No-Arbitrage Option Bounds: Calls, Puts, and Parity Checks

Apply cash-flow bounds and put-call parity to matched option quotes, with the correct carry, exercise style, executable prices, and contract terms.

For educational purposes only; not investment advice. Investing may result in loss.

Direct answer

No-arbitrage option bounds are price inequalities implied by portfolios that dominate or replicate an option’s cash flows. For matched European calls and puts on a non-dividend-paying stock, the basic relationships are:

max(0, S₀ − Ke^(−rT)) ≤ c ≤ S₀

max(0, Ke^(−rT) − S₀) ≤ p ≤ Ke^(−rT)

c − p = S₀ − Ke^(−rT)

Here c and p are current option values, S₀ is spot, K is strike, r is a nonnegative continuously compounded rate, and T is time to expiration. Put-call parity and the bounds require the same underlying claim, strike, expiration, exercise style, deliverable, and settlement terms. A screen quote beyond a bound is a reason to investigate, not proof of executable risk-free profit.

Why the bounds hold

An option holder may decline to exercise, so a long option cannot have negative value. A European call cannot cost more than the stock under these assumptions, and a European put cannot cost more than the present value of K.

For the call lower bound, compare the stock with a call plus a zero-coupon claim costing Ke^(−rT). At expiration that package is worth max(S_T − K, 0) + K = max(S_T, K), which is never below S_T. For the put lower bound, stock plus put has the same dominating terminal value. The law of one price therefore gives the stated inequalities; parity follows because stock plus put and call plus the strike claim have identical terminal cash flows.

With deterministic continuous dividend yield q, replace spot by the prepaid-forward value S₀e^(−qT):

max(0, S₀e^(−qT) − Ke^(−rT)) ≤ c ≤ S₀e^(−qT)

max(0, Ke^(−rT) − S₀e^(−qT)) ≤ p ≤ Ke^(−rT)

c − p = S₀e^(−qT) − Ke^(−rT)

For known discrete cash dividends, use the present value of the dividends whose entitlement belongs to the stock holder during the option’s life. Uncertain dividends are estimates, not locked cash flows. Negative rates, borrow constraints, taxes, or different borrowing and lending curves require bounds built from the relevant attainable discount factors rather than mechanically reusing the simplified formulas.

American exercise changes the result. With a nonnegative rate and an ordinary non-dividend-paying stock, useful simple bounds are max(0, S₀ − K) ≤ C ≤ S₀ and max(0, K − S₀) ≤ P ≤ K. European put-call equality does not generally carry over because either American option may be exercised early.

Worked violation check

Suppose a non-dividend-paying stock is $100, a one-year European call has K = $90, and PV(K) = $86. The call lower bound is $14. If its executable ask is $5, and the stock can actually be shorted at $100, the textbook trade is:

short one share for +$100;
buy one call for −$5;
invest −$86 in a zero-coupon claim that pays $90 at expiration.

The initial cash inflow is +$9. At expiration, the remaining cash flow is max(S_T − 90, 0) + 90 − S_T = max(90 − S_T, 0), so it is never negative. With a standard 100-share multiplier, the apparent initial inflow is $900 before stock-borrow charges, dividends, bid-ask spreads, fees, margin, tax, and execution risk.

As a separate diagnostic, if displayed values are c = $5 and p = $1, then c − p = $4 while S₀ − PV(K) = $14, a $10 parity gap. That calculation still does not establish a trade: every leg must be priced on its executable side, synchronized, available in sufficient size, and legally and operationally deliverable.

Quote-validation checklist

Match the exact underlying or reference claim, option root, strike, expiration, call or put, and exercise style.
Confirm multiplier, deliverable, corporate-action adjustment, currency, and physical or cash settlement.
Use synchronized executable bids and asks for the direction and size of every leg, not last prices, marks, or midpoints.
Match the stock or forward timestamp to the option quotes and account for different trading hours.
Apply the relevant discount factor, day count, compounding convention, and actual borrowing or lending rate.
Include known dividends and ex-dates; stress uncertain, special, reduced, or omitted distributions.
Verify stock locate, borrow availability, fee, rebate, collateral, recall, buy-in, and payment-in-lieu treatment.
Distinguish European from American exercise and model early exercise and assignment over the full holding period.
Use the official settlement reference and timeline, including last trading time, exercise cutoff, and cash or security posting.
Include commissions, exchange charges, taxes, margin, capital usage, and opportunity cost.
Control partial fills, rejected legs, halts, quote staleness, insufficient size, and legging exposure.
Treat a bound violation first as a data, contract, or model exception; reconcile all positions and cash flows before calling it arbitrage.

Common misconceptions

“Any screen violation is free money.” Stale, crossed, one-sided, or non-executable quotes are common causes.
“Intrinsic value is always the lower bound.” Discounting, dividends, and exercise style determine the applicable bound.
“Any call and put can be compared.” The claims and contract terms must match exactly.
“Midpoint parity proves arbitrage.” Each required purchase occurs at an ask and each sale at a bid.
“The risk-free rate is my funding rate.” Actual borrowing, lending, collateral, and margin terms can differ.
“American and European options use the same equality.” Early-exercise rights break the simple European replication.
“A price below a volatility model is a no-arbitrage violation.” Model value is stronger and more assumption-dependent than a cash-flow bound.
“Positive initial cash is enough.” Future funding, delivery, exercise, assignment, and settlement obligations must also be locked.