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Calendar Arbitrage: Total Variance, Forward Volatility, and False Signals

For educational purposes only; not investment advice.

Direct answer

Section titled “Direct answer”

Calendar arbitrage in an option surface means prices across expirations violate conditions needed to prevent a portfolio with favorable initial cash flow and no negative future payoff under the modeled contract assumptions. A common diagnostic is that implied total variance at comparable forward moneyness should not decrease with maturity:

w(k,T) = σ_imp(k,T)² T

For T₂ > T₁, an arbitrage-aware surface generally requires w(k,T₂) ≥ w(k,T₁) at fixed log-forward moneyness k, subject to the model and quote conventions. Comparing headline IVs at the same nominal strike is not enough.

Total variance and forward variance

Section titled “Total variance and forward variance”

Annualized IV mixes the amount of variance with the time over which it accumulates. Total variance separates them. Between two maturities, implied forward variance is:

v_fwd = [w(k,T₂) − w(k,T₁)] / (T₂ − T₁)

A negative value is inconsistent with a standard arbitrage-free diffusion interpretation and signals that quotes, forwards, interpolation, or contract adjustments need investigation.

The comparison must hold moneyness relative to each maturity’s forward, not blindly hold strike fixed. Rates, dividends, borrow, and discrete cash distributions shift forwards. European and American prices cannot be mixed without treating early exercise. Different settlement conventions, stale timestamps, crossed markets, and corporate actions can create artificial violations.

A long calendar spread is not automatically an arbitrage. Its near option expires first, leaving a live far option whose value depends on future spot and volatility. Static surface conditions and a discretionary calendar-spread trade are related concepts but not the same payoff claim.

One-month versus three-month variance

Section titled “One-month versus three-month variance”

At matched forward moneyness, suppose one-month IV is 40% and three-month IV is 20%. Using T₁=1/12 and T₂=3/12:

w₁ = 0.40² × 1/12 ≈ 0.0133

w₂ = 0.20² × 3/12 = 0.0100

v_fwd = (0.0100 − 0.0133) / (2/12) ≈ −0.0200

Negative forward variance is a strong inconsistency signal. Before calling it tradable arbitrage, rebuild the comparison with synchronized executable quotes, correct forwards, dividends, and an arbitrage-constrained fit.

By contrast, if one-month IV is 30%, then w₁ = 0.30²/12 = 0.0075, below w₂=0.0100. Three-month IV is still lower than one-month IV, yet total variance increases and this calendar test passes.

Surface verification checklist

Section titled “Surface verification checklist”
  • Use synchronized executable bid and ask quotes for matching contract families.
  • Convert strikes to forward moneyness for each maturity using consistent rates, dividends, and borrow.
  • Compare total variance, not annualized IV alone.
  • Separate European and American exercise and account for discrete dividends and early-exercise value.
  • Fit the surface with calendar and butterfly no-arbitrage constraints rather than differentiating raw midpoints.
  • Test bid, mid, and ask surfaces and report an uncertainty range.
  • Check expiration settlement, time zones, holiday calendars, and exact year fractions.
  • Include spread, fees, margin, funding, depth, and legging risk before claiming executable profit.

Common misconceptions

Section titled “Common misconceptions”
  • “Far-month IV below near-month IV is arbitrage.” Total variance can still increase.
  • “The same strike means the same economic moneyness.” Different forwards change moneyness.
  • “Negative forward variance guarantees a trade.” Bad timestamps, spreads, and fitting can create it.
  • “Buying far and selling near locks a profit.” The near leg expires first and leaves market exposure.
  • “American stock options obey the simple European test.” Dividends and early exercise alter values.
  • “A smooth surface is arbitrage free.” Smoothness alone does not impose calendar or butterfly constraints.
Section titled “Related topics”

Authoritative sources

Section titled “Authoritative sources”