Put-Call Parity With Known Cash Dividends

Why Known Dividends Change Put-Call Parity

A share bought today may pay cash dividends before a European option expires. The shareholder receives those dividends; a portfolio made only of options and a risk-free payment at expiration does not. Therefore, the familiar parity relation without dividends needs an adjustment when the dividend amounts and payment dates are known.

The adjustment uses the present value of the dividends, not their total undiscounted amount. The date of each payment matters: a dividend paid sooner has a different present value from the same amount paid later. This tutorial considers European calls and puts on the same underlying asset, with the same strike and expiration.

Definition: Suppose an underlying asset has known cash dividends $D_j$ paid at times $t_j$, where $0<t_j<T$. At valuation time 0, the present value of these dividends is $\sum_j D_jv^{t_j}$, where $T$ is the options’ expiration time in years, $v=1/(1+i)$, and $i$ is the effective annual risk-free rate. With $C$ and $P$ denoting the European call and put prices, $K$ the common strike, and $S_0$ the current asset price, dividend-adjusted put-call parity is $$ C+Kv^T+\sum_j D_jv^{t_j}=P+S_0. $$

The equivalent rearrangement $C-P=S_0-\sum_jD_jv^{t_j}-Kv^T$ can be useful when comparing option prices. Under the assumptions of the parity result, either equation must hold if prices are free of arbitrage. Use the same rate convention and time units for every discount factor. With a single flat effective annual rate, a cash flow at time $t$ years is discounted by $v^t$; for example, a payment at six months is discounted by $v^{0.5}$.

Assumptions and Cash-Flow Logic

The identity assumes a frictionless market with no arbitrage: trading is possible at the stated prices, borrowing and lending use the same risk-free rate, and there are no taxes, transaction costs, or short-sale restrictions. The call and put are European, have the same underlying asset, strike $K$, and expiration $T$, and have no early-exercise feature. All dividends before expiration are known cash amounts paid at specified dates. The stock price $S_0$ is the current price before any future dividend payments in the schedule.

These conditions matter. An uncertain dividend cannot simply be inserted as though its amount and timing were known. Nor should dividends paid after expiration be included: they do not affect the option’s cash flows before expiration. Questions about American exercise, transaction costs, or dividend-date conventions require additional care beyond this equality.

List the dividend cash flows.
Record each known amount $D_j$ and its payment time $t_j$, making sure each payment occurs strictly before expiration.

Discount to time 0.
Calculate $D_jv^{t_j}$ for each payment and add the present values. Do not discount all dividends from expiration unless they are actually paid then.

Apply parity with consistent signs.
Use $C+Kv^T+\sum_jD_jv^{t_j}=P+S_0$, then rearrange for the unknown price or value.

Check the result.
Substitute the calculated value back into both sides. If prices are quoted and the sides differ, the difference identifies a potential parity violation under the stated assumptions.

Why the Dividend Adjustment Holds

The identity follows by comparing two portfolios that have the same cash flows at expiration and during the dividend dates. The left-hand portfolio consists of a call, an investment worth $Kv^T$ today that pays $K$ at expiration, and separate risk-free investments worth $D_jv^{t_j}$ today that pay $D_j$ at the dividend dates. Its initial cost is $C+Kv^T+\sum_jD_jv^{t_j}$.

The right-hand portfolio consists of a put and one share, at initial cost $P+S_0$. The share pays the dividends when they occur. The dividend investments in the left-hand portfolio provide matching cash on those dates. At expiration, let the share price be $S_T$. The left portfolio’s option-and-strike payoff is $\max(S_T-K,0)+K$; the right portfolio’s stock-and-put payoff is $S_T+\max(K-S_T,0)$. Both equal $\max(S_T,K)$. Thus the portfolios have identical cash flows at every relevant date. If their initial costs differed, an investor could buy the cheaper portfolio and sell the more expensive one with no difference in later cash flows, creating an arbitrage. Their initial costs must therefore be equal.

Worked Example: Solving for a Call Price With a Dividend

A share is priced at $100. A known dividend of $4 will be paid one year from now. A European call and put have strike $K=\$95$ and expire in two years. The effective annual risk-free rate is $i=10\%$, and the put price is $8. Find the call price. Assume the market meets the frictionless conditions stated above.

Method. Apply dividend-adjusted parity and solve for $C$. The dividend is paid at $t=1$, not at expiration $T=2$.

The discount factor is $v=1/1.10=10/11$. The discounted strike is

Kv^T=95\left(\frac{10}{11}\right)^2 =\frac{9500}{121} \approx 78.5124.

The dividend’s present value is

4v^1=4\left(\frac{10}{11}\right) =\frac{40}{11} \approx 3.6364.

Rearrange parity to isolate the call:

C=P+S_0-Kv^T-4v^1 =8+100-78.5124-3.6364 \approx 25.8512.

Rounded to the nearest cent, the call price is $25.85. Check both sides using the unrounded value: $C+Kv^T+4v^1\approx25.8512+78.5124+3.6364=108.0000$, while $P+S_0=8+100=108$. The equality holds. The common mistake this method avoids is subtracting the dividend without discounting it from its actual payment date.

Using Parity to Detect an Arbitrage

Parity is also a diagnostic for quoted prices. Calculate each side using the same inputs and compare them. A difference is not automatically an executable arbitrage in a real market: the frictionless assumptions may fail, or quotes may include spreads and costs. Under the assumptions in this tutorial, however, unequal sides imply that one replicated portfolio is overpriced relative to the other.

Worked Example: Finding the Direction of a Parity Trade

A share costs $60. A known $3 dividend is paid six months from now. European options with strike $58 expire in one year. The effective annual risk-free rate is $i=0\%$; the call costs $7 and the put costs $11. Determine whether parity holds and, if it does not, describe the zero-risk cash-flow strategy under the stated assumptions.

Method. Compare the initial costs of the portfolios in the parity identity. With $i=0\%$, all the discount factors are 1, so the present value of the dividend is $3 and the present value of the strike is $58.

C+Kv^T+Dv^{0.5}=7+58+3=68, \qquad P+S_0=11+60=71.

The right-hand portfolio costs $3 more than the left-hand portfolio. Buy the less expensive left-hand portfolio: long one call, hold $58 in a risk-free investment that pays $58 at expiration, and hold $3 in a risk-free investment that pays $3 at the dividend date. Short the right-hand portfolio: short one put and one share. The initial net cash flow is $71-68=\$3$ received.

At the dividend date, the $3 investment pays the $3 dividend owed on the short share, so the intermediate cash flows cancel. At expiration, if $S_T\ge58$, the long call and the $58 payment produce $S_T$; the short share requires paying $S_T$, and the put expires worthless. If $S_T<58$, the call expires worthless and the $58 payment remains; the short put costs $58-S_T$ to settle, while the short share position gains $S_T$ relative to its delivery value. Equivalently, the net terminal payoff is

\max(S_T-58,0)+58-\max(58-S_T,0)-S_T=0.

Thus the strategy has a $3 initial receipt and zero net cash flow at the dividend date and expiration. Parity does not hold; the right-hand side exceeds the left-hand side by $3.00. The common mistake this trade avoids is to buy the more expensive portfolio or to ignore the dividend obligation on the short share.

Solving for a Put When There Are Multiple Dividends

With several known dividends, discount each amount separately and add the resulting present values. Do not combine the payments first and then discount their total from one date: different payment times generally imply different discount factors. The strike is discounted from expiration, while each dividend is discounted from its own payment date.

Worked Example: Two Dividends and an Unknown Put Price

A share costs $100. It will pay known dividends of $2 at the end of year 1 and $4 at the end of year 2. A European call with strike $K=\$90$ expires at the end of year 3 and costs $30. Find the price of the matching European put. The effective annual risk-free rate is $i=10\%$.

Method. Discount the strike for three years and each dividend for its own payment period. Then rearrange parity to solve for $P$.

Here $v=10/11$. The strike present value is

90v^3=90\left(\frac{10}{11}\right)^3 =\frac{90000}{1331} \approx67.6183.

The dividend present values are

2v^1=\frac{20}{11}\approx1.8182, \qquad 4v^2=\frac{400}{121}\approx3.3058.

Their total present value is $1.8182+3.3058\approx5.1240$. Rearranging $C+Kv^T+\sum_jD_jv^{t_j}=P+S_0$ gives

P=30+67.6183+5.1240-100 \approx2.7423.

Rounded to the nearest cent, the put price is $2.74. To check, the left side of parity is $30+67.6183+5.1240\approx102.7423$; the right side is $2.7423+100=102.7423$. The common mistake this calculation avoids is discounting both dividends from year 3 instead of discounting each from its payment date.

Common Mistakes and Practical Checks

The adjustment is compact, but several errors can change the answer materially. Before accepting a result, check the dividend dates, the side of the equation on which the dividend present value appears, and whether the option contracts truly match.

Omitting dividends. Using $C+Kv^T=P+S_0$ when known dividends occur before expiration leaves out cash received by the shareholder.
Using the wrong sign. In the form $C+Kv^T+\text{PV(dividends)}=P+S_0$, the dividend present value is added on the call-and-strike side. When solving for $P$, it is included in $P=C+Kv^T+\text{PV(dividends)}-S_0$.
Discounting from the wrong date. The strike is paid at expiration; a dividend is paid on its stated date. Their discount periods need not be equal.
Mixing time units. If $i$ is an effective annual rate, use time in years in $v^t$. A six-month payment has $t=0.5$, not $t=6$.
Comparing nonmatching options. The call and put must have the same underlying, strike, and expiration for this identity to apply as stated.

Key takeaway: For European options with known cash dividends before expiration, add the present value of those dividends to the call-and-discounted-strike side of parity. Discount each dividend from its own payment date, then check the equality or use the cost difference to determine the trade direction.

Check Your Understanding

Use the dividend-adjusted parity relation. Unless stated otherwise, assume European options, known dividends, and the frictionless conditions described above.

A share costs $80. A known dividend of $2 is paid six months from now. Matching options have $K=\$75$, $T=1$, and the effective annual rate is $0\%$. If the call costs $9, find the put price. (Answer: $6.00)
A share costs $105. A known dividend of $5 is paid at the end of year 1. Matching options have $K=\$100$, $T=2$, and $i=10\%$. If the call costs $20, find the put price, rounded to the nearest cent. (Answer: $2.19)
A share costs $40. A $1 dividend is paid at the end of year 1. Matching options have $K=\$42$, $T=2$, and $i=0\%$. If the call costs $4 and the put costs $5, state whether parity holds and give the difference between the two sides. (Answer: Parity does not hold; the left side exceeds the right side by $2.00.)
A call costs $13 and a put costs $8. The strike is $50, expiration is in two years, $i=10\%$, and a known $2 dividend is paid at the end of year 1. Find the current share price, rounded to the nearest cent. (Answer: $48.14)
A share costs $70. A known $2 dividend is paid six months from now. Matching options have $K=\$68$, $T=1$, and $i=0\%$. The call costs $6 and the put costs $9. Identify which parity portfolio is overpriced and state the initial arbitrage receipt under the assumptions above. (Answer: The put-and-share portfolio is overpriced; buy the call-and-strike-and-dividend portfolio and short the put and share, receiving $3.00 initially.)

What You'll Learn