Dividends and European Call Values

Why Dividends Matter to a European Call

A European call gives its owner the right, but not the obligation, to buy a share for the exercise price at expiration. A shareholder, by contrast, may receive cash dividends before expiration. When a share goes ex-dividend, its price generally falls by approximately the amount of the cash distribution, all else equal. The call holder does not receive that dividend, so an expected distribution tends to reduce the value of the call.

A practical estimate for known cash dividends is to subtract their present value from the current stock price and use the result in the Black–Scholes call formula. This is an approximation for discrete cash dividends, not an exact universal formula. Its assumptions and limitations matter: the dividend dates and amounts must be known, and the adjusted-price method is less reliable when dividends are large relative to the stock price.

Definition: For known cash dividends $D_j$ payable at times $t_j$ before expiration $T$, the present value of the dividends under a constant force of interest $\delta$ is $$ \operatorname{PV}(\text{dividends})=\sum_j D_j e^{-\delta t_j}. $$ The adjusted stock price is $S_0^{*}=S_0-\operatorname{PV}(\text{dividends})$. It is the estimated current value of the stock component after allowing for the cash distributions.

The Adjusted-Stock-Price Method

Use the method for a European call with current stock price $S_0$, exercise price $K$, and time to expiration $T$. Assume the cash dividends $D_j$ are fixed and known, the force of interest $\delta$ is constant, and volatility $\sigma$ is constant. The Black–Scholes model assumes that, apart from dividends, the stock follows a lognormal process. The adjusted-stock-price method treats the present value of the known dividends as a separate cash component and applies the non-dividend-paying Black–Scholes formula to the remaining stock value.

The method also assumes no taxes, transaction costs, or other market frictions. In particular, it does not model every detail of the stock-price jump at each discrete ex-dividend date. Therefore, its result is an estimate. It is not an exact closed-form valuation for every stock with discrete dividends.

d_1^{*}= \frac{\ln(S_0^{*}/K)+(\delta+\sigma^2/2)T}{\sigma\sqrt{T}}, \qquad d_2^{*}=d_1^{*}-\sigma\sqrt{T}

C_0\approx S_0^{*}\Phi(d_1^{*})-Ke^{-\delta T}\Phi(d_2^{*}),

Here $C_0$ is the estimated call value and $\Phi(z)$ is the cumulative distribution function of a standard normal random variable. Use the dividend-adjusted price in both $d_1^{*}$ and the first term of the call formula. The exercise price is discounted to expiration using the same constant force of interest.

Why Subtract the Present Value of Dividends?

The adjustment has a simple economic rationale. The current stock price reflects the value of both the future stock and the cash distributions expected before expiration. A dollar of dividend payable later is worth less today than a dollar payable now. Under the method, the present value of those cash distributions is separated from the stock component, leaving $S_0^{*}$ as the starting value used to estimate the call’s exposure to the stock at expiration.

Discount each dividend.
A cash dividend $D_j$ paid at time $t_j$ has present value $D_j e^{-\delta t_j}$ when the force of interest is constant.

Remove the dividend component.
Subtract the sum of the dividend present values from $S_0$, giving $S_0^{*}$.

Recalculate the call inputs.
Use $S_0^{*}$ to calculate $d_1^{*}$ and $d_2^{*}$, then evaluate the adjusted Black–Scholes call formula.

The formula also explains the direction of the estimated effect. With the other inputs fixed, the Black–Scholes call value increases with the stock price: its sensitivity to the stock price, or call delta, is positive. Subtracting a positive dividend present value lowers the adjusted stock price, so it lowers the estimated call value. The reduction is generally not equal to the present value of the dividends dollar for dollar, because the call’s value changes according to its sensitivity to the underlying price.

Worked Valuations

Worked Example: One Dividend and the Estimated Call Reduction

A European call has $S_0=\$100$, $K=\$100$, and $T=1$ year. A known cash dividend of $\$5$ will be paid in six months. The force of interest is $\delta=0$, and volatility is $\sigma=0.20$ per year. Estimate the call value with the dividend and compare it with the no-dividend estimate. Use standard normal probabilities rounded to four decimals, and round final dollar values to cents.

Method. Discount the dividend, adjust the stock price, then apply the Black–Scholes call formula. Since $\delta=0$, the dividend’s present value is $\$5e^0=\$5$, so

S_0^{*}=100-5=\$95.

For the adjusted call,

d_1^{*}=\frac{\ln(95/100)+(0+0.20^2/2)(1)}{0.20} =\frac{-0.0513+0.0200}{0.20} \approx -0.1565, \qquad d_2^{*}\approx-0.3565.

Using $\Phi(-0.1565)\approx0.4378$, $\Phi(-0.3565)\approx0.3607$, and $Ke^{-\delta T}=\$100$,

C_0\approx95(0.4378)-100(0.3607) =41.5910-36.0700 =\$5.5210 \approx \$5.52.

For comparison, without a dividend the spot price remains $\$100$. Then $d_1=0.1000$, $d_2=-0.1000$, and the corresponding probabilities are approximately $0.5398$ and $0.4602$. Thus,

C_{\text{no div}}\approx100(0.5398)-100(0.4602) =53.9800-46.0200 =\$7.9600 \approx \$7.96.

The estimated reduction is $\$7.9600-\$5.5210=\$2.4390$, or $\$2.44$ to the nearest cent. It is smaller than the $\$5$ dividend because the call value does not fall dollar for dollar with the stock price.

Worked Example: Two Dividends Before Expiration

A European call has $S_0=\$120$, $K=\$115$, and $T=1$ year. Known dividends of $\$4$ and $\$6$ are payable in three months and nine months, respectively. Let $\delta=0.04$ and $\sigma=0.20$. Estimate the call value and compare it with the value estimated by ignoring dividends. Use normal probabilities rounded to four decimals and round final dollar values to cents.

Method. Discount each dividend to time zero. The first present value is $4e^{-0.04(0.25)}=4e^{-0.01}\approx\$3.9602$; the second is $6e^{-0.04(0.75)}=6e^{-0.03}\approx\$5.8227$. Therefore,

\operatorname{PV}(\text{dividends})\approx3.9602+5.8227=\$9.7829, \qquad S_0^{*}\approx120-9.7829=\$110.2171.

The adjusted parameters are

d_1^{*}= \frac{\ln(110.2171/115)+(0.04+0.20^2/2)(1)}{0.20} \approx0.0876, \qquad d_2^{*}\approx-0.1124.

Using $\Phi(0.0876)\approx0.5349$, $\Phi(-0.1124)\approx0.4552$, and $115e^{-0.04}\approx\$110.4908$,

C_0\approx110.2171(0.5349)-110.4908(0.4552) =58.9551-50.2954 =\$8.6597 \approx \$8.66.

Ignoring the dividends gives $d_1\approx0.5128$, $d_2\approx0.3128$, with probabilities $0.6959$ and $0.6228$. Thus,

C_{\text{no div}}\approx120(0.6959)-110.4908(0.6228) =83.5080-68.8137 =\$14.6943 \approx \$14.69.

The adjusted estimate is lower by approximately $\$14.6943-\$8.6597=\$6.0346$, or $\$6.03$ to the nearest cent. The two dividends must be discounted at their separate payment times; treating both as if they were paid immediately would overstate their combined present value.

Worked Example: A Dividend Before a Longer Expiration

A European call has $S_0=\$80$, $K=\$75$, and $T=2$ years. A known $\$3$ cash dividend is payable in one year. Let $\delta=0.03$ and $\sigma=0.25$. Estimate the call value using the adjusted-stock-price method. Use normal probabilities rounded to four decimals and round the final value to cents.

Method. The dividend present value is $3e^{-0.03(1)}\approx\$2.9113$, so $S_0^{*}\approx80-2.9113=\$77.0887$. The call has two years to expiration, but the dividend is discounted for only one year; these are different time periods.

d_1^{*}= \frac{\ln(77.0887/75)+(0.03+0.25^2/2)(2)} {0.25\sqrt{2}} \approx0.4242, \qquad d_2^{*}=0.4242-0.25\sqrt{2}\approx0.0706.

Using $\Phi(0.4242)\approx0.6643$, $\Phi(0.0706)\approx0.5281$, and $75e^{-0.03(2)}\approx\$70.6323$,

C_0\approx77.0887(0.6643)-70.6323(0.5281) =51.2100-37.3009 =\$13.9091 \approx \$13.91.

This is an estimate under the adjusted-stock-price approximation. It uses the dividend’s payment time for discounting the dividend and the option’s expiration time for discounting the exercise price.

Assumptions, Limits, and Common Traps

The dividend adjustment is most useful when dividend amounts and dates are known and the resulting adjustment is modest. A discrete cash dividend changes the stock price at its ex-dividend date, while the adjusted-stock-price method represents the effect through a single time-zero subtraction. A large dividend can make this representation a poor approximation, and the formula should not be presented as an exact result in that case.

The method also depends on the option being European. It values exercise only at expiration; it does not analyze whether early exercise is available or desirable. Do not transfer this valuation procedure to an American call without separately considering its exercise rights and the relevant model.

A common error is to subtract the stated dividend amount rather than its present value. For example, if a $\$6$ dividend is paid in nine months and $\delta=0.04$, the correct adjustment is $6e^{-0.04(0.75)}\approx\$5.8227$, not $\$6$. Subtracting $\$6$ treats a future payment as though it were paid today and lowers the estimated adjusted stock price too much.

Another trap is to adjust the stock price but leave the rest of the calculation inconsistent. In the adjusted formula, recompute $d_1^{*}$ and $d_2^{*}$ from $S_0^{*}$, retain the same exercise price and expiration, and discount the exercise price using the stated interest assumption. Do not subtract the dividend present value again from the final call value; that would count the dividend effect twice.

Key takeaway: For known cash dividends, discount each payment to time zero, subtract their total present value from the current stock price, and use the adjusted price in the European Black–Scholes call formula. This estimates the effect under stated assumptions; it is not an exact general formula for discrete dividends. This treatment is aligned with the SOA VEE Accounting and Finance syllabus.

Check Your Understanding

For each problem, use the adjusted-stock-price method when a call value is requested. Round final dollar answers to cents. Any stated normal probabilities are rounded to four decimals.

A share is priced at $\$90$. It will pay dividends of $\$2$ in six months and $\$3$ in 18 months. With $\delta=0.04$, calculate the total dividend present value and adjusted stock price. (Answer: dividend present value $\$4.79$; adjusted stock price $\$85.21$)
A European call has $S_0=\$100$, $K=\$100$, $T=1$, $\delta=0$, and $\sigma=0.20$. A known $\$4$ dividend is paid in six months. For the adjusted calculation, use $\Phi(d_1^{*})=0.4585$ and $\Phi(d_2^{*})=0.3805$. Find the adjusted call value. (Answer: $\$5.97$)
A European call has $S_0=\$150$, $K=\$145$, $T=1.5$, $\delta=0.05$, and $\sigma=0.30$. Dividends of $\$5$ are paid at 0.5 and 1 year. For the adjusted calculation, use $\Phi(d_1^{*})=0.6177$ and $\Phi(d_2^{*})=0.4729$. Also use $145e^{-0.05(1.5)}=\$134.5227$. Find the adjusted call value. (Answer: $\$23.09$)
A share will pay one $\$8$ dividend. Compare its present value if paid in three months with its present value if paid in nine months, using $\delta=0.08$. State which timing produces the lower adjusted stock price for the same $S_0$. (Answer: present values $\$7.84$ and $\$7.53$, respectively; the three-month payment produces the lower adjusted stock price.)
A European call has $S_0=\$110$, $K=\$105$, $T=1$, $\delta=0.03$, and $\sigma=0.25$. Dividends of $\$3$ and $\$2$ are paid at 0.25 and 0.75 years. For the adjusted calculation, use $\Phi(d_1^{*})=0.5978$, $\Phi(d_2^{*})=0.4990$, and $105e^{-0.03}=\$101.8968$. Find the adjusted call value. (Answer: $\$11.96$)

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