Option-chain structure

Calendar Arbitrage

Examine when option maturity inequalities hold, why dividends and rates matter, and how calendar spreads differ from executable no-arbitrage violations.

Mechanics & mathematicsHypothetical examples

What Is It?

Calendar arbitrage concerns inconsistent option values across expiries under a valid maturity comparison. A calendar spread typically buys one expiry and sells another. It is usually a trade on price behavior, volatility term structure and time decay, not a position with a guaranteed nonnegative payoff.

A longer maturity does not universally mean a higher European premium at a fixed strike. Dividends, interest rates, option type, forward levels and exercise rights all matter. Any maturity inequality must state the conditions that make one claim dominate the other.

How It Works

For European calls on a non-dividend-paying asset with nonnegative deterministic interest rates and standard frictionless financing, the longer-dated call at the same strike is worth at least the shorter-dated call. The right to postpone paying the strike, together with the relevant convex-payoff argument, supports this conditional ordering.

European puts do not inherit that universal ordering: discounting a distant strike payment can reduce their value. Dividends can also reverse the raw fixed-strike call comparison. For American options with otherwise identical terms, the longer contract includes all exercise opportunities of the shorter one; that exercise-set inclusion supports a maturity ordering without borrowing the European proof.

Under a zero-rate, zero-dividend martingale model, a European call's convex payoff has increasing conditional expectation as maturity extends. If a later call is sold too cheaply relative to an earlier call, its time-T₁ value still dominates its intrinsic payoff. This clean benchmark helps explain the conditional arbitrage, but does not justify applying the result to every carry regime.

Mathematical Formula

C(K,T_2)\ge C(K,T_1),\quad T_2>T_1\quad(q=0,\ r\ge0)

This fixed-strike European call comparison assumes no dividends, nonnegative deterministic rates, the same underlying and standard frictionless markets. Do not copy it to dividend-paying calls or all European puts.

w(k,T)=\sigma_{\mathrm{imp}}(k,T)^2T,\qquad k=\ln\!\left(\frac{K}{F(0,T)}\right)

Surface analysis often compares total implied variance at fixed forward log-moneyness with consistent discount/forward normalization. In the zero-rate, zero-dividend setting, no calendar arbitrage implies nondecreasing total variance at fixed strike/moneyness. With carry, the correct normalized comparison must be derived; comparing arbitrary fixed-strike implied volatilities is not a universal test.

Worked Example

Illustrative prices · not market data

Use the clean European-call benchmark: zero rates, no dividends, the same strike 100, and expiries T₁ < T₂. Assume the near call can be sold at 6 and the far call bought at 5.50.

Buy the far call and sell the near call to collect 0.50 at inception.
At T₁, the far call is worth at least max(Sₜ₁ − 100, 0) under this benchmark. That covers the near call's cash settlement if the far call can be sold at its model-consistent liquidation value.
Deduct 0.20 of total opening and assumed closing costs, measured with zero rates, to leave a modeled minimum surplus of 0.30.
If dividends or different exercise/settlement terms are introduced, this particular dominance argument must be re-derived rather than reused.

The assumed ability to monetize the later European option at T₁ is essential; it cannot simply be exercised early. A normal debit calendar can lose as implied volatility, spot and remaining time value change.

Execution Considerations

Match strikes and settlement terms and map intervening dividends and corporate actions. Compare forward and discount conventions consistently; far and near expiries may refer to different forwards even for the same underlying.

Plan the near-expiry event explicitly. A short American option can be assigned before the European long can be exercised. Liquidity for closing the later option, stock borrow after assignment and margin treatment all affect the achievable package economics.

Risks & Limitations

Changes in volatility term structure, skew and underlying price can hurt an ordinary calendar. There is no fixed common terminal payoff because the two legs expire at different times.

Stale settlement marks, ex-dividend jumps, early assignment and carry assumptions can produce false calendar flags. A maturity comparison at one strike does not validate an entire volatility surface.

Frequently Asked Questions

Must every longer-dated option cost more?

No. European fixed-strike comparisons depend on option type, rates and dividends. American exercise-set inclusion is a different argument requiring otherwise identical terms.

Does falling implied volatility across expiries prove arbitrage?

No. Compare the appropriate price constraints or total variance under a valid normalization and carry model. Volatility alone omits the maturity factor.

Can I exercise a European far call when the near call expires?

No. It must be monetized through a sale or another feasible hedge. Assuming early exercise would invalidate the European strategy.

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References & Further Reading

Background sources for the concepts above. Apply the assumptions stated in this guide to each formula.

Options Industry Council: Put/Call Parity

Educational material only; not investment advice or an offer to trade. Examples are hypothetical. Trading involves execution, financing and capital-loss risks; no return is guaranteed.