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

For educational purposes only; not investment advice.

Direct answer

No-arbitrage option bounds are price inequalities implied by portfolios whose future cash flows dominate or replicate an option. They are basic data and model checks: a quote outside the applicable bound suggests inconsistent inputs, contract terms, timestamps, or prices. It is not automatically an executable risk-free trade.

For European calls (c) and puts (p) on a non-dividend-paying stock, with spot (S_0), strike (K), nonnegative continuously compounded rate (r), and time (T):

[ max(0,S_0-Ke^(-rT))le cle S_0 ]

[ max(0,Ke^(-rT)-S_0)le ple Ke^(-rT) ]

and put-call parity is:

[ c-p=S_0-Ke^(-rT). ]

The same strike, expiration, underlying, exercise style, settlement, and synchronized executable prices are essential.

Why the bounds exist

An option cannot have a negative value because its holder can decline to exercise. A call cannot be worth more than acquiring the underlying today under the stated assumptions. A European put cannot exceed the present value of receiving (K) at expiration.

The call lower bound follows by comparing a call plus a zero-coupon investment that grows to (K) with the stock. At expiration, (c)’s payoff plus (K) is (max(S_T,K)), never less than (S_T). The dominating portfolio cannot cost less than the stock without contradicting the law of one price. An analogous comparison gives the put lower bound.

With continuous dividend yield (q), replace spot in European parity with the prepaid-forward value (S_0e^(-qT)):

[ c-p=S_0e^(-qT)-Ke^(-rT). ]

Then the call lower bound is (max(0,S_0e^(-qT)-Ke^(-rT))), and its corresponding upper bound is (S_0e^(-qT)) in that continuous-yield model. For known discrete dividends, use the present value of the contractually relevant dividends with care; uncertain dividends are model inputs, not locked cash flows.

American exercise changes the relationships. Simple nonnegative-rate bounds include (Cgemax(0,S_0-K)), (Pgemax(0,K-S_0)), (Cle S_0), and (Ple K), but European equality generally becomes an inequality range because either side may be exercised early.

A theoretical violation and its cash flows

Assume a non-dividend-paying stock is $100. A one-year European call has strike $90, and the present value of $90 is $86. The lower bound is:

[ cge S_0-PV(K)=$100-$86=$14. ]

If an executable call Ask were $5 while stock could be shorted at $100 and the bond purchased for $86, a textbook portfolio would:

short one share for +$100;
buy one call for −$5;
invest −$86 to receive $90 at expiration.

The initial net cash is +$9. At expiration, the call plus $90 is worth (max(S_T,90)); after covering the short share, the terminal amount is (max(90-S_T,0)), never negative. That is the logic behind the bound.

For a standard 100 multiplier, the apparent initial amount is $900 before dividends, stock-borrow charges, Bid/Ask spreads, fees, margin, and execution risk. If the call’s $5 was a stale Last rather than an executable Ask, or the stock could not be borrowed, the trade would not exist as described.

Parity provides a second check. If the matching put is $1, then (c-p=4), while (S_0-PV(K)=14); the $10 discrepancy should trigger a review of all four executable legs and contract terms before any conclusion.

Quote-validation checklist

Confirm exact underlying, option root, strike, expiration, call or put, exercise style, settlement, multiplier, and deliverable.
Use synchronized executable Bid and Ask prices for the direction of every required trade, not Last, Mark, or midpoints.
Match the stock or forward timestamp to the option quotes.
Use the relevant financing and lending rates; a single risk-free rate may not represent actual borrowing and investing costs.
Include expected or known dividends, ex-dates, special distributions, and stock-loan treatment.
Verify stock availability, borrow rate, recall risk, and short-sale constraints before relying on a short-stock leg.
Distinguish European from American exercise. Put-call equality cannot be copied unchanged to American equity options.
Account for early exercise and assignment, especially around dividends and deep-in-the-money options.
Apply the correct present-value convention and day count. Check whether rates and time are decimal or percentage, simple or compounded.
Use the actual deliverable after splits, mergers, spin-offs, and special dividends.
Include fees, exchange charges, taxes where applicable, margin, capital requirements, and opportunity cost.
Require enough displayed or executable size across all legs; one small quote does not scale to the intended quantity.
Use simultaneous complex execution where available. Legging can turn a theoretical lock into directional exposure.
Treat negative rates, hard-to-borrow stocks, uncertain dividends, and settlement mismatches as changes to assumptions, not small footnotes.
Use bounds first as quality controls for chains, theoretical values, and code. Investigate violations before labeling them arbitrage.

Common misconceptions

“Any screen violation is free money.” Stale or non-executable quotes are common explanations.
“Intrinsic value is the European lower bound.” Before expiration, discounting and dividends matter; American intrinsic bounds differ.
“Put-call parity applies to any call and put.” Strike, expiration, underlying, style, and settlement must match.
“Midpoint prices prove an arbitrage.” Every required trade must be executable on the correct side and size.
“The risk-free rate is the trader’s funding rate.” Actual borrowing, lending, and margin economics can differ.
“Dividends can be ignored.” They change forward value, parity, early-exercise incentives, and short-stock cash flows.
“American and European options share the same equality.” Early exercise introduces additional value and inequality ranges.
“An option below a model value violates no-arbitrage.” Model value depends on volatility; bounds are weaker cash-flow restrictions.
“A positive initial cash flow is sufficient.” Future obligations, borrowing, assignment, settlement, and execution must be locked too.
“Bounds predict where the option should trade.” They exclude impossible regions under assumptions; they do not estimate fair volatility value.