Put-Call Parity With Dividends And Carry

Why Dividends and Carry Change Parity

Put-call parity links the prices of European calls and puts with the same underlying asset, strike, and expiration. When the underlying pays income before expiration, owning it provides cash flows that an option holder does not receive. The parity equation must account for that difference. Carrying expenses work in the opposite direction: owning the underlying requires additional payments, so their present value increases the adjusted cost of the asset position.

The essential exam discipline is to list every cash flow by date before writing the equation. A distribution paid halfway through the option term is not discounted as though it were paid at expiration, and a distribution received by the asset holder is not ignored merely because the option payoff depends on the asset price at expiration.

Definition: Let $C_0$ and $P_0$ be the time-0 prices of European calls and puts on the same asset, with strike $K$ and expiration $T$. Let $S_0$ be the asset's time-0 price. Write $D(0,t)$ for the time-0 price of $1 payable at time $t$. Under a constant effective annual rate $i$, $D(0,t)=(1+i)^{-t}=v^t$. A known cash distribution reduces the adjusted asset value; a known cash carrying expense increases it.

Parity With Known Cash Distributions and Expenses

Suppose the asset pays known cash distributions $I_j$ at dates $t_j$ before expiration, and requires known cash carrying expenses $H_k$ at dates $s_k$. Assume the amounts and dates are fixed, the same underlying asset is used for both options, and the relevant discount factors are known. The adjusted asset value is the spot price less the present value of distributions plus the present value of expenses.

C_0-P_0 = S_0-\sum_j I_jD(0,t_j)+\sum_k H_kD(0,s_k)-KD(0,T).

If there are no carrying expenses, omit the expense sum. If there are no distributions, the equation reduces to the familiar parity relationship $C_0-P_0=S_0-KD(0,T)$. A distribution after expiration is not included: it does not occur during the life of these options.

Match the options.
Confirm that the call and put are European and have the same asset, strike $K$, and expiration $T$.

List interim asset cash flows.
Record each distribution received and each carrying expense paid, with its date. Treat distributions as positive income and expenses as positive outflows.

Discount each amount separately.
Use $D(0,t)$ for a cash flow at its own date $t$, and $D(0,T)$ for the strike paid at expiration.

Apply parity and check signs.
Subtract the present value of income, add the present value of expenses, and subtract the present value of the strike.

Why the Adjustment Holds

The parity adjustment follows from matching terminal payoffs while accounting for interim cash flows. Consider a position that delivers one unit of the asset at $T$ but has no net interim distributions or carrying payments. Its time-0 cost is $S_0-\sum_j I_jD(0,t_j)+\sum_k H_kD(0,s_k)$: distributions reduce the cost of obtaining the asset exposure, while expenses increase it. This is the value of the prepaid asset position under the stated assumptions.

At expiration, for any asset price $S_T$, the call payoff plus $K$ equals the put payoff plus the asset:

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

The equality holds in both cases. If $S_T\ge K$, both sides equal $S_T$. If $S_T<K$, both sides equal $K$. Thus a call plus a risk-free payment of $K$ at $T$ has the same expiration value as a put plus delivery of the asset at $T$. The initial value of the strike payment is $KD(0,T)$; the initial value of the prepaid asset position is its adjusted cost above. Equating the two initial values gives the parity equation. The argument relies on matched terminal delivery and correctly offset interim cash flows; it does not apply unchanged if the options have different expirations or exercise terms.

Worked Example: One Distribution Before Expiration

A hypothetical non-dividend-reinvested share is priced at $S_0=\$110$. A European call and put both have strike $K=\$100$ and expire in one year. The asset will pay a known cash distribution of $\$5$ at $t=0.5$. The effective annual interest rate is $i=10\%$, with annual compounding. If the put costs $\$6.50$, find the call price. Round the final amount to the nearest cent.

Method. Use parity with the present value of the known distribution subtracted. The common trap is to omit the distribution and apply the no-income equation.

First discount the distribution from its actual payment date:

D(0,0.5)=(1.10)^{-0.5}=0.9534626, \qquad PV(\text{distribution})=\$5(0.9534626)=\$4.7673.

Next discount the strike from expiration:

D(0,1)=(1.10)^{-1}=0.9090909, \qquad PV(K)=\$100(0.9090909)=\$90.9091.

Therefore,

C_0-P_0 = \$110-\$4.7673-\$90.9091 = \$14.3236.

Since $P_0=\$6.50$, $C_0=\$6.50+\$14.3236=\$20.8236$, so the call price is \$20.82. As a check, the call exceeds the put by about $\$14.32$, the adjusted asset value less the present value of the strike. Omitting the distribution would instead give $C_0=\$6.50+\$110-\$90.9091=\$25.59$, which is too high by the distribution's present value.

Carry Costs and Continuous Proportional Income

A known fixed carrying payment, such as a specified storage charge payable on a given date, enters as a positive present-value term. The first example below used a known cash flow schedule; the same sign logic applies whether the expense occurs once or at several dates. Do not treat an expense as a distribution: an expense raises the cost of maintaining the asset exposure.

A different common model represents income and carry as continuous rates proportional to the asset value. Let $q$ be the continuously compounded proportional income yield and $c$ the continuously compounded proportional carrying-cost rate. Under this standard model, with constant force of interest $\delta$, no-arbitrage parity is

C_0-P_0=S_0e^{-(q-c)T}-Ke^{-\delta T}.

The net yield $q-c$ summarizes the offset between income received and proportional carrying cost. If income exceeds carrying cost, $q-c>0$, so the adjusted asset term is less than $S_0$. If carrying cost exceeds income, $q-c<0$, so the adjusted asset term is greater than $S_0$. This exponential form assumes the stated continuous proportional-rate model; it should not be substituted for separately specified fixed cash distributions or expenses.

Worked Example: Multiple Distributions and a Carrying Payment

A hypothetical asset costs $\$80$. A European call and put have strike $K=\$75$ and expire in one year. The asset pays known distributions of $\$1$ at $t=0.25$ and $\$2$ at $t=0.75$. A known carrying charge of $\$0.60$ is paid at $t=0.5$. The effective annual interest rate is $i=4\%$, with annual compounding. If the put price is $\$4.25$, find the call price to the nearest cent.

Method. Discount each dated cash flow separately, subtract the distribution values, and add the carrying-charge value. The trap is to use one discount factor for cash flows that occur on different dates.

\begin{aligned} PV(I_1)&=\$1(1.04)^{-0.25}=\$0.990243,\\ PV(I_2)&=\$2(1.04)^{-0.75}=\$1.942024,\\ PV(H)&=\$0.60(1.04)^{-0.5}=\$0.588348,\\ PV(K)&=\$75(1.04)^{-1}=\$72.115385. \end{aligned}

The adjusted parity difference is

C_0-P_0 = \$80-\$0.990243-\$1.942024+\$0.588348-\$72.115385 = \$5.540696.

Thus $C_0=\$4.25+\$5.540696=\$9.790696$, or \$9.79 to the nearest cent. The signs are sensible: distributions lower the adjusted asset value, while the carrying charge raises it. For comparison, incorrectly discounting both distributions and the expense from $t=1$ would give a parity difference of about $\$5.58$, not $\$5.54$. Each amount must be discounted from its own stated date.

Worked Example: Continuous Income and Proportional Carry

A hypothetical asset has $S_0=\$100$. A European call and put have $K=\$105$ and expire in $T=2$ years. The asset has a continuous proportional income yield $q=2\%$ and a continuous proportional carrying-cost rate $c=1\%$. The force of interest is $\delta=6\%$. If the put costs $\$8.00$, find the call price to the nearest cent.

Method. Use the continuous proportional carry form of parity. The net yield is $q-c=0.02-0.01=0.01$, and the two discounting exponents use different rates: $q-c$ for the adjusted asset term and $\delta$ for the strike.

\begin{aligned} S_0e^{-(q-c)T} &=\$100e^{-(0.01)(2)} =\$100e^{-0.02} =\$98.019867,\\ Ke^{-\delta T} &=\$105e^{-(0.06)(2)} =\$105e^{-0.12} =\$93.126646. \end{aligned}

Therefore $C_0-P_0=\$98.019867-\$93.126646=\$4.893221$. Adding the put price gives $C_0=\$8.00+\$4.893221=\$12.893221$, or \$12.89. A check is that the net income yield lowers the adjusted asset term below $\$100$, while discounting the strike lowers its time-0 value below $\$105$.

Common Parity Errors

Parity is especially sensitive to cash-flow timing and sign conventions. Before accepting an equation, check the model and the dates as well as the arithmetic.

Omitting income. For known distributions, use $S_0-PV(\text{distributions})$, not $S_0$. The first worked example showed that omission overstates the call-put difference by the present value of the distribution.
Adding income instead of subtracting it. A distribution benefits the owner of the asset, so the prepaid asset exposure costs less than the spot price. A carrying expense has the opposite sign.
Discounting every cash flow to expiration. Use the date of each payment. Only the strike is necessarily paid at $T$ in this parity setup.
Mixing rate models. Use effective annual discounting for an effective annual rate and continuous discounting for a force of interest. Do not use $e^{-\delta t}$ merely because it looks convenient when the given rate is effective.
Using parity outside its assumptions. The call and put must match in underlying, strike, and expiration, and the equation here is for European options. Different exercise terms or uncertain, unmodeled cash flows require additional analysis.

This equation is useful both for valuation checks and for locating an inconsistent cash-flow assumption. A calculated discrepancy is a signal to recheck dates, signs, rate conventions, and option terms; establishing and executing an arbitrage trade requires a separate analysis.

Check Your Understanding

For each problem, identify the relevant cash-flow model, apply parity, and round the final dollar amount to the nearest cent.

A hypothetical asset is priced at $\$60$. European options with $K=\$58$ expire in one year. The asset pays a known $\$1.50$ distribution at $t=0.5$, and $i=5\%$ effective annually. Find $C_0-P_0$. (Answer: $\$3.30$)
A hypothetical asset costs $\$125$. Its European call and put have $K=\$120$, $T=1$, and $i=6\%$ effective annually. It pays distributions of $\$3$ at $t=0.25$ and $\$2$ at $t=0.75$. If $C_0=\$14$, find $P_0$. (Answer: $\$7.08$)
A hypothetical asset has $S_0=\$90$, $K=\$92$, and $T=1.5$. Its continuous proportional income yield is $q=3\%$, its proportional carrying-cost rate is $c=0.5\%$, and the force of interest is $\delta=4\%$. Find $C_0-P_0$. (Answer: $\$0.05$)
A hypothetical asset costs $\$90$. European options with $K=\$88$ expire in one year. The asset pays a known $\$2$ distribution at $t=0.5$ and requires a known carrying payment of $\$0.40$ at $t=0.25$. If $i=8\%$ effective annually and $P_0=\$6$, find $C_0$. (Answer: $\$12.99$)
A hypothetical asset has $S_0=K=\$100$, $T=2$, continuous proportional income yield $q=2.5\%$, and proportional carrying-cost rate $c=0.5\%$. The force of interest is $\delta=5\%$. If $C_0=\$9$, find $P_0$. (Answer: $\$3.40$)

What You'll Learn