TheQuantBateman

01

INTUITION

Build the state before the equation.

Vega describes a parallel local volatility move. Vanna and volga explain how delta changes with volatility and how vega curves with volatility—terms that become material during combined spot and surface shocks.

01

Greek units must match the scenario units.

02

Vanna is a cross derivative, so sign conventions depend on definitions.

03

Surface risk is bucketed; one scalar vega hides strike and maturity structure.

02

WHY MARKETS CARE

The product exists before the model.

Volatility books are explained and hedged through bucketed first- and second-order sensitivities before full repricing closes the residual.

INSTRUMENTS

vanilla option books

risk reversals

barriers

volatility swaps

QUOTE CONVENTION

Platform convention: vega per one volatility point, theta per calendar day and rho per 100bp. State the exact vanna definition used.

03

MATHEMATICS

Notation, units and exact claims.

ν=∂V/∂σ: raw vegaVanna=∂²V/(∂S∂σ)Volga=∂²V/∂σ²Δσ_pts: volatility-point move

QUANT NOTESYNTHETIC · EDUCATIONAL

Vega point unit

P&L for a one percentage-point volatility move.

\nu_{1pt}=0.01\,\frac{\partial V}{\partial\sigma}

Read the equation together with its financial domain and convention.

QUANT NOTESYNTHETIC · EDUCATIONAL

Second-order spot-vol expansion

Cross and curvature terms matter when spot and volatility move together.

\Delta V\approx\Delta\,\Delta S+\nu\,\Delta\sigma+\tfrac12\Gamma(\Delta S)^2+\operatorname{Vanna}\,\Delta S\Delta\sigma+\tfrac12\operatorname{Volga}(\Delta\sigma)^2

Read the equation together with its financial domain and convention.

QUANT NOTESYNTHETIC · EDUCATIONAL

Black–Scholes volga

The sign and magnitude vary by moneyness and maturity.

\operatorname{Volga}=\nu\frac{d_1d_2}{\sigma}

Read the equation together with its financial domain and convention.

04

DERIVATION

Do not jump to the final expression.

DERIVATION

A two-factor Taylor P&L explain

Expand option value in spot and volatility around the current state while holding stated conventions fixed.

01
Choose state variables
Treat V=V(S,σ,t,...) locally; for a surface, σ means a stated bucket or deformation.
02
Write first-order terms
Delta and raw vega multiply changes in spot and decimal volatility.
$dV\approx V_SdS+V_\sigma d\sigma$
03
Add pure curvature
Gamma and volga capture second-order effects in each factor.
04
Add the mixed derivative
Symmetry of smooth mixed partials combines the two half cross terms into vanna times dS dσ.
$V_{S\sigma}dSd\sigma$
05
Convert units before attribution
If vega is per point, multiply by Δσ in points; if raw, multiply by decimal Δσ.
NUMERICAL CHECKA 5-point move is 0.05 in raw-vega units and 5 in point-vega units.

Higher-order Greeks are local approximations whose usefulness depends on clear units, surface buckets and comparison with full repricing.

05

MODEL / PRICING

Fit, compute, then challenge the assumptions.

METHOD

Compute analytical Greeks where possible; otherwise use symmetric bumps sized above numerical noise and below nonlinear regime changes.

CALIBRATION

Greeks are evaluated on a calibrated surface/model snapshot. Recalibrated Greeks and frozen-parameter Greeks answer different questions.

LIMITATIONS

Local expansion fails for large discontinuous moves.

Bucket interpolation changes aggregated risk.

Cross-Greek definitions vary across systems.

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

Spot-volatility P&L expansion

Apply a second-order explain with explicit decimal-volatility units.

REUSABLE EXAMPLE

01from __future__ import annotations

02

03def pnl(delta: float, gamma: float, raw_vega: float, vanna: float, volga: float, ds: float, dvol: float) -> float:

04 values = [delta, gamma, raw_vega, vanna, volga, ds, dvol]

05 if not all(map(lambda x: abs(x) < float("inf"), values)):

06 raise ValueError("inputs must be finite")

07 return delta*ds + 0.5*gamma*ds**2 + raw_vega*dvol + vanna*ds*dvol + 0.5*volga*dvol**2

08

09explained = pnl(0.55, 0.018, 38.0, -0.22, 12.0, -4.0, 0.05)

10assert abs(explained - (-0.078)) < 1e-12

11print(round(explained, 6))

07

INTERACTIVE LAB

Move the state. Challenge the equation.

SPOT × VOLATILITY P&L GEOMETRY

Volatility P&L explain lab

Shock spot and surface level; compare first order, cross/curvature terms and full repricing.

SYNTHETIC · CONTROLLED SCENARIOS

SCENARIO INTENSITY70%

Vega / point0.380

Cross stateLong vega

Max residual3.47

08

FRONT OFFICE

Where the model meets the book.

ON THE DESK

“The unexplained P&L often has a Greek; it may not have the Greek you reported.”

VISIBLE INPUTS

position state

surface buckets

spot/forward

bump sizes

recalibration policy

CALIBRATION

Greeks are evaluated on a calibrated surface/model snapshot. Recalibrated Greeks and frozen-parameter Greeks answer different questions.

RISK

bucketed vega

vanna

volga

surface gamma

DAILY WORKFLOW

freeze state
compute Greeks
observe moves
explain P&L
full reprice

Production failure modes

point/raw vega mix
asymmetric bumps
unstable recalibration
double-counted buckets

09

MACRO CONNECTION

Map the transmission channel.

MACRO CONNECTION

Spot-vol correlation drives cross-Greek P&L

Risk-off regimes combine falling spot and rising downside volatility, making vanna and skew exposure central to the book response.

Risk-off shock

spot falls

Surface

level and skew rise

Vanna/volga

add nonlinear P&L

Hedge residual

requires full repricing

10

COMMON PITFALLS

Most failures begin outside the formula.

01

Mixing raw and point vega.

02

Reporting one scalar vega for a structured surface book.

03

Using oversized finite-difference bumps.

04

Treating Taylor explain as exact 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 ↗

Vega point unit

Second-order spot-vol expansion

Black–Scholes volga

A two-factor Taylor P&L explain

Choose state variables

Write first-order terms

Add pure curvature

Add the mixed derivative

Convert units before attribution

Volatility, Monte Carlo and stochastic-volatility lectures

Current volatility structures, processes, calibration helpers and tests