The closed form in two terms, the calendar-day and trading-day conventions, and why an at-the-money option's decay is anything but linear.
Compute Black-Scholes theta, which is an annual rate, and divide it by 365 for a calendar day or by 252 for a trading day. For an illustrative one-year at-the-money call at 24 per cent volatility and a 4 per cent rate, theta is -6.5877 a year, or -0.018048 a day: 180.48 a day on 10,000 options. The number is not constant. With 10 days left the same at-the-money option loses 0.0847 a day, 4.7 times as fast.
Worked in full in Quantitative Finance by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
The option is Kestrel 100, the illustrative European call that runs through Quantitative Finance. No dividends.
| Input | Value |
|---|---|
| Spot and strike | 100.00 |
| Rate, continuously compounded | 4.0% |
| Volatility | 24.0% |
| Time to expiry | 1 year |
| Price | 11.454068 |
| d1 / d2 | 0.2867 / 0.0467 |
| Position | 10,000 calls |
Theta is the derivative of the price with respect to calendar time, with everything else held fixed. For a call it has two parts: the decay of the option's time value, and the interest on the strike that will be paid later.
Θcall = − S × n(d1) × σ / (2√T) − r × K × e−rT × N(d2)
Decay term: − 100 × 0.382882 × 0.24 / (2 × 1) = − 9.1892 / 2 = −4.5946
Carry term: − 0.04 × 96.0789 × 0.518611 = −1.9931
Annual theta = −4.5946 − 1.9931 = −6.5877
Excel: =(-S*NORM.S.DIST(d1,FALSE)*sig/(2*SQRT(T)) - r*K*EXP(-r*T)*NORM.S.DIST(d2,TRUE))/365
n(d1) is the normal density, not the cumulative probability: in Excel, NORM.S.DIST(d1,FALSE). Using TRUE there is the most common spreadsheet error with theta, and it produces a plausible-looking wrong number.
Per calendar day = −6.5877 / 365 = −0.018048, or −180.48 on 10,000 calls
Per trading day = −6.5877 / 252 = −0.026142, or −261.42 on the lot
Check it by repricing. With one calendar day less, the call is worth 11.436010, a fall of 0.018058. The analytic and repriced figures differ in the fifth decimal because theta is a slope and the repricing is a one-day step.
The two conventions answer different questions. Calendar-day theta matches how the model's clock runs, so Friday to Monday costs three days, 541.45 on the lot. Trading-day theta spreads the year's decay over the days you can trade, which some desks prefer for daily P&L. Neither is wrong, but a risk report that mixes them overstates or understates decay by 45 per cent.
The decay term alone, 0.012588 a day, is what a delta-hedged position actually pays: the carry term is offset by the interest on the hedge. That is the figure to set against gamma, worked in the gamma-theta break-even move.
Dividing the price by the days left gives 0.031381 a day for the year, nearly twice the true figure for today. The decay is back-loaded.
| Days left | Strike 90 price | Theta/day | Strike 100 price | Theta/day | Strike 110 price | Theta/day |
|---|---|---|---|---|---|---|
| 365 | 17.2443 | -0.0166 | 11.4541 | -0.0180 | 7.2382 | -0.0172 |
| 180 | 13.8585 | -0.0205 | 7.6722 | -0.0238 | 3.7420 | -0.0210 |
| 90 | 11.8802 | -0.0236 | 5.2352 | -0.0317 | 1.7248 | -0.0239 |
| 30 | 10.4544 | -0.0217 | 2.9072 | -0.0511 | 0.3026 | -0.0206 |
| 10 | 10.1028 | -0.0119 | 1.6392 | -0.0847 | 0.0123 | -0.0051 |
| 5 | 10.0494 | -0.0099 | 1.1479 | -0.1175 | 0.0003 | -0.0004 |
| 1 | 10.0099 | -0.0099 | 0.5066 | -0.2560 | 0.0000 | 0.0000 |
Three patterns sit in the table. At the money, theta accelerates all the way to expiry: 2.8 times the one-year rate at 30 days, 4.7 times at 10. Time value there is roughly proportional to the square root of time left, so of the 2.9072 the option is worth at 30 days, 1.6392 is still there at 10 days, and 56.4 per cent of the value goes in the last third of the time. Out of the money, theta peaks and then fades, because there is almost nothing left to lose: the 110 strike is worth 0.0123 at 10 days. Deep in the money, theta settles on the carry term, r × K a year: 0.04 × 90 / 365 = 0.0099 a day for the 90 strike, because the option has become a forward on the stock.
| Volatility | Theta per day |
|---|---|
| 16% | -0.014256 |
| 24% | -0.018048 |
| 32% | -0.021909 |
Higher volatility means more time value to lose, so the decay term grows roughly in proportion. Theta is the rent for the option's convexity: a book long gamma pays it daily, and a book short gamma collects it. That is the trade, not a free income.
All five Greeks, with a finite-difference estimate beside each, are in the free workbooks for this book. The price itself is built step by step in the Black-Scholes worked example.
Most systems divide the annual theta by 365, so a weekend shows as three days of decay. The illustrative one-year call has annual theta of -6.5877: -0.018048 per calendar day, or -0.026142 if divided by 252 trading days. On 10,000 options that is -180.48 against -261.42, so always check which convention a risk report uses before comparing books.
For an at-the-money option the time value is roughly proportional to the square root of time left, so the slope steepens as time runs out. The illustrative call loses 0.018048 a day with a year left, 0.0511 with 30 days and 0.0847 with 10. Of the 2.9072 it is worth at 30 days, 1.6392 is still there at 10 days: 56.4 per cent of the value goes in the last third of the time.
Often, because the interest term flips sign. For the illustrative put on the same terms the decay term is the same -4.5946 a year, but the carry term is +1.8501, so put theta is -2.7445 a year or -0.007519 a day, against -0.018048 for the call. A deep in-the-money European put can even have positive theta.
Theta and the other four Greeks, each with a finite-difference check, are in the chapter 8 workbook of 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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