TheQuantBateman

01

INTUITION

Separate the quote from the quantity inferred from it.

Realized and implied volatility answer different questions. One summarizes a path under the physical measure; the other translates an option price under a model and embeds risk premia, convexity and supply-demand effects.

01

Match horizon before comparing the two numbers.

02

The spread is not a pure forecast error.

03

Variance is the cleaner additive object for risk-premium analysis.

02

WHY MARKETS CARE

Start from executable inputs and conventions.

The realized-implied relationship drives volatility carry, variance-swap analysis, option relative value and model backtesting.

INSTRUMENTS

listed options

variance swaps

VIX-style indices

delta-hedged option strategies

QUOTE CONVENTION

Compare annualized total variance over the same future window. State whether VRP means implied minus expected realized or the reverse.

03

MATHEMATICS

Transform quotes without losing units or arbitrage constraints.

Formula · Full derivation

Variance risk premium convention

\operatorname{VRP}_t(T)=\mathbb E_t^{\mathbb Q}[RV_{t,t+T}^{2}]-\mathbb E_t^{\mathbb P}[RV_{t,t+T}^{2}]

A positive number under this convention compensates sellers of variance risk.

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Full derivation

From option value to a variance premium

The two measures differ because pricing weights states by marginal utility and hedging demand, not historical frequency alone.

01
Fix a future window
Pair today’s implied quote with realized variance accumulated over exactly the option’s remaining horizon.
$[t,t+T]$
02
Express both sides in variance
Total variance is time-additive under idealized partitioning and avoids comparing square roots prematurely.
$w_t(T)=IV_t^2(T)T$
03
Separate probability measures
Physical expectation describes future outcomes; risk-neutral expectation prices hedgeable cash flows after discounting.
$\mathbb E^{\mathbb P}[RV^2]\ne\mathbb E^{\mathbb Q}[RV^2]$
04
Define the premium
Subtract the physical expected variance from its risk-neutral counterpart and keep the sign convention attached.
$VRP=E^{Q}[RV^2]-E^{P}[RV^2]$
05
Backtest without leakage
Use only information available at the option timestamp, preserve bid/offer and roll the realized window forward.

Implied-minus-realized is a priced risk spread plus forecast error and implementation effects—not a free return statistic.

Inputs

IV_t(T): option-implied volatility
RV_{t,t+T}: subsequent realized volatility
w=σ²T: total variance
P and Q: physical and risk-neutral measures

Assumptions and limits

Overlapping windows induce serial dependence.
Implied volatility is strike dependent.
Transaction costs and tail losses dominate simple carry stories.

Formula · Definition

Ex-post variance spread

\widehat{VRP}_{t,T}=IV_t^2(T)-RV_{t,t+T}^2

A realized outcome useful for backtests, not the ex-ante expected premium itself.

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05

MODEL / PRICING

Invert, fit, and reprice the market instruments.

METHOD

Construct constant-maturity implied variance and pair it with non-overlapping or carefully adjusted subsequent realized windows.

CALIBRATION

Estimate the physical expectation with a documented forecast; infer risk-neutral variance from option prices or a model-consistent index.

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

Horizon-matched variance spread

Compute ex-post variance spreads without mixing volatility and variance units.

REUSABLE EXAMPLE

01from __future__ import annotations

02

03import numpy as np

04

05def variance_spread(implied_vol: np.ndarray, realized_vol: np.ndarray) -> np.ndarray:

06 if implied_vol.shape != realized_vol.shape or np.any(implied_vol < 0) or np.any(realized_vol < 0):

07 raise ValueError("volatility arrays must align and be non-negative")

08 return implied_vol**2 - realized_vol**2

09

10iv = np.array([0.20, 0.24, 0.31])

11rv = np.array([0.17, 0.28, 0.25])

12spread = variance_spread(iv, rv)

13assert np.allclose(spread, np.array([0.0111, -0.0208, 0.0336]))

14print(np.round(spread * 10_000, 1))

07

INTERACTIVE LAB

Move the quote and inspect every linked representation.

IMPLIED VARIANCE − SUBSEQUENT REALIZED VARIANCE

Volatility carry laboratory

Shock implied level and subsequent regimes; compare variance spread, carry and tail outcomes.

SYNTHETIC · CONTROLLED SCENARIOS

SCENARIO INTENSITY70%

Mean VRP188.1 bp²

IV22.00%

RV state17.00%

Annualized variance by Forward months

Carry: Implied variance remains above realized.

implied variance
realized variance
variance spread

Forward months: 0. implied variance: 0.0523. realized variance: 0.0316. variance spread: 0.0207.

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

“Short volatility earns a premium until the distribution collects it back.”

VISIBLE INPUTS

constant-maturity IV

realized estimator

bid/offer

roll convention

transaction costs

CALIBRATION

Estimate the physical expectation with a documented forecast; infer risk-neutral variance from option prices or a model-consistent index.

RISK

gap risk

short convexity

skew carry

roll-down

DAILY WORKFLOW

align horizons
normalize variance
estimate carry
stress tails
attribute outcome

Production failure modes

look-ahead leakage
overlapping samples
mid-only returns
unmatched maturities

09MACRO CONNECTIONOpen the transmission channel.

MACRO CONNECTION

Insurance demand and the variance premium

Risk aversion and demand for crash protection can keep risk-neutral tail variance above the physical expectation.

01Macro uncertaintytransmits

raises protection demand

02Option premiumtransmits

embeds state prices

03Implied variancetransmits

moves above physical expectation

04Volatility carryoutput

compensates tail exposure

10COMMON PITFALLSOpen the failure checklist.

01

Calling IV the market’s unbiased forecast.

02

Subtracting volatilities instead of comparing variance.

03

Ignoring skew and maturity.

04

Backtesting with overlapping observations without robust inference.

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 ↗

From option value to a variance premium

Fix a future window

Express both sides in variance

Separate probability measures

Define the premium

Backtest without leakage

Insurance demand and the variance premium

Volatility, Monte Carlo and stochastic-volatility lectures

Current volatility structures, processes, calibration helpers and tests