European Option Bounds Implied By Parity

Why Option Bounds Matter

Put-call parity links the prices of a European call and put with the same underlying, strike, and expiration. It also lets us test whether individual option prices are plausible, even when we do not know the correct option value. The test is a set of no-arbitrage bounds: a price outside those bounds would allow a portfolio with a positive initial receipt and no possible future loss.

The bounds depend on the underlying and the financing cost of the strike. In particular, the present value of the strike need not be less than the strike itself: when interest rates are negative, it can be greater. The formulas below accommodate either sign of the rate.

Assumptions: The options are European calls and puts on the same non-dividend-paying stock, with current price $S_0$, strike $K>0$, and expiration time $T$. The stock price at expiration, $S_T$, is nonnegative. The market is free of arbitrage, and borrowing and lending use the same risk-free discount factor $D(0,T)>0$. There are no transaction costs or restrictions on short selling. Write $C$ and $P$ for the current call and put prices. If the effective annual risk-free rate $i$ is constant, then $D(0,T)=v^T=(1+i)^{-T}$, where $v=1/(1+i)$ and $i>-1$.

The Bounds From Parity and Payoff Comparisons

For these options, European put-call parity is

$$ C-P=S_0-KD(0,T). $$

The term $KD(0,T)$ is the current cost of a risk-free payment of $K$ at expiration. The parity equation equates a long call and a short put with a portfolio consisting of one share of stock and a short position in that payment. To get useful price limits, we combine parity with the fact that option prices cannot be negative, and compare option payoffs with the payoffs of stock and risk-free investments.

Theorem: Under the stated assumptions, the European call and put prices satisfy

\max\!\left(S_0-KD(0,T),0\right) \le C \le S_0,

\max\!\left(KD(0,T)-S_0,0\right) \le P \le KD(0,T).

The lower bounds follow directly from parity and the nonnegativity of option prices. Since $P\ge 0$, parity gives $C=S_0-KD(0,T)+P\ge S_0-KD(0,T)$. Since $C\ge 0$, the call must satisfy both lower limits, so its lower bound is their maximum. Rearranging parity gives $P=KD(0,T)-S_0+C\ge KD(0,T)-S_0$. Together with $P\ge0$, this gives the put lower bound.

The upper bounds require an additional no-arbitrage comparison. At expiration, a call pays no more than one share of stock, and a put pays no more than $K$. A share costs $S_0$, while a risk-free promise to pay $K$ at expiration costs $KD(0,T)$. If either option cost more than its corresponding investment, the difference could be received upfront while the investment covers the option's possible payment at expiration. That would be an arbitrage.

Compare expiration payoffs.
$\max(S_T-K,0)\le S_T$, so the call payoff is no greater than the stock payoff. Also, $\max(K-S_T,0)\le K$, so the put payoff is no greater than a risk-free payment of $K$.

Compare current prices.
In a no-arbitrage market, an asset that always pays no more than another asset cannot cost more than that other asset. Therefore $C\le S_0$ and $P\le KD(0,T)$.

Combine the limits.
Parity and nonnegative prices supply the lower bounds; payoff dominance supplies the upper bounds. Both ingredients are needed for the full intervals.

The upper-bound argument can also be checked by constructing the arbitrage that would result if a price exceeded its limit. If $C>S_0$, sell the call and buy a share. The initial receipt is $C-S_0>0$, and the expiration value of the share minus the call payoff is $\min(S_T,K)\ge0$. If $P>KD(0,T)$, sell the put and buy a risk-free payment of $K$. The initial receipt is $P-KD(0,T)>0$, and the expiration value of the payment minus the put payoff is $\min(S_T,K)\ge0$. These comparisons rely on the stock price being nonnegative.

Worked Examples

Worked Example: Computing Both Bounds at a Positive Rate

A non-dividend-paying stock is priced at $S_0=\$120$. A European call and put have strike $K=\$110$ and expire in two years. The effective annual risk-free rate is $i=5\%$, compounded annually. Find the no-arbitrage bounds for both prices. If the call is quoted at $\$24.00$, find the put price implied by parity and check it against its bounds.

Method and assumptions. Use the constant effective annual rate to discount the strike for two years. The options are European and matched in underlying, strike, and expiration, so parity applies.

Step 1: Discount the strike. The one-year discount factor is $v=1/(1.05)$, so

D(0,2)=v^2=\frac{1}{1.05^2} =\frac{1}{1.1025} \approx 0.90702948.

Thus the present value of the strike is

KD(0,2)=110(0.90702948)\approx \$99.77324.

Step 2: Calculate the lower bounds. The call's parity lower limit is $S_0-KD(0,2)=120-99.77324=\$20.22676$, which is positive. The put's parity lower limit is $KD(0,2)-S_0=99.77324-120=-\$20.22676$, so nonnegativity makes its lower bound zero.

Step 3: State the full intervals. The call lies between $\$20.22676$ and $\$120$; the put lies between zero and $\$99.77324$. To the nearest cent, these are $\$20.23\le C\le\$120.00$ and $\$0.00\le P\le\$99.77$.

Step 4: Apply parity to the quoted call. Rearranging parity gives $P=C-S_0+KD(0,2)$. Hence

P=24.00-120+99.77324=\$3.77324\approx \$3.77.

This price is within the put bounds. The check is consistent: $C-P=24.00-3.77324=\$20.22676$, equal to $S_0-KD(0,2)$. Common mistake avoided: subtracting the undiscounted strike from the stock price would give $120-110=\$10$, not the parity lower bound. The strike must be discounted to expiration's present value.

Worked Example: Bounds When the Risk-Free Rate Is Negative

A non-dividend-paying stock is priced at $S_0=\$80$. Matched European options have strike $K=\$100$, expire in one year, and the effective annual risk-free rate is $i=-4\%$. Find both price intervals. If the put is quoted at $\$25.00$, find the call price implied by parity.

Method and assumptions. Use the effective annual rate even though it is negative. Since $1+i=0.96>0$, the discount factor is positive and parity remains valid.

Step 1: Discount the strike.

D(0,1)=\frac{1}{1-0.04}=\frac{1}{0.96} =\frac{25}{24}\approx1.04166667, \qquad KD(0,1)=100\left(\frac{25}{24}\right) =\$104.16667.

Step 2: Calculate the lower and upper limits. The call lower bound is $\max(80-104.16667,0)=0$, and its upper bound is $\$80$. The put lower bound is $\max(104.16667-80,0)=\$24.16667$, and its upper bound is $\$104.16667$. To the nearest cent, $0\le C\le\$80.00$ and $\$24.17\le P\le\$104.17$.

Step 3: Use parity for the quoted put. Rearranging gives $C=P+S_0-KD(0,1)$, so

C=25+80-104.16667=\$0.83333\approx\$0.83.

The implied call is within its bounds, and $C-P=0.83333-25=-\$24.16667=S_0-KD(0,1)$. Common mistake avoided: assuming the present value of the strike must be below $\$100$. With a negative rate, the discount factor exceeds one, so the strike's present value is $\$104.16667$; the put's upper bound can therefore exceed the strike amount.

Worked Example: Diagnosing a Call Price Below Its Lower Bound

A non-dividend-paying stock is priced at $S_0=\$50$. A European call and put have strike $K=\$60$, expire in one year, and the effective annual risk-free rate is $i=25\%$. The call is quoted at $\$1.50$. Determine whether the quote is consistent with no-arbitrage bounds. If not, give a portfolio with an arbitrage.

Method and assumptions. First compute the discounted strike and the call lower bound. For a violation, use a long call, a risk-free payment of $K$, and a short share to compare initial and expiration cash flows.

Step 1: Compute the discount factor and call lower bound.

D(0,1)=\frac{1}{1.25}=0.8, \qquad KD(0,1)=60(0.8)=\$48,

so the call must cost at least $\max(50-48,0)=\$2.00$. The quoted price of $\$1.50$ is below the bound.

Step 2: Construct the portfolio. Buy one call for $\$1.50$, buy a risk-free zero-coupon investment that pays $\$60$ for $\$48$, and short one share to receive $\$50$. The initial net receipt is

50-1.50-48=\$0.50.

At expiration, the call and risk-free investment together pay $\max(S_T-60,0)+60$; covering the short share costs $S_T$. The net expiration payoff is

\max(S_T-60,0)+60-S_T=\max(60-S_T,0)\ge0.

The portfolio receives $\$0.50$ initially and has a nonnegative expiration payoff for every $S_T\ge0$, so it is an arbitrage under the assumptions. Common mistake avoided: checking only whether the call price is nonnegative. A positive option price can still violate its stronger parity lower bound.

Interpreting and Using the Intervals

The bounds are necessary conditions, not a formula for the exact option price. A price inside its interval is not automatically correct or free of every possible arbitrage; it simply passes this particular test. Parity imposes a further relationship between matched call and put prices. For example, prices can each fall within their individual intervals but still fail parity as a pair.

The role of the discount factor is especially important. Under negative rates, $KD(0,T)$ can exceed $K$, so the put's upper bound can exceed the strike. This does not mean the put can pay more than $K$ at expiration. It means that funding a certain payment of $K$ at expiration costs more than $K$ today under the stated negative-rate discounting. The upper bound compares current prices, while the option's payoff is received in the future.

These formulas are specific to the assumptions stated above. A dividend-paying stock changes the parity relationship and can change the relevant bounds. An American option can be exercised before expiration, so the European equality and bounds should not be transferred without a separate argument. Likewise, if the underlying can take negative values, the payoff comparisons used for the upper limits need not hold as written.

Key takeaway: For European options on a non-dividend-paying, nonnegative stock, parity and nonnegative prices give the lower bounds; payoff dominance gives the upper bounds. Discount the strike using the applicable positive discount factor, including when the interest rate is negative.

Check Your Understanding

For each problem, use the stated discount factor or rate and assume the options are European and the stock pays no dividends. Money answers are rounded to the nearest cent where needed.

A stock is priced at $\$90$. Matched options have strike $\$100$ and discount factor $D(0,T)=0.90$. Find the call and put price intervals. (Answer: $0\le C\le\$90.00$; $0\le P\le\$90.00$)
A stock is priced at $\$140$. Matched options have strike $\$120$ and discount factor $D(0,T)=0.95$. Find both price intervals. (Answer: $\$26.00\le C\le\$140.00$; $0\le P\le\$114.00$)
A stock is priced at $\$75$. Matched options have strike $\$80$ and discount factor $D(0,T)=1.05$. Find both intervals and explain whether the discount factor is consistent with a negative rate. (Answer: $0\le C\le\$75.00$; $\$9.00\le P\le\$84.00$; yes, a discount factor greater than one is consistent with a negative rate)
A stock is priced at $\$100$. Matched options have strike $\$105$ and discount factor $D(0,T)=0.96$. If the call costs $\$8.00$, find the parity-implied put price and check both option prices against their bounds. (Answer: $P=\$8.80$; call bounds are $\$0.00$ to $\$100.00$, and put bounds are $\$0.80$ to $\$100.80$, so both prices are within bounds)
A stock is priced at $\$64$. A European call with strike $\$70$ has discount factor $D(0,T)=0.90$ and is quoted at $\$0.60$. Determine whether the quote violates a bound and state an arbitrage portfolio with its initial receipt. (Answer: The call lower bound is $\$1.00$, so the quote violates it; buy the call and a risk-free payment of $\$70$, and short one share, for an initial receipt of $\$0.40$ and a nonnegative expiration payoff)

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