Diagnosing Option Prices with Basic Relationships

Diagnosing Quotes, Not Just Calculating Prices

An option quote can look plausible in isolation yet conflict with another quote or with a basic no-arbitrage relationship. A useful audit therefore asks several questions in sequence: does the price respect the option's immediate-exercise value, does it move in the right direction when a contract factor changes, and do related call, put, stock, and bond positions satisfy the appropriate parity or bounds?

These tests do not require a particular option-pricing formula or probability distribution. They do require careful matching: compare options with the same underlying, maturity, exercise style, and relevant contract terms, and use the relationship that fits the exercise style and dividend assumptions. A mathematical discrepancy indicates a trading opportunity only if the quoted prices are simultaneously available and transaction costs, financing, and short-sale constraints do not eliminate it.

Definition: A quote is inconsistent with a no-arbitrage relationship when, under the stated assumptions, it violates a necessary price bound, a price-ordering relationship, or a parity equality or inequality. A corrective trade buys the cheaper side of the relationship and sells the more expensive side, with the positions chosen so that their future cash flows offset.

Start With Bounds and Matched Comparisons

An American option can be exercised at any time up to expiration. Its value cannot be less than the amount received from exercising immediately: a call has intrinsic value $\max(S_0-K,0)$, and a put has intrinsic value $\max(K-S_0,0)$. This lower bound does not depend on the interest-rate sign because it compares the option with an immediately available exercise.

Upper bounds need assumptions. In the standard setting of nonnegative interest rates and nonnegative dividends, an American call cannot be worth more than the stock, $S_0$, and an American put cannot be worth more than the strike, $K$. The call bound relies on the stated nonnegative carry assumptions; the put bound relies on nonnegative interest rates. In particular, do not use the put upper bound $K$ when rates may be negative: receiving $K$ before expiration can then be worth more than receiving $K$ immediately.

Assumption-dependent American bounds: For a non-dividend-paying stock and nonnegative effective annual interest rate $i$, an American put satisfies $\max(K-S_0,0)\leq P_A\leq K$. With nonnegative interest rates and nonnegative dividends, an American call satisfies $\max(S_0-K,0)\leq C_A\leq S_0$. The intrinsic-value lower bounds apply generally; the upper bounds must not be detached from their assumptions.

Price direction is another quick diagnostic. Holding other terms fixed, a call's value is nondecreasing in $S_0$ and nonincreasing in $K$; a put's value is nonincreasing in $S_0$ and nondecreasing in $K$. With the same underlying, strike, and maturity otherwise fixed, a higher nonnegative interest rate tends to raise a European call value and lower a European put value. A larger dividend yield has the opposite direction for calls and puts. These directions are comparisons, not formulas for the size of a price change.

A quote table can fail a direction test even when every price lies within its own broad bounds. For example, if two calls have the same underlying and maturity but the lower-strike call is quoted below the higher-strike call, the ordering is wrong. A candidate should first verify that exercise style and all terms really match; a difference in dividends, maturity, or contract specification can invalidate a direct comparison.

Worked Example: Bounds and Strike Ordering

A hypothetical non-dividend-paying stock is at $S_0=\$72$. The effective annual interest rate is $i=3\%$, and two American calls expire on the same date. The call with $K=\$65$ is quoted at $\$9.80$, while the call with $K=\$70$ is quoted at $\$10.40$. An American put with $K=\$70$ is quoted at $\$5.60$. Check the bounds and ordering, then specify a trade suggested by any violation. Assume frictionless trading and simultaneous execution.

Method: check individual bounds, then compare matched strikes. The $K=\$65$ call has intrinsic value $\max(72-65,0)=\$7$, and its quote of $\$9.80$ exceeds that lower bound. Its upper bound is $S_0=\$72$, so it passes that check. The $K=\$70$ call has intrinsic value $\max(72-70,0)=\$2$; its quote of $\$10.40$ is also between $\$2$ and $\$72$.

The $K=\$70$ put has intrinsic value $\max(70-72,0)=\$0$, and, because $i\geq0$, its upper bound is $K=\$70$. Its quote of $\$5.60$ passes those checks.

The call quotes nevertheless violate strike ordering: with all other terms matched, the lower-strike call must be worth at least as much as the higher-strike call. Here $\$9.80<\$10.40$. Buy the $K=\$65$ call and sell the $K=\$70$ call. The initial net receipt is $\$10.40-\$9.80=\$0.60$. At expiration, for every stock price $S_T$, the difference in payoffs is $\max(S_T-65,0)-\max(S_T-70,0)\geq0$. The lower-strike call also dominates at earlier exercise dates, so this is a nonnegative-payoff position under the stated matching assumptions. The check is consistent: the trade receives $\$0.60$ initially and cannot have a negative option payoff at expiration. The mistake this method avoids is assuming that individually plausible bounds guarantee consistency between quotes.

European Put–Call Parity as a Diagnostic

For European options with the same underlying, strike, and expiration, put–call parity compares a call-plus-bond portfolio with a put-plus-stock portfolio. With no dividends before expiration and a constant effective annual rate $i$, define $v=1/(1+i)$. The bond position costs $K v^T$ today and pays $K$ at time $T$.

Theorem: For European options on a non-dividend-paying stock, with common strike $K$ and maturity $T$, no-arbitrage requires $C_E+K v^T=P_E+S_0$, or equivalently $C_E-P_E=S_0-Kv^T$. This is an equality under the stated assumptions, not merely an approximate price tendency.

The equality follows from matching future cash flows. At expiration, a call plus a payment of $K$ has value $\max(S_T-K,0)+K$. A put plus a share has value $\max(K-S_T,0)+S_T$. If $S_T\geq K$, both expressions equal $S_T$; if $S_T<K$, both equal $K$. Equal future payoffs must have equal values today in a no-arbitrage market, which gives the parity equality.

\max(S_T-K,0)+K = \max(K-S_T,0)+S_T

To diagnose quotes, calculate both sides rather than relying on a remembered sign. If $C_E+K v^T$ is larger, the call-plus-bond side is rich; if $P_E+S_0$ is larger, the put-plus-stock side is rich. Sell the rich side and buy the cheap side. The two sides' future payoffs match, so the initial difference is the potential arbitrage receipt.

Worked Example: Finding the Rich Side of Parity

A hypothetical non-dividend-paying stock is quoted at $S_0=\$100$. A European call and put both have $K=\$100$ and expire in one year. The effective annual interest rate is $i=5\%$; the call quote is $\$12.00$, and the put quote is $\$8.00$. Find the discrepancy and the corresponding positions. Assume the quotes are executable and ignore transaction costs.

Method: discount the strike and compare parity portfolios. The one-year discount factor is $v=1/1.05=0.95238095$, so the present value of the strike is $K v^T=100(0.95238095)=\$95.238095$.

The call-plus-bond side costs $C_E+K v^T=12+95.238095=\$107.238095$. The put-plus-stock side costs $P_E+S_0=8+100=\$108.00$. Thus the put-plus-stock side is more expensive by $108-107.238095=\$0.761905$, or $\$0.76$ to the cent.

Buy the call and the bond, and short the put and one share. The initial net receipt is $\$108.00-\$107.238095=\$0.761905$. At expiration, the long call plus the bond's $\$100$ payment equals the short put plus short share obligation in value, by the payoff identity above; equivalently, the two parity sides have identical expiration payoffs. The net expiration payoff is therefore zero, leaving the initial receipt before costs. This check confirms both the trade direction and the calculated difference. The common mistake is to short the cheaper portfolio rather than the rich one.

Adjust Parity for Known Cash Dividends

When a stock will pay known cash dividends before a European option expires, the stock's owner receives those payments, while a call holder does not. Parity must include a risk-free claim that pays each dividend on its payment date. If the known dividends are $D_j$ at times $t_j$, and the same effective annual rate $i$ applies, their present value is $\sum_j D_jv^{t_j}$.

C_E+K v^T+\sum_j D_jv^{t_j}=P_E+S_0

The justification is the same cash-flow matching argument: a stock position receives each dividend, so the other parity portfolio must include claims to those dividends. At expiration, the call payoff plus the strike payment still matches the put payoff plus the terminal stock value; before expiration, the dividend claims match the cash distributions received by the stock. With a continuous dividend yield $q$ and constant force of interest $\delta$, the corresponding European relation is $C_E-P_E=S_0e^{-qT}-Ke^{-\delta T}$. Do not mix that continuous-yield version with a discrete known-dividend calculation.

Worked Example: A Cash Dividend Changes the Parity Check

A hypothetical stock is quoted at $S_0=\$80$. A known cash dividend of $\$2$ will be paid six months from now. European calls and puts have $K=\$80$ and expire in one year. The effective annual interest rate is $i=4\%$; the call is quoted at $\$7.40$ and the put at $\$6.10$. Calculate the parity discrepancy and corrective positions. Assume the dividend and rates are known and the quotes can be traded simultaneously.

Method: discount the strike and dividend separately. The strike's present value is $80/(1.04)=\$76.923077$. The dividend's present value is $2/(1.04)^{0.5}$. Since $(1.04)^{0.5}\approx1.019804$, its present value is approximately $2/1.019804=\$1.961161$.

The call-plus-bond-plus-dividend-claim side costs $7.40+76.923077+1.961161=\$86.284238$. The put-plus-stock side costs $6.10+80=\$86.10$. The first side is richer by $86.284238-86.10=\$0.184238$, which rounds to $\$0.18$ to the cent.

Buy the put and stock; sell the call, the risk-free claim paying $\$80$ at expiration, and the claim paying $\$2$ at the dividend date. The initial receipt is $\$86.284238-\$86.10=\$0.184238$. The stock's dividend receipt offsets the short dividend claim, and the expiration payoffs match by the call–put payoff identity. The net future payoff is zero under the assumptions, confirming that the call-and-claims side is the rich side. The mistake avoided here is omitting the dividend claim or discounting the six-month dividend as though it were paid at expiration.

American Quotes Require Bounds, Not European Equality

European parity is an equality because the two portfolios have matched cash flows under the specified assumptions. American options can be exercised early, so the same equality generally does not apply to their quotes. For a non-dividend-paying stock and nonnegative interest rates, a useful American relationship is $S_0-K\leq C_A-P_A\leq S_0-Kv^T$. It is an interval, not a point prediction. The interval's assumptions matter: if rates can be negative, the stated upper endpoint should not be applied without re-deriving an appropriate bound.

A reliable diagnostic separates three levels of evidence. First, reject any quote that violates an applicable individual bound. Second, check matched-contract price ordering. Third, use parity only when its exercise-style and dividend assumptions fit, or use the corresponding American inequality when those conditions hold. If the test fails, write down the exact rich and cheap portfolios before describing a trade; this catches sign errors and makes the cash-flow logic auditable.

Common exam trap: Treating every mismatch as European put–call parity arbitrage. An American option may be worth more because of early exercise rights, and a cash dividend changes the European parity portfolio. Also, the American put upper bound $P_A\leq K$ requires nonnegative rates; applying it mechanically with negative rates can produce a false violation.

Match the contracts.
Confirm the underlying, strike, maturity, exercise style, dividend terms, and quote units.

Check bounds and directions.
Use intrinsic values and only those upper bounds whose rate and dividend assumptions are satisfied; compare otherwise identical contracts.

Apply the correct parity relationship.
Use European equality with the appropriate dividend adjustment, or an American inequality when its assumptions hold.

Construct and verify the trade.
Buy the cheaper side, sell the richer side, calculate the initial difference, and check that future cash flows offset.

Check Your Understanding

For each problem, state the applicable relationship and keep the assumptions attached to the conclusion.

A hypothetical American call on a non-dividend-paying stock has $S_0=\$73$, $K=\$70$, and a quote of $\$2.40$. What bound does it violate, and by how much? (Answer: Its intrinsic-value lower bound is $\$3$; the quote is $\$0.60$ below the bound.)
A non-dividend-paying stock is at $\$120$. Matched European options have $K=\$125$, $T=0.5$, $i=4\%$, $C_E=\$9.20$, and $P_E=\$12.00$. Which parity side is rich, and by how much to the nearest cent? (Answer: $K v^T=125/(1.04)^{0.5}\approx\$122.57$; call-plus-bond costs $\$131.77$, while put-plus-stock costs $\$132.00$. Put-plus-stock is rich by $\$0.23$.)
A stock at $\$50$ will pay a known $\$1$ dividend in six months. Matched European options have $K=\$48$, $T=1$, $i=5\%$, $C_E=\$5$, and $P_E=\$3$. Which parity side is rich, and by how much to the nearest cent? (Answer: $5+48/1.05+1/(1.05)^{0.5}\approx\$51.69$, while $3+50=\$53.00$; put-plus-stock is rich by $\$1.31$.)
For otherwise identical European options on a non-dividend-paying stock, the effective annual interest rate rises while $S_0$, $K$, and $T$ stay fixed. What directions should the call and put prices move under the standard model? (Answer: The call value is nondecreasing and the put value is nonincreasing as the interest rate rises.)
A hypothetical American put has $K=\$100$ and is quoted at $\$101$, while the effective annual interest rate is negative. Can the upper bound $P_A\leq K$ alone establish that the quote is inconsistent? (Answer: No. That American put upper bound requires nonnegative interest rates, so a different valid bound or model is needed.)

What You'll Learn