Options fundamentals

Options Arbitrage

Study European and American options, payoff replication, carry and discounting, with guides to parity, boxes, butterflies, calendars and pricing bounds.

Mechanics & mathematicsHypothetical examples

What Is It?

Options arbitrage tests whether option contracts and their replicating portfolios have compatible prices. A call gives its holder a right to buy; a put gives a right to sell, at strike K under the contract's exercise and settlement rules. The premium is the price paid for that right, not the strike paid or received on exercise.

European options permit exercise only at expiry. American options permit exercise before expiry, which changes replication, bounds and cash-flow timing. A pricing rule derived for European contracts cannot simply be applied to American quotes or to options with different settlement currencies.

How It Works

At expiry a vanilla call pays the positive part of spot minus strike; a put pays the positive part of strike minus spot. Before expiry, premium reflects the distribution of future prices as well as rates, carry and exercise rights. Intrinsic value is the immediate exercise value; time value is premium minus intrinsic. A European premium can be below immediate intrinsic under some carry conditions because immediate exercise is unavailable.

No-arbitrage relationships arise from matching payoffs rather than forecasting price direction. Put–call parity connects calls, puts, the underlying and cash. A box connects two strikes to a fixed terminal payment. Vertical spreads, butterflies and pricing bounds test strike consistency. Calendar comparisons require extra care about dividends, rates and exercise opportunities.

The underlying's prepaid forward value accounts for cash flows received before expiry. For deterministic continuous yield q it is S₀ exp(−qT); for known discrete dividends it is S₀ minus their present value. Unknown dividends, borrowing restrictions and asymmetric financing widen practical valuation bands.

Mathematical Formula

C_T=(S_T-K)^+,\qquad P_T=(K-S_T)^+,\qquad x^+=\max(x,0)

These are per-unit terminal payoffs for vanilla options. Multiply by the actual contract multiplier; do not assume every contract represents 100 shares.

D(0,T)=e^{-rT},\qquad C-P=A_0-KD(0,T)

European parity uses matched underlying, expiry, strike and settlement. A₀ is the prepaid forward value; D discounts a unit cash payment. Deterministic r is used here for the exponential form.

Worked Example

Illustrative prices · not market data

Consider European options on a non-dividend-paying asset with S₀ = 100, K = 100, one year to expiry and r = 5% continuously compounded.

The discounted strike is 100 × exp(−0.05) = 95.1229, so parity requires C − P = 4.8771.
A call at 8.8771 and a put at 4 are consistent with parity; each can still include substantial option time value.
If the executable call bid instead becomes 9.50 while the put ask stays 4 and spot can be bought at 100, a conversion has a 0.6229 initial gross gap against borrowing 95.1229.
With 0.20 of combined costs measured today, 0.4229 remains in the ideal matched European replication.

A volatility model disagreement alone would not establish this replication. All prices must be simultaneously executable and collateral, funding and settlement must support the intended cash flows.

Execution Considerations

Read the contract specification before building a spread: underlying, multiplier, expiry timestamp, exercise style, premium currency and settlement method must agree. Multi-leg limit orders can constrain total price, but availability and fill guarantees vary by venue.

The options interface groups parity, box and option-chain results, with chain categories for vertical, butterfly, calendar and pricing-bound checks. Available results depend on the connected service and its reported engine status. The pricing relationships in these articles do not establish that every theoretical violation is detected or executable.

Risks & Limitations

Wide spreads, stale quotes, thin depth, margin requirements and failed legs can consume a small pricing gap. Exercise and assignment can create stock positions and unexpected cash requirements, especially around dividends and expiry.

Volatility changes matter for ordinary spread trades even when a static payoff identity is model-independent. A hedged terminal payoff does not remove interim cash-flow, counterparty or settlement risk.

Frequently Asked Questions

Is a cheap option automatically mispriced?

No. Premium depends on strike, expiry, exercise rights, expected distributions and market conventions. Compare equivalent claims and executable bounds.

Does put–call parity require Black–Scholes?

No. European parity follows from payoff replication under its financing and carry assumptions. A particular volatility model is unnecessary.

Is every butterfly or calendar spread an arbitrage?

No. Most are ordinary risk-taking strategies. A no-arbitrage violation requires a valid price inequality, its assumptions and feasible execution.

Explore the Existing Tool

Dashboard available · connection required

The options dashboard displays parity, box, and option-chain results when its data service is connected. Check the current connection and reported engine status; this guide does not certify individual checks or live service availability. Paper-trading results are not live fills.

Open Options Arbitrage Scanner

References & Further Reading

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

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.