TheQuantBateman

01

INTUITION

Separate the quote from the quantity inferred from it.

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

Start from executable inputs and conventions.

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

Transform quotes without losing units or arbitrage constraints.

Formula · Definition

Surface coordinate

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

Forward log-moneyness makes strike geometry comparable across maturities.

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Formula · Definition

Total variance

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

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

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Formula · Full derivation

Risk-neutral density

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

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

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Formula · Full derivation

Dupire local variance

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

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

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Full 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.

Inputs

k = log(K/F_T): log-forward moneyness
w(k,T) = σ_imp²(k,T)T: total implied variance
C(K,T): call-price surface
σ_loc(K,T): local volatility

Assumptions and limits

Sparse wings force extrapolation choices.
A perfect static fit does not guarantee realistic forward dynamics.
Volatility-space interpolation can hide option-price arbitrage.

05

MODEL / PRICING

Invert, fit, and reprice the market instruments.

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.

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.

06PYTHON IMPLEMENTATIONOpen the implementation and checks.

ARCHITECTURE

Store source quotes and fitted state separately.
Evaluate one deterministic grid in the framework-free quant layer.
Drive heatmap, smile, term and 3D views from the same values.
Always provide exact readouts and a 2D/table alternative to 3D.

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

GUIDED EXPERIMENT

Change one assumption with intent

Load a coherent example, observe the linked calculation, then return to unrestricted manual exploration.

New here? Start with one example and change only the suggested input.

First try

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

09MACRO CONNECTIONOpen 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.

01Inflation / growth surprisetransmits

changes policy and earnings distribution

02Rates and forwardtransmits

move carry and moneyness coordinates

03Risk sentimenttransmits

changes downside protection demand

04Surfaceoutput

reprices level, skew and term structure

10COMMON PITFALLSOpen the failure checklist.

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.

11SOURCES / FURTHER READINGOpen sources and continue the track.

research

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 ↗

implementation reference

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 ↗

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

A surface is a map of event pricing

Lectures 04 and 07 — implied and stochastic volatility

ql/termstructures/volatility and QuantLib 1.42 release notes