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Free lesson · Volatility

Variance exposure and the difference from arbitrage

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Start with the idea

A variance contract pays for realized squared movement relative to its agreed strike, so the payoff is nonlinear in realized volatility.

Symbols, units & horizon
  • N_var: signed variance notional in currency per unit of annualized decimal variance
  • RV_ann: contract-defined realized annualized variance, decimal fraction squared
  • K_var: agreed variance strike in matching units
  • Π: expiry currency payoff before separate costs

When and why to use this

Stress nonlinear variance exposure and distinguish implied-premium hypotheses from guaranteed payoffs.

A variance contract pays for realized squared movement relative to its agreed strike, so the payoff is nonlinear in realized volatility.

Write the contract’s annualization, sampling, caps and treatment of disrupted observations before computing payoff. The illustrative notional here is currency per unit of decimal annualized variance; market contracts may quote a different convention such as vega notional or percentage-squared units.

A short variance position can collect a premium in ordinary periods and suffer large losses when realized variance jumps. Calling it volatility arbitrage does not make it riskless; expected risk compensation and hedge implementation remain uncertain.

Π=Nvar(RVann−Kvar)
Contract variance-payoff definition

Variance exposure and the difference from arbitrage

  1. Compute realized variance using the exact contract sampling and annualization.
  2. Subtract the agreed variance strike in the same squared units.
  3. Multiply by signed variance notional; a short position uses negative notional.
Work it by hand

N_var=$100,000 per variance unit, realized volatility .30 and strike volatility .20 imply variance .09 versus .04. Long payoff=100000×(.09−.04)=$5,000; short payoff is −$5,000.

Apply it in a strategy

  • Freeze inputs at the stated decision time and record their units.
  • Stress nonlinear variance exposure and distinguish implied-premium hypotheses from guaranteed payoffs.
  • Recompute the example, then change the material assumption and explain the difference.

Research deliverable

Variance exposure and the difference from arbitrage: produce the worked calculation, a timestamped input record and a written decision addressing this limitation: Jumps, sampling conventions, collateral and expensive dynamic replication can invalidate a seemingly attractive expected spread.

These are synthetic mechanics examples, not historical performance or paper replications. Module evidence and research boundaries record the 12 September 2026 review.

Evidence and boundaries · reviewed 12 September 2026

The records below distinguish research status, access depth and data dates. Abstract-only review identifies research questions; it does not establish a replicated empirical claim.

Further reading: François, Gauthier, Godin & Pérez-Mendoza: Deep Hedging with Options Using the Implied Volatility Surface ↗

arXiv research preprint; no journal status established by this review. Version: v3, 12 August 2025. Review: 2026-09-12; Abstract and version metadata only. Markets: S&P 500 index options and simulated markets. Data dates: Current abstract reports historical out-of-sample straddles 2020–2023; training dates not inspected. Limitation: Version 1 search excerpts describe a different data window. This record follows v3. Claimed hedge rankings are not taught as established results; simulator, trading costs and data construction require full-text review and replication.

Further reading: Gatheral & Jacquier: Arbitrage-free SVI volatility surfaces ↗

Foundational research manuscript; publication mapping not verified here. Version: v4, 21 March 2013. Review: 2026-09-12; Abstract and version metadata only. Markets: SPX options illustration. Data dates: Exact quote date not inspected. Limitation: Supports the importance of static arbitrage constraints; the elementary convexity check below is not a full SVI calibration or sufficient global surface validation.

Research sources, review dates and limitations

Connect the ideas: Sensitivity and approximation

Retrieve: A local sensitivity describes how a model responds near a specified input.

Check the change: The input, its units and what is held fixed differ across slope, duration and option sensitivities.

Differential calculus → Partial derivatives → Rates, credit & macro → Stochastic calc → Options

Explain it yourself: What must you check before using a small-move approximation for a large scenario?

Self-assessed. Write your explanation before opening this comparison.

Check the expansion point, units, held-fixed inputs, curvature and model domain; compare with a full repricing under the same scenario.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

# Python 3.10+; standard library and NumPy only.
# Synthetic teaching inputs; conventions and units are defined in the notation above.
def variance_payoff(notional,realized_variance,strike_variance):
    if min(realized_variance,strike_variance)<0: raise ValueError('Nonnegative variances required')
    return notional*(realized_variance-strike_variance)

assert abs(variance_payoff(100000,.3**2,.2**2)-5000)<1e-8
print(variance_payoff(100000,.3**2,.2**2))

Continue learning

Volatility: Measurement, Surfaces & Variance Risk — all lessons
  1. Realized variance starts with squared returns
  2. EWMA as a causal variance baseline
  3. A multi-horizon realized-variance forecast
  4. Implied volatility is a model inversion
  5. Term structure through total and forward variance
  6. Strike convexity and a butterfly consistency check
  7. Vega requires a volatility-unit convention
  8. Variance exposure and the difference from arbitrage

Quantitative finance and development glossary · Python resources and libraries · Research sources and limitations