TheQuantBateman

01

INTUITION

Build the state before the equation.

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

The product exists before the model.

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

Notation, units and exact claims.

IV_t(T): option-implied volatilityRV_{t,t+T}: subsequent realized volatilityw=σ²T: total varianceP and Q: physical and risk-neutral measures

QUANT NOTESYNTHETIC · EDUCATIONAL

Variance risk premium convention

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

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

Read the equation together with its financial domain and convention.

QUANT NOTESYNTHETIC · EDUCATIONAL

Ex-post variance spread

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

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

Read the equation together with its financial domain and convention.

04

DERIVATION

Do not jump to the final expression.

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.

05

MODEL / PRICING

Fit, compute, then challenge the assumptions.

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.

LIMITATIONS

Overlapping windows induce serial dependence.

Implied volatility is strike dependent.

Transaction costs and tail losses dominate simple carry stories.

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

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 state. Challenge the equation.

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%

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

09

MACRO CONNECTION

Map 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.

Macro uncertainty

raises protection demand

Option premium

embeds state prices

Implied variance

moves above physical expectation

Volatility carry

compensates tail exposure

10

COMMON PITFALLS

Most failures begin outside the formula.

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.

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 ↗

Variance risk premium convention

Ex-post variance spread

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

Volatility, Monte Carlo and stochastic-volatility lectures

Current volatility structures, processes, calibration helpers and tests