON 3 · NUMPY / SCIPY

Vectorized educational surface

Build one deterministic surface grid with explicit parameters, units and sanity checks.

REUSABLE EXAMPLE

01from __future__ import annotations

02

03from dataclasses import dataclass

04import numpy as np

05from numpy.typing import NDArray

06

07FloatArray = NDArray[np.float64]

08

09@dataclass(frozen=True)

10class SurfaceShape:

11 atm: float = 0.20

12 skew: float = -0.18

13 curvature: float = 0.55

14 term_slope: float = 0.025

15

16def educational_surface(

17 log_moneyness: FloatArray,

18 maturity: FloatArray,

19 shape: SurfaceShape,

20) -> FloatArray:

21 """Return a deterministic teaching surface; never market data."""

22 if np.any(maturity <= 0.0):

23 raise ValueError("maturity must be positive")

24 vol = (

25 shape.atm

26 + shape.skew * log_moneyness

27 + shape.curvature * log_moneyness**2

28 + shape.term_slope * np.log1p(maturity)

29 )

30 return np.maximum(vol, 0.01)

31

32tenors = np.array([7 / 365, 30 / 365, 0.25, 0.5, 1.0, 2.0])

33moneyness = np.linspace(0.70, 1.30, 25)

34k = np.log(moneyness)[None, :]

35t = tenors[:, None]

36surface = educational_surface(k, t, SurfaceShape())

37

38assert surface.shape == (tenors.size, moneyness.size)

39assert np.isfinite(surface).all() and np.all(surface > 0.0)

40print(f"ATM 1Y: {surface[4, 12]:.2%}")

07

INTERACTIVE LAB

One surface, four linked views.

VOL SURFACE LAB · ONE LINKED STATE

Implied volatility surface

Inspect the same deterministic grid as a heatmap, rotatable surface, smile or term slice.

SYNTHETIC / EDUCATIONALNOT MARKET DATA · NOT A PREDICTION

SELECTED NODE0.50Y

STRIKE / MONEYNESS100.0 / 1.00

IMPLIED VOL21.01%

SCENARIO SPOT100.0

LOW 18.6%HIGH 36.2%

K/S
0.70K/S
0.76K/S
0.82K/S
0.88K/S
0.94K/S
1.00K/S
1.06K/S
1.12K/S
1.18K/S
1.24K/S
1.30

7D

30D

0.25Y

0.50Y

1.00Y

2.00Y

Maturity ↓ · Moneyness → · Select any cell for exact state.

Accessible numeric surface grid

Maturity	0.70	0.76	0.82	0.88	0.94	1.00	1.06	1.12	1.18	1.24	1.30
0.02y	33.46%	29.13%	25.79%	23.25%	21.37%	20.05%	19.19%	18.71%	18.57%	18.72%	19.11%
0.08y	33.61%	29.28%	25.94%	23.40%	21.52%	20.20%	19.34%	18.86%	18.72%	18.87%	19.26%
0.25y	33.97%	29.64%	26.30%	23.76%	21.88%	20.56%	19.70%	19.22%	19.09%	19.23%	19.62%
0.50y	34.43%	30.10%	26.75%	24.21%	22.34%	21.01%	20.15%	19.68%	19.54%	19.69%	20.08%
1.00y	35.15%	30.82%	27.47%	24.93%	23.06%	21.73%	20.87%	20.40%	20.26%	20.41%	20.80%
2.00y	36.16%	31.83%	28.48%	25.95%	24.07%	22.75%	21.88%	21.41%	21.27%	21.42%	21.81%

LOADING VISUAL ENGINEBuilding one deterministic surface grid…

08

FRONT OFFICE

Where the model meets the book.

ON THE DESK

“The mark is a surface. The hedge is a view about how that surface moves.”

VISIBLE INPUTS

bid/offer vanilla grid

spot and forwards

discount/dividend curves

corporate-action calendar

expiry/settlement conventions

CALIBRATION

Fit liquid nodes more strongly, preserve bid/offer awareness and inspect residuals in premium as well as volatility space.

RISK

vega buckets

vanna and volga

skew/term scenarios

spot-vol correlation

calendar roll

DAILY WORKFLOW

ingest and timestamp quotes
normalize strikes/deltas
clean crossed or stale nodes
fit
reprice quotes
publish surface and risk
monitor drift

Production failure modes

stale spot or forward
calendar-arbitrage from interpolation
wing extrapolation instability
quote holes
parameter jumps across recalibrations
cache and version mismatch

09

MACRO CONNECTION

Map the transmission channel.

MACRO CONNECTION

A surface is a map of event pricing

Macro regimes affect the level, asymmetry and timing of protection demand rather than moving every option uniformly.

Inflation / growth surprise

changes policy and earnings distribution

Rates and forward

move carry and moneyness coordinates

Risk sentiment

changes downside protection demand

Surface

reprices level, skew and term structure

10

COMMON PITFALLS

Most failures begin outside the formula.

01

Treating a smooth plot as proof of no arbitrage.

02

Mixing spot and forward moneyness.

03

Fitting mid quotes without spreads or liquidity weights.

04

Using a static surface as a claim about future smile dynamics.

05

Showing synthetic scenarios as market observations.

11

SOURCES / FURTHER READING

Attribution with implementation authority.

researchBSD-3-Clause

Lectures 04 and 07 — implied and stochastic volatility

Research path for the volatility progression; all explanations and implementation are original.

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

OPEN ORIGINAL SOURCE ↗LICENSE ↗

implementation referenceQuantLib permissive license

ql/termstructures/volatility and QuantLib 1.42 release notes

Current implementation reference for smile sections, SABR and variance-surface representations.

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

OPEN ORIGINAL SOURCE ↗LICENSE ↗

Surface coordinate

Total variance

Risk-neutral density

Dupire local variance

From a call-price surface to local volatility

Assume risk-neutral local dynamics

Write the forward density equation

Differentiate option prices by strike

Differentiate by maturity

Isolate local variance

Lectures 04 and 07 — implied and stochastic volatility

ql/termstructures/volatility and QuantLib 1.42 release notes