The escrowed dividend model step by step, the parity check that proves it, and when an American call is worth exercising before the ex-date.
To price a European option on a stock that pays a known cash dividend, subtract the present value of the dividend from the spot price and run Black-Scholes on what is left. For a one-year at-the-money option on a stock at 100, with a 4 per cent rate, 24 per cent volatility and a dividend of 3.00 paid in six months, the adjusted spot is 97.0594 and the call is worth 9.7224 instead of 11.4541, 15.1 per cent less; the put rises from 7.5330 to 8.7420. Ignoring the dividend is the costliest of the dividend errors covered below.
Worked in full in Quantitative Finance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
On the ex-dividend date the share price drops by roughly the dividend. The holder of a call does not receive the dividend, so the part of today's price that will be paid out before expiry never reaches the option. The escrowed dividend model treats the share as two pieces: a riskless claim worth the present value of the dividend, and a risky remainder that follows the usual lognormal process. Only the remainder goes into the formula. Everything else in the standard Black-Scholes calculation is unchanged.
The escrowed model is a practical approximation, not an exact one. Applying one volatility to the risky remainder can misprice options, most visibly when dividends are large relative to the share price, maturities are long, or the option is American. Merton's continuous dividend yield version is the usual alternative, especially for an index that pays many small dividends, and American options on single stocks can instead be priced on a tree that drops the price by the cash dividend on the ex-date.
| Input | Symbol | Value |
|---|---|---|
| Spot price | S | 100 |
| Strike | K | 100 |
| Maturity, years | T | 1 |
| Risk-free rate, continuously compounded | r | 4.00% |
| Volatility | σ | 24.00% |
| Cash dividend | D | 3.00 |
| Time to ex-dividend date, years | tD | 0.5 |
S* = S − D × e−r×tD
d1 = [ln(S*/K) + (r + σ²/2) × T] / (σ × √T), d2 = d1 − σ × √T
Call = S* × N(d1) − K × e−rT × N(d2), Put = K × e−rT × N(−d2) − S* × N(−d1)
Excel, adjusted spot: =S-D*EXP(-r*tD), then the usual call formula with that cell in place of S
Step 1: the present value of the dividend. The discount factor to the ex-date is e−0.04 × 0.5 = 0.9802, so the dividend is worth 3.00 × 0.9802 = 2.9406 today. The adjusted spot is 100 − 2.9406 = 97.0594.
Step 2: d1 and d2. ln(97.0594 / 100) = −0.0298. The drift term is (0.04 + 0.0288) × 1 = 0.0688. So d1 = (−0.0298 + 0.0688) / 0.24 = 0.1623 and d2 = 0.1623 − 0.24 = −0.0777. Without the dividend they were 0.2867 and 0.0467: the dividend lowers the forward price, so the option is less in the money in forward terms.
Step 3: the prices. N(d1) = 0.5645 and N(d2) = 0.4690. The present value of the strike is 96.0789. Call = 97.0594 × 0.5645 − 96.0789 × 0.4690 = 54.7868 − 45.0644 = 9.7224. Put = 96.0789 × 0.5310 − 97.0594 × 0.4355 = 51.0146 − 42.2726 = 8.7420.
Step 4: the parity check. With a dividend, put-call parity becomes C − P = S − PV(D) − K × e−rT. Here 9.7224 − 8.7420 = 0.9805, and 100 − 2.9406 − 96.0789 = 0.9805. If the two sides differ, the dividend has been treated inconsistently somewhere. The put-call parity worked example covers the identity without dividends.
The dividend takes 1.7316 off the call, 15.1 per cent of its value, and adds 1.2090 to the put, 16.0 per cent. The hedge changes too: delta falls from 0.6128 to 0.5645, so a desk short 10,000 calls holds 5,645 shares rather than 6,128. A trader who priced the 10,000 calls without the dividend would pay 114,541 for options worth 97,224, an overpayment of 17,316.
A continuous dividend yield gives exactly the same answer if it is matched to the same present value: q = −ln(97.0594 / 100) = 2.9847 per cent prices the call at 9.7224. The two models differ only when the yield is set some other way, which is where the shortcuts come in.
| Dividend | Adjusted spot | Call | Put | Call delta |
|---|---|---|---|---|
| 0.00 | 100.0000 | 11.4541 | 7.5330 | 0.6128 |
| 1.00 | 99.0198 | 10.8611 | 7.9202 | 0.5970 |
| 2.00 | 98.0396 | 10.2838 | 8.3231 | 0.5809 |
| 3.00 | 97.0594 | 9.7224 | 8.7420 | 0.5645 |
| 4.00 | 96.0792 | 9.1773 | 9.1770 | 0.5478 |
| 5.00 | 95.0990 | 8.6487 | 9.6286 | 0.5308 |
Each unit of dividend costs the call between 0.53 and 0.59, close to its delta, because a dividend is economically a fall in the spot that everyone knows about in advance. At a dividend of 4.00 the at-the-money call and put are worth the same, because the forward price has fallen to the strike.
An American call on a stock with no dividend is never worth exercising early, so the European price is enough. On a dividend payer it may be worth exercising just before the ex-date. Black's approximation prices two European calls and takes the higher: one held to expiry on the adjusted spot, 9.7224, and one that expires just before the ex-date on the full spot, which is worth 7.7316. Holding wins, so the American call is worth about 9.7224 and early exercise does not pay.
A quick screen tells you when to look harder: early exercise can only be optimal if the dividend exceeds the interest earned on the strike between the ex-date and expiry, here 100 × (1 − e−0.04 × 0.5) = 1.9801. A 3.00 dividend passes the screen but the time value left in the option still outweighs it. With a dividend of 8.0, the held call falls to 7.1641, below the 7.7316 of exercising before the ex-date, and early exercise becomes the better choice.
Take the present value of every dividend paid before expiry off the spot, then price as usual: 100 − 2.9406 = 97.0594, and a call of 9.7224 rather than 11.4541. Check the result with the dividend version of parity, and for American calls compare holding with exercising before the ex-date. The same option without the dividend, and the hedging that follows from its delta, is built cell by cell in the free workbooks for the book.
It gives the same European price if the yield is matched to the dividend's present value. A 3.00 dividend in six months on a 100 stock, with a 4 per cent rate, corresponds to a continuous yield of 2.9847 per cent, and both methods price the one-year at-the-money call at 9.7224. Setting the yield at dividend over spot, 3.00 per cent, gives 9.7140 instead.
The present value of the dividend comes off the stock side: C minus P equals S minus PV(D) minus the present value of the strike. In the worked case, 9.7224 minus 8.7420 is 0.9805, which equals 100 minus 2.9406 minus 96.0789. If the identity does not close, the dividend has been treated differently in the call and the put.
Only when the dividend exceeds the interest earned on the strike until expiry and the remaining time value. Black's approximation compares a call held to expiry on the adjusted spot with one exercised just before the ex-date. With a 3.00 dividend holding is worth 9.7224 against 7.7316, so no early exercise; with a dividend of 8.0 holding falls to 7.1641 and exercising wins.
This article is one calculation from Quantitative Finance. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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