No-Arbitrage Bounds for European Calls

Why Call Prices Have Bounds

A call gives its holder the right, but not the obligation, to buy a share for a specified exercise price at a specified time. Its payoff can never be negative, and it cannot exceed the value of the share at expiration. Those payoff facts, together with the ability to hold stock and borrow or lend at a risk-free rate, put limits on a call’s price today.

These limits are no-arbitrage bounds, not a complete pricing model. A price inside the bounds may or may not be a reasonable market price; the bounds alone do not select one precise value. The results below apply to a European call on a non-dividend-paying stock.

Definition: Let $S_0$ be the stock price today, $K$ the call’s exercise price, $c$ the call price today, and $S_T$ the stock price at expiration $T$. A European call’s expiration payoff is $\max(S_T-K,0)$. For a non-dividend-paying stock, the no-arbitrage bounds are $$ \max\left(S_0-\frac{K}{(1+i)^n},\,0\right)\leq c\leq S_0, $$ when the maturity is $n$ whole years and $i$ is the effective annual risk-free rate. The discounted exercise price $K/(1+i)^n$ is the amount that must be invested or borrowed today to have $K$ at expiration.

The formula assumes that the stock and call refer to the same underlying share and expiration date, that the call is European, and that the stock pays no dividends before expiration. It also assumes no transaction costs, taxes, or restrictions on short selling, and that an investor can borrow and lend at the same risk-free rate. These idealized conditions allow us to compare positions with different initial costs but ordered expiration payoffs. If the stock pays dividends, or if borrowing and lending rates differ, the stated bounds need adjustment.

The notation here uses an effective annual rate $i$ with whole-year maturity $n$. If the risk-free accumulation over the contract’s actual horizon is represented by a discount factor $D(0,T)$, the discounted exercise price is $KD(0,T)$, and the lower bound is $\max(S_0-KD(0,T),0)$. The rate and time units must describe the same horizon.

Deriving the Lower and Upper Bounds

The key idea is to compare expiration payoffs. If one portfolio costs less today but is guaranteed to be worth at least as much as another portfolio at expiration, the cheaper portfolio would create an arbitrage. The comparisons below use only stock, the call, and risk-free cash.

Lower-bound comparison.
Compare a call plus cash that grows to $K$ with one share of stock. At expiration, the call-plus-cash position is worth $\max(S_T-K,0)+K=\max(S_T,K)$, which is at least $S_T$. Therefore, its current cost $c+K/(1+i)^n$ cannot be less than the stock price $S_0$.

Keep the call’s nonnegative payoff in view.
The payoff $\max(S_T-K,0)$ is never negative. A call price cannot be negative under the no-arbitrage assumptions. Combining this with the first comparison gives the lower bound $\max(S_0-K/(1+i)^n,0)$.

Upper-bound comparison.
At expiration, one share is worth $S_T$, while the call is worth $\max(S_T-K,0)\leq S_T$. Thus stock costs at least as much as the call it dominates, so $S_0\geq c$.

The lower-bound comparison can be written directly as an inequality:

\max(S_T-K,0)+K=\max(S_T,K)\geq S_T \quad\Longrightarrow\quad c+\frac{K}{(1+i)^n}\geq S_0.

Rearranging gives $c\geq S_0-K/(1+i)^n$. Since $c\geq 0$ as well, the stronger combined statement is $c\geq\max(S_0-K/(1+i)^n,0)$. For the upper bound, the expiration payoff inequality $\max(S_T-K,0)\leq S_T$ implies $c\leq S_0$. If either price inequality failed, the comparisons would produce a position with a positive initial receipt and no negative expiration payoff.

Applying the Bounds

For each contract, first find the present value of the exercise price. Then calculate the lower and upper bounds and compare the quoted call price with the resulting interval. If a quoted price is outside the interval, the payoff comparisons also indicate how to form the arbitrage position.

Worked Example: A Call Quoted Below Its Lower Bound

A non-dividend-paying share is priced at $S_0=\$120$. A one-year European call has exercise price $K=\$100$, and the effective annual risk-free rate is $i=25\%$. The call is quoted at $\$35$. Determine whether this price respects the bounds and, if not, construct the arbitrage.

Method and discounting. The maturity is one year, so the present value of the $\$100$ exercise price is $$ \frac{K}{(1+i)^n}=\frac{\$100}{(1.25)^1}=\$80. $$ The lower bound is $\max(\$120-\$80,0)=\$40$, and the upper bound is $\$120$. The quoted price of $\$35$ is below the lower bound by $\$5$.

Arbitrage position today. Buy the call for $\$35$, lend $\$80$ at the risk-free rate, and short one share, receiving $\$120$. The initial net receipt is $$ \$120-\$35-\$80=\$5. $$

Expiration check. The loan grows to $\$100$. The call plus the loan is worth $\max(S_T-\$100,0)+\$100=\max(S_T,\$100)$. The short share requires payment of $S_T$, so the net expiration payoff is $\max(S_T,\$100)-S_T$, which is nonnegative for every possible $S_T$. The initial $\$5$ receipt is therefore not offset by a possible loss at expiration.

The method is to buy the underpriced call, pair it with the cash position used in the lower-bound comparison, and short the stock. The common mistake is to discount the exercise price and then subtract that present value from the quoted call price. The bound compares the present value of the exercise price with the stock price: the correct lower bound is $\$120-\$80=\$40$.

Worked Example: A Call Quoted Above Its Upper Bound

A non-dividend-paying share costs $S_0=\$80$. A European call on the share is quoted at $c=\$84$, with an exercise price of $K=\$70$ and a two-year maturity. The effective annual risk-free rate is $i=10\%$. Test the quote and verify any proposed arbitrage at expiration.

Method. The upper bound is the current share price, $\$80$, so the $\$84$ quote exceeds it by $\$4$. The interest rate and maturity are not needed to calculate this upper bound.

Arbitrage position today. Sell the call for $\$84$ and use $\$80$ to buy one share. The initial net receipt is $\$84-\$80=\$4$.

Expiration check. The share covers the short call if the call is exercised. The expiration value of the share minus the call payoff is $$ S_T-\max(S_T-\$70,0)=\min(S_T,\$70)\geq 0. $$ If $S_T\leq \$70$, the call expires without value and the share is worth $S_T$. If $S_T>\$70$, the share is delivered for $\$70$ under the call and the difference $S_T-\$70$ covers the call obligation. In either case, the expiration payoff is nonnegative, in addition to the $\$4$ received today.

This is the upper-bound comparison: stock costs $\$80$ and has an expiration value at least as large as the call payoff. The common mistake is to subtract the discounted exercise price from the stock price when testing the upper bound. Discounting $K$ is part of the lower-bound calculation; the upper bound here is simply $c\leq S_0$.

Worked Example: Checking a Price Inside the Bounds

A non-dividend-paying share costs $S_0=\$150$. A two-year European call has exercise price $K=\$140$, and the effective annual risk-free rate is $i=5\%$. A dealer quotes the call at $\$26$. Determine whether the quote violates either no-arbitrage bound.

Method and present value. The two-year accumulation factor is $$ (1.05)^2=1.1025. $$ The present value of the exercise price is $$ \frac{\$140}{1.1025}=\$126.9841\ldots. $$

Bounds. Before rounding, the lower bound is $$ \$150-\$126.9841\ldots=\$23.0159\ldots, $$ and the upper bound is $\$150$. Rounded to cents, the interval is $\$23.02\leq c\leq \$150.00$. The quote of $\$26$ is inside this interval, so it does not violate either bound.

Check and interpretation. The quoted price exceeds the lower bound by approximately $\$2.98$, and it is below the upper bound by $\$124.00$. Thus neither comparison identifies an arbitrage at this quote. Being inside the interval does not prove that $\$26$ is the call’s exact or unique fair value; it means only that these bounds do not rule it out.

The method is to discount the exercise price for the full two years before finding the lower bound. The common mistake is to use $\$140/1.05$, which discounts for only one year and gives the wrong lower bound. Calculations were carried without rounding until the final bounds, which are reported to two decimal places.

Interpreting the Results and Avoiding Traps

The bounds are useful because they can reject impossible prices without estimating volatility or predicting the stock’s future price. The lower bound becomes positive when the stock price exceeds the present value of the exercise price. When it does not, the call’s nonnegative payoff still gives a lower bound of zero. The upper bound does not depend on the exercise price or interest rate: a call cannot be worth more than the share that covers its payoff at expiration.

A no-arbitrage bound should not be mistaken for a forecast. For example, a price between the lower and upper bounds is not guaranteed to be attractive, accurately estimated, or suitable for a particular investor. The bounds establish necessary consistency conditions under the stated assumptions; they do not determine the premium within the interval.

Discount the exercise price over the contract’s full term. Use $K/(1+i)^n$ when $i$ is an effective annual rate and $n$ is the number of years.
Keep the two bounds distinct. The lower bound uses both the stock price and discounted exercise price; the upper bound is the stock price.
Do not ignore the zero floor. A negative value of $S_0-K/(1+i)^n$ does not imply a negative call price.
Check the expiration payoff, not just the initial cash flow. An arbitrage construction must have a nonnegative payoff for every possible stock price at expiration.
Check the assumptions. Dividends, transaction costs, or different borrowing and lending rates can change the comparison.

Key takeaway: For a European call on non-dividend-paying stock, the no-arbitrage interval is $\max(S_0-K/(1+i)^n,0)\leq c\leq S_0$. A price below the lower bound can be challenged with a long call, lending, and a short stock position; a price above the upper bound can be challenged with a short call and a long stock position.

Check Your Understanding

For each problem, use an effective annual risk-free rate and assume no dividends, transaction costs, or short-sale restrictions. Amounts are per share. Round money to cents where needed.

A non-dividend-paying stock is priced at $\$95$. A one-year European call has $K=\$90$, and the effective annual risk-free rate is $5\%$. Find the no-arbitrage interval. (Answer: $\$9.29\leq c\leq \$95.00$.)
A stock costs $\$72$. A two-year European call has $K=\$80$, and the effective annual risk-free rate is $10\%$. If the call is quoted at $\$5$, determine whether the quote violates a bound and give the initial arbitrage receipt. (Answer: The lower bound is $\$72-\$80/(1.10)^2=\$5.88$; the quote is too low, and buying the call, lending $\$66.12$, and shorting the stock produces an initial receipt of about $\$0.88$.)
A stock costs $\$210$. A two-year European call has $K=\$225$, and the effective annual risk-free rate is $4\%$. Find the lower bound and state whether a quote of $\$3$ violates either bound. (Answer: The lower bound is $\max(\$210-\$225/(1.04)^2,0)=\$1.97$; the $\$3$ quote is within the interval $[\$1.97,\$210]$.)
A stock costs $\$64$, and a European call on it is quoted at $\$66$, with $K=\$55$ and a one-year maturity. Identify the arbitrage position and its initial receipt. (Answer: The quote exceeds the $\$64$ upper bound; buy the stock and sell the call for an initial receipt of $\$2$, with expiration payoff $\min(S_T,\$55)\geq0$.)
A stock costs $\$130$. A three-year European call has $K=\$120$, and the effective annual risk-free rate is $6\%$. A dealer quotes $\$28$. Test the quote and state the initial receipt from the lower-bound arbitrage if applicable. (Answer: $120/(1.06)^3=\$100.75$, so the lower bound is $\$29.25$; the quote is too low, and buying the call, lending about $\$100.75$, and shorting the stock gives an initial receipt of about $\$1.25$.)

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