TheQuantBateman

01

INTUITION

Separate the quote from the quantity inferred from it.

A term structure shows where the option market locates uncertainty in time. Volatility levels are not additive; total variance is the object that carries across adjacent horizons.

01

A flat volatility curve still implies increasing total variance.

02

Scheduled events create localized variance increments.

03

Negative forward variance signals inconsistent inputs or interpolation.

02

WHY MARKETS CARE

Start from executable inputs and conventions.

Term structure controls calendar spreads, event trades, variance forwards and the maturity dimension of every surface.

INSTRUMENTS

calendar spreads

variance swaps

forward-start options

earnings and event options

QUOTE CONVENTION

This lesson uses annualized implied volatility and total variance w(T)=σ²(T)T. Expiries use a consistent ACT/365-like year fraction.

03

MATHEMATICS

Transform quotes without losing units or arbitrage constraints.

Formula · Definition

Total variance

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

Variance accumulated to maturity in the chosen strike or delta slice.

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Formula · Short derivation

Forward variance

\xi(T_1,T_2)=\frac{w(T_2)-w(T_1)}{T_2-T_1}

Average variance priced for the forward interval.

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Short derivation

Extracting a forward variance interval

Assume integrated variance adds over non-overlapping time intervals.

01
Express each maturity in total variance
Convert annualized volatility quotes before taking differences.
$w_i=\sigma_i^2T_i$
02
Partition integrated variance
Cumulative variance to T2 equals cumulative variance to T1 plus the forward interval.
$w(T_2)=w(T_1)+\xi(T_1,T_2)(T_2-T_1)$
03
Solve for the forward rate
Rearrange the partition identity.
$\xi=\frac{w_2-w_1}{T_2-T_1}$
04
Require non-negative increments
A negative forward interval is economically and numerically suspect for a diffusion-style variance representation.
$w_2-w_1\ge0$
05
Interpret event bumps
A discrete event variance can be added to baseline integrated variance, causing short maturities spanning the event to look elevated.

Term structure becomes coherent when represented as cumulative and forward variance, with the maturity clock and coordinate held fixed.

Inputs

w(T)=σ²(T)T: total variance
ξ(T1,T2): forward variance
T1<T2: maturity interval
ΔT=T2-T1

Assumptions and limits

Fixed-strike and fixed-delta slices differ as forward moves.
Sparse long maturities require extrapolation.
Event variance is not directly observable.

Formula · Definition

Calendar condition

w(k,T_2)\ge w(k,T_1)\quad\text{for }T_2>T_1

A practical fixed-coordinate condition used to detect negative forward variance.

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05

MODEL / PRICING

Invert, fit, and reprice the market instruments.

METHOD

Fit monotone total variance through liquid expiries, preserve event knots, and extract forward intervals for interpretation.

CALIBRATION

Use spreads and liquidity weights; do not smooth away scheduled events or force a parametric curve that creates negative increments.

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.

06PYTHON IMPLEMENTATIONOpen the implementation and checks.

ARCHITECTURE

Validate domains and units at the boundary.
Keep the numerical kernel framework-free and deterministic.
Return diagnostics with values.
Test analytical limits and failure states.

PYTHON 3 · NUMPY / SCIPY

Forward variance extraction

Convert a volatility term structure into non-negative forward intervals.

REUSABLE EXAMPLE

01from __future__ import annotations

02

03import numpy as np

04

05def forward_variances(times: np.ndarray, vols: np.ndarray) -> np.ndarray:

06 if times.shape != vols.shape or np.any(np.diff(times) <= 0) or np.any(vols < 0):

07 raise ValueError("times must increase and vols must be non-negative")

08 total = vols**2 * times

09 forwards = np.diff(total) / np.diff(times)

10 if np.any(forwards < -1e-12):

11 raise ValueError("negative forward variance")

12 return forwards

13

14t = np.array([30, 90, 180, 365]) / 365.0

15vol = np.array([0.28, 0.24, 0.225, 0.22])

16fwd = forward_variances(t, vol)

17assert np.isfinite(fwd).all() and np.all(fwd >= 0.0)

18print(np.round(np.sqrt(fwd), 4))

07

INTERACTIVE LAB

Move the quote and inspect every linked representation.

TOTAL AND FORWARD VARIANCE

Forward-variance lab

Add an event bump or invert the front end; inspect total and forward variance by interval.

SYNTHETIC · CONTROLLED SCENARIOS

SCENARIO INTENSITY70%

1Y total variance0.0378

Min forward0.0337

StateNormal

Annualized variance by Maturity (years)

Normal: Gently rising cumulative uncertainty.

total variance
forward variance

Maturity (years): 0.1Y. total variance: 0.0026. forward variance: 0.0512.

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

“The front expiry is not high; it contains a different piece of time.”

VISIBLE INPUTS

expiry timestamps

forward coordinate

ATM/slice vols

event calendar

spread weights

CALIBRATION

Use spreads and liquidity weights; do not smooth away scheduled events or force a parametric curve that creates negative increments.

RISK

calendar vega

event variance

roll-down

forward-vol exposure

DAILY WORKFLOW

normalize expiries
convert total variance
fit monotone curve
extract forwards
stress events

Production failure modes

day-count mismatch
event timestamp error
negative forwards
over-smoothed knots

09MACRO CONNECTIONOpen the transmission channel.

MACRO CONNECTION

Events redistribute variance through time

Policy decisions, elections and earnings concentrate uncertainty in intervals rather than lifting every maturity equally.

01Scheduled catalysttransmits

localizes uncertainty

02Total variancetransmits

adds an event increment

03Term curvetransmits

forms a bump or inversion

04Calendar tradeoutput

isolates the interval

10COMMON PITFALLSOpen the failure checklist.

01

Subtracting volatilities to obtain forward volatility.

02

Comparing expiries with inconsistent clocks.

03

Smoothing through known events.

04

Calling an inverted volatility curve arbitrage by itself without inspecting total variance.

11SOURCES / FURTHER READINGOpen sources and continue the track.

research

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 ↗

implementation reference

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 ↗

Extracting a forward variance interval

Express each maturity in total variance

Partition integrated variance

Solve for the forward rate

Require non-negative increments

Interpret event bumps

Events redistribute variance through time

Volatility, Monte Carlo and stochastic-volatility lectures

Current volatility structures, processes, calibration helpers and tests