Deriving The Discrete-Dividend Adjustment

Why a Discrete Dividend Changes Parity

For European options on a stock that pays known cash dividends, the stock price today includes the value of dividend cash flows that will be paid before expiration. A call or put holder does not receive those cash flows merely by holding the option. Consequently, the stock term in put-call parity must be adjusted for the present value of the dividends.

The adjustment is not the sum of the dividend amounts themselves. Each dividend is paid at a particular time, so its value today depends on how long it takes to receive it. The derivation below accounts for those payments separately and shows why each is discounted from its own payment date.

Definition: A known discrete cash dividend is a specified cash amount $D_j$ paid by a stock at a specified time $t_j$. For European options expiring at time $T$, the dividends relevant to the parity adjustment are those with $0<t_j<T$. Under a constant effective annual interest rate $i$, their present value is $$ \operatorname{PV}(\text{dividends})=\sum_{j=1}^{m}\frac{D_j}{(1+i)^{t_j}}. $$ The time $t_j$ is measured in years from today. A dividend paid after expiration is not included.

Assumptions and the Parity Formula

Consider a European call and a European put on the same stock, with the same strike price $K$ and expiration time $T$. Let $C_0$ and $P_0$ be their prices today, and let $S_0$ be the stock price today. Assume the dividend amounts and payment dates before expiration are known, and that the same risk-free discounting convention applies to the strike and dividends. With a constant effective annual rate $i$, the parity relation is

C_0-P_0 = S_0-\sum_{j=1}^{m}\frac{D_j}{(1+i)^{t_j}} -\frac{K}{(1+i)^T}.

This equality assumes frictionless trading, no taxes or transaction costs, and the ability to trade the stock, options, and risk-free securities in the quantities required. The options are European, so they can be exercised only at expiration. The result does not require a probability distribution for the future stock price: the payoff comparison holds for every possible expiration price $S_T$, given the stated cash flows. If dividends are uncertain, if the options have different strikes or expirations, or if early exercise is possible, this derivation does not establish this equality as written.

Derivation by Matching Cash Flows

The key is to compare two portfolios that have identical cash flows at every date. In particular, a stockholder receives the dividends before expiration, while a call or put holder does not. A set of risk-free securities that pays the same amounts on the dividend dates can be used to account for those interim cash flows.

Build the first portfolio.
Buy one European call and invest $K/(1+i)^T$ today in a risk-free security that pays $K$ at expiration. Its initial cost is $C_0+K/(1+i)^T$.

Build the second portfolio.
Buy one European put and one share, and sell risk-free securities that pay $D_j$ on each dividend date $t_j$. The securities sold have a total value today of $\sum_j D_j/(1+i)^{t_j}$, so the portfolio’s initial cost is $P_0+S_0-\sum_j D_j/(1+i)^{t_j}$.

Match the interim dividend cash flows.
At time $t_j$, the share in the second portfolio receives $D_j$. The short risk-free security requires a payment of $D_j$ at that same time. These amounts cancel, leaving no net interim cash flow. The first portfolio has no dividend-related interim cash flow.

Match the expiration payoffs.
At expiration, the first portfolio pays $\max(S_T-K,0)+K=\max(S_T,K)$. The share and put in the second portfolio pay $S_T+\max(K-S_T,0)=\max(S_T,K)$. The short dividend securities have already been settled on their payment dates.

The portfolios have the same cash flows at every date, so the no-arbitrage principle requires their initial costs to be equal:

C_0+\frac{K}{(1+i)^T} = P_0+S_0-\sum_{j=1}^{m}\frac{D_j}{(1+i)^{t_j}}.

Rearranging gives the stated parity formula. This construction explains the sign of the dividend adjustment: the stock-and-put portfolio receives the dividends, so matching it to the call-and-bond portfolio requires subtracting an equal-value stream of dividend payments.

Worked Applications

Worked Example: Find a Call Price with Two Dividends

A stock is priced at $S_0=\$104$. A European call and put have strike $K=\$100$ and expire in two years. The effective annual interest rate is $i=4\%$. The stock will pay known dividends of $\$3$ at $t=0.5$ and $\$2$ at $t=1.5$. If the put price is $\$7.25$, find the call price to the nearest cent. Assume the market satisfies the frictionless no-arbitrage assumptions above; no probability distribution for $S_T$ is needed.

Method. Apply the derived parity relation, discounting each dividend from its own payment date and the strike from expiration.

Step 1: Present value of the dividends.

\operatorname{PV}(\text{dividends}) = \frac{3}{(1.04)^{0.5}}+\frac{2}{(1.04)^{1.5}} \approx 2.941742+1.885732 =\$4.827474.

Step 2: Present value of the strike.

\frac{K}{(1+i)^T} = \frac{100}{(1.04)^2} = \frac{100}{1.0816} \approx \$92.455621.

Step 3: Find the call-minus-put difference, then the call.

C_0-P_0 = 104-4.827474-92.455621 \approx \$6.716905,

so $C_0\approx 7.25+6.716905=\$13.966905$. Rounded only at the final step to two decimal places, the call price is $\$13.97$.

Check. The call exceeds the put by about $\$6.72$, which is the stock price less the present values of the two dividends and the strike. A common mistake is to subtract the undiscounted dividends, $\$3+\$2=\$5$, which would give a call-minus-put difference of only $\$6.544379$. That is incorrect because the dividend payments occur in the future.

Worked Example: Infer the Stock Price from Option Quotes

A European call is quoted at $\$21.40$, and a European put on the same stock is quoted at $\$5.10$. Both have strike $K=\$100$ and expire in two years. The effective annual interest rate is $i=5\%$, and the stock will pay a known $\$5$ dividend at the end of year 1. Find the stock price implied by parity, to the nearest cent. Assume frictionless trading, known dividends, and European exercise; the option payoff comparison is valid for every $S_T$.

Method. Rearrange parity to solve for $S_0$. Add the present values of the dividend and strike to $C_0-P_0$.

Step 1: Calculate the quoted option-price difference.

C_0-P_0=21.40-5.10=\$16.30.

Step 2: Discount the dividend and strike.

\operatorname{PV}(\text{dividend})=\frac{5}{1.05}=\$4.761905, \qquad \operatorname{PV}(\text{strike})=\frac{100}{(1.05)^2} =\frac{100}{1.1025} \approx \$90.702948.

Step 3: Solve for the implied stock price.

S_0 = (C_0-P_0)+\operatorname{PV}(\text{dividend})+\operatorname{PV}(\text{strike}) \approx16.30+4.761905+90.702948 =\$111.764853.

Rounded only at the final step to two decimal places, the implied stock price is $\$111.76$.

Check. Substituting this value into the right side of parity gives $111.764853-4.761905-90.702948=\$16.30$, matching the quoted call-minus-put difference. The common sign error is to subtract the present value of the dividend when solving for $S_0$; that adjustment belongs on the right side of the original formula, but must be added when isolating $S_0$.

Worked Example: Check a Quoted Call-Minus-Put Difference

A stock is priced at $\$150$. European options with strike $K=\$140$ expire in $T=1.5$ years. The effective annual interest rate is $i=8\%$. Known dividends of $\$6$ and $\$3$ will be paid at $t=0.5$ and $t=1$, respectively. The quoted call is $\$21.40$ and the put is $\$5.10$. Calculate the parity value of $C_0-P_0$ and the difference between the quote and parity, to the nearest cent. Assume the stated dividends and payment dates are certain.

Method. First calculate the dividend and strike present values, then compare their parity result with the quoted option difference.

Step 1: Discount each dividend separately.

\operatorname{PV}(\text{dividends}) = \frac{6}{(1.08)^{0.5}}+\frac{3}{1.08} \approx 5.773503+2.777778 =\$8.551280.

Step 2: Discount the strike to today.

\operatorname{PV}(\text{strike}) = \frac{140}{(1.08)^{1.5}} \approx \$124.736169.

Step 3: Calculate parity and the quoted difference.

(C_0-P_0)_{\text{parity}} = 150-8.551280-124.736169 \approx \$16.712550, $$ $$ (C_0-P_0)_{\text{quoted}} = 21.40-5.10 =\$16.30.

The quote is $16.712550-16.30=\$0.412550$, or $\$0.41$ below the parity value after rounding to the nearest cent.

Check. Each dividend is discounted from its own date, while the strike is discounted from the option expiration. The calculation compares call minus put with call minus put; comparing one option price alone with the parity difference would not be a like-for-like check. This difference indicates a mismatch with the stated parity equality under its assumptions; evaluating an executable trade would additionally require market trading details.

Timing, Interpretation, and Common Traps

The derivation is useful because it identifies exactly what must be matched: not just the expiration payoff, but also every cash flow before expiration. For each dividend, the stock in the replication receives $D_j$ at $t_j$, and the corresponding short risk-free claim pays $D_j$ at that same time. Its value today is $D_j/(1+i)^{t_j}$, which is why the dividend adjustment uses a separate discount factor for each payment.

Keep the following distinctions clear when applying the formula:

Use dividend payment dates. A dividend paid at $t=0.5$ is discounted for half a year, not for a full year or for the full option term.
Include only payments before expiration. A payment at or after expiration is not an interim cash flow received by the share during the option’s life under this construction.
Discount the strike from expiration. The strike present value uses $T$, not any dividend date.
Preserve the signs. In $C_0-P_0=S_0-\operatorname{PV}(\text{dividends})-\operatorname{PV}(K)$, both present values are subtracted. When solving for $S_0$, both are added to $C_0-P_0$.
Do not treat option holders as shareholders. The dividends belong to the holder of the stock, not to the holder of a European call or put.

The parity formula is therefore more than a memorized adjustment. It is a cash-flow identity: buying the stock and put, while offsetting the stock’s interim dividends, replicates the call and a risk-free payment of the strike at expiration. This reasoning is aligned with the SOA Exam FM (Financial Mathematics) syllabus and provides a disciplined way to check signs and timing.

Check Your Understanding

For each problem, assume European options, known discrete dividends, the stated effective annual rate, and frictionless no-arbitrage trading. Round money to the nearest cent.

A stock is priced at $\$80$. A European call and put have strike $\$75$ and expire in one year. The effective annual rate is $4\%$, and a known dividend of $\$2$ is paid in six months. Find the parity value of $C_0-P_0$. (Answer: $\$5.92$)
A European call costs $\$15.50$, and a put on the same stock costs $\$0.69$. Both have strike $\$125$ and expire in two years. The effective annual rate is $5\%$, and a known $\$4$ dividend is paid at the end of year 1. Find the stock price implied by parity. (Answer: $\$132.00$)
A European call costs $\$9$, and a put with the same strike and expiration costs $\$13$. The strike is $\$90$, expiration is one year, and the effective annual rate is $5\%$. A known $\$3$ dividend is paid in six months. Find the implied stock price. (Answer: $\$84.64$)
A stock is priced at $\$200$. European options with strike $\$180$ expire in two years. The effective annual rate is $6\%$, and known dividends of $\$5$ and $\$4$ are paid at $t=0.5$ and $t=1.5$. Find the parity value of $C_0-P_0$. (Answer: $\$31.28$)
A stock is priced at $\$96$. A European call and put have strike $\$100$ and expire in 1.5 years. The effective annual rate is $8\%$, and a known $\$7$ dividend is paid at the end of year 1. If the put costs $\$10$, find the parity call price. (Answer: $\$10.42$)

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