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Key Takeaways
- Option-price bounds connect the contract payoff to stock prices, exercise rights and financing.
- For a non-dividend-paying stock, the European call floor is max(0, stock price minus discounted strike).
- An American option’s immediate-exercise right establishes its intrinsic-value floor.
- European call prices decrease and remain convex across strikes at the same expiration.
- A violation pays only at executable prices, so price every leg of the package at its bid or offer on the contract’s actual deliverable.
What are no-arbitrage bounds?
No-arbitrage bounds are the model-free upper and lower limits an option price must respect, set by intrinsic value, the underlying price, the discounted strike, and consistency across strikes. They connect an option’s payoff with stock and financing trades. When executable prices break a bound and its replication assumptions hold, a complete package of option, stock and cash trades produces a gross gain at those prices.
Every check starts with the exact underlying, strike, expiration, exercise style and deliverable.
European call and put price bounds
For a European call on a non-dividend-paying stock, the lower bound is max(0, S - K × exp(-rT)). The stock price provides an upper bound. A European put lies between max(0, K × exp(-rT) - S) and K × exp(-rT). Each bound comes from a stock-and-cash trade that replicates or dominates the option’s payoff, priced at one common borrowing and lending rate.
Formula
For a non-dividend-paying stock: max(0, S − K × exp(−rT)) ≤ European call ≤ S; max(0, K × exp(−rT) − S) ≤ European put ≤ K × exp(−rT). American call ≥ max(0, S − K); American put ≥ max(0, K − S). European calls: 0 ≤ C(K1) − C(K2) ≤ (K2 − K1) × exp(−rT). For equally spaced strikes: C(K1) + C(K3) ≥ 2 × C(K2).- S is a nonnegative stock price, K is a nonnegative strike, r is the continuously compounded financing rate and T is time in years.
- These stock-and-cash replication bounds use a common borrowing and lending rate and unrestricted stock trading.
- Compare the same underlying, deliverable, exercise style and expiration. The strike-grid formulas here describe European calls.
- Use the prepaid forward value in place of S for the European lower bounds when incorporating known dividends.
- Prices are per share; the worked standard contract uses a 100-share multiplier.
Worked Example
XYZ has an executable stock bid of $100 and an American 80 call offered at $19. One contract delivers 100 shares, and the stated exercise-and-short-sale package is available. Separately, European 30-day 95, 100 and 105 calls have executable package prices of $7.30, $5.00 and $2.55 at a 4% annual financing rate.
- American call intrinsic value: $100 − $80 = $20 per share
- Buy one 80 call: $1,900 debit
- Sell 100 shares short at $100: $10,000 credit
- Exercise the call at $80 and deliver the 100 shares to close the short: $8,000 debit
- Completed package: $10,000 − $1,900 − $8,000 = $100 gross profit
- European 95/100 vertical: $7.30 − $5.00 = $2.30, below the $4.98 discounted-width bound
- European butterfly: $7.30 − 2 × $5.00 + $2.55 = −$0.15 per share
- The butterfly receives $15 initially and has expiration payoff from $0 to $500 per 1:2:1 package
The immediate-exercise package closes both the option and short-stock position with $100 gross profit at the stated executable prices. The separate European butterfly has a $15 opening credit and a nonnegative expiration payout. Each calculation uses its own exercise style and cash-flow timing.
How immediate exercise sets the American option floor
An American call’s intrinsic-value floor is max(0, S - K). An American put’s is max(0, K - S). Immediate exercise pays the same intrinsic value whatever the stock costs to borrow, so this floor needs no financing input.
In the example, stock is available to sell at $100 and the American 80 call is offered at $19. Buying one call costs $1,900. Selling 100 shares short receives $10,000. Exercising costs $8,000 and supplies the shares to close the short. The completed transactions produce $100 gross profit at those prices.
Exercising at once completes the package. Holding the call and the short stock instead brings in financing, borrow and dividend cash flows that change the result.
European call vertical spread and butterfly bounds across strikes
European call prices decrease as the strike rises. For two strikes, the price difference is bounded by the discounted strike width. With a $5 width, 4% annual rate and 30 days, that ceiling is $4.98.
The example’s 95/100 vertical costs $2.30. Its 95/100/105 butterfly instead receives $0.15 per share: $7.30 minus twice $5.00 plus $2.55. Its expiration payout ranges from zero to $5 per share. A credit for that nonnegative European payout identifies inconsistent package prices.
How to test a quoted option price against the bounds
Check executable bids and offers for every leg. Match adjusted deliverables, include the relevant dividends and funding, and record each settlement date. Known dividends enter European lower bounds through the prepaid forward value.
These relationships also identify vertical spreads, butterflies and box spreads in the chain, and the options profit calculator compares each structure’s entry price and expiration outcomes.
Limitations of no-arbitrage bounds
- A bound sets the range; a valuation model picks the price inside it from added inputs such as volatility.
- Apply your actual financing and borrow costs. The replication behind the bounds assumes one common borrowing and lending rate and unrestricted stock trading; different rates, borrowing costs or limits on short sales loosen them in practice.
- Test a violation at executable bids and offers. An apparent violation in displayed quotes can disappear once every leg, fee and funding cost is included.
- Apply the vertical and butterfly tests to European calls that share one expiration. American contracts add exercise cash flows before expiration, and dividends or adjusted deliverables change the inputs.
Bottom line
No-arbitrage bounds set the price range an option must stay within, given the stock, the strike, financing, and the prices at neighboring strikes. Use them as a consistency check on executable quotes before any valuation model, and price every leg and cash flow of the complete package before treating a quote as an arbitrage.
Clear answers
Frequently Asked Questions
Sources and Further Reading
- OIC: Put/Call Parity
- Alex Shapiro, NYU Stern: Options, Valuation and (No) Arbitrage, Lecture Notes 15
- Gatheral and Jacquier: Arbitrage-Free SVI Volatility Surfaces (arXiv 1204.0646)
- Cboe Global Indices: American and European-Style Theoretical Options Calculation Methodology
- OIC: Options Exercise FAQ
How entries are sourced, calculated and updated: editorial methodology.