A delta-hedged option pays for its time decay with the squared daily move. The break-even is one formula, and which theta goes into it decides whether it agrees with the implied volatility.
Take the square root of two times the daily theta divided by gamma, with theta measured net of the interest on the delta hedge. On a one-year at-the-money call at 24 per cent volatility (gamma 0.015953, theta −0.018048 a day), the break-even daily move is 1.2562 on a spot of 100. Using raw theta gives 1.5042, which overstates the move the option needs by 19.7 per cent.
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 call in Quantitative Finance. Its holder is long the call and short delta in shares, rebalanced daily, so the position has no exposure to direction. It earns when the stock moves and pays theta when it does not. All figures are illustrative.
| Input or output | Value |
|---|---|
| Spot and strike | 100.00 |
| Rate, continuous | 4.0% |
| Expiry, years | 1 |
| Implied volatility | 24.0% |
| Price | 11.454068 |
| Delta | 0.612816 |
| Gamma | 0.015953 |
| Theta per calendar day | −0.018048 |
Over one day, a delta-hedged option gains half of gamma times the squared move, and loses one day of theta. Set the two equal and solve for the move.
½ × Γ × (ΔS)² = Θday
Break-even move = √(2 × Θday / Γ)
Raw theta: √(2 × 0.018048 / 0.015953) = 1.5042
Excel: =SQRT(2*ABS(ThetaDay)/Gamma)
That is the textbook answer, and it is slightly wrong for a hedged book, because the hedge has a cash leg. Short 0.612816 shares at 100 raises 61.2816; the call cost 11.4541; the remaining 49.8276 sits on deposit and earns 4 per cent. That is 0.005461 a day, and it offsets part of the decay.
Net daily cost = |Θday| − r × (Δ × S − C) / 365 = 0.018048 − 0.005461 = 0.012588
Break-even net of financing = √(2 × 0.012588 / 0.015953) = 1.2562
The net figure is not a coincidence. The Black-Scholes equation says that half of σ²S²Γ equals theta net of financing. Substitute into the break-even and gamma cancels:
Break-even move = σ × S / √365 = 0.24 × 100 / 19.1050 = 1.2562
So the hedged option breaks even when the stock moves by one implied standard deviation a day, 1.2562 per cent of spot. The raw-theta break-even of 1.5042, put back into the same formula, corresponds to a volatility of 28.74 per cent. A trader reading raw theta would conclude the option needs 28.74 per cent of realised volatility to pay for itself, when 24 per cent is enough.
| Move in the stock | Gamma gain | Net P&L per option | Net P&L on the lot |
|---|---|---|---|
| 0.00 | 0.000000 | −0.012588 | −126 |
| 0.50 | 0.001994 | −0.010594 | −106 |
| 1.00 | 0.007977 | −0.004611 | −46 |
| 1.2562 | 0.012588 | 0.000000 | 0 |
| 1.5042 | 0.018048 | 0.005460 | 55 |
| 2.00 | 0.031907 | 0.019319 | 193 |
| 2.50 | 0.049854 | 0.037267 | 373 |
The gain is quadratic and the cost is fixed, so the position loses a little on most quiet days and makes it back on a few large ones. A move of 2.50, about twice the break-even, earns nearly three times what a flat day costs.
Gamma falls and theta rises as implied volatility rises, and the break-even moves in proportion to volatility.
| Implied volatility | Gamma | Theta per day | Break-even, raw theta | Break-even, net |
|---|---|---|---|---|
| 16% | 0.023613 | −0.014256 | 1.0989 | 0.8375 |
| 20% | 0.019069 | −0.016133 | 1.3008 | 1.0468 |
| 24% | 0.015953 | −0.018048 | 1.5042 | 1.2562 |
| 28% | 0.013689 | −0.019978 | 1.7085 | 1.4656 |
| 32% | 0.011971 | −0.021909 | 1.9132 | 1.6750 |
The net column is always σ × 100 / √365; the raw column always overshoots it. On today's gamma, a realised volatility of 28 per cent earns the lot about 45.46 a day on average, and 20 per cent loses about 38.46. Those are first-order figures: gamma and theta move as the stock and the calendar move, and discrete rebalancing adds noise around them.
The break-even is a statement about expected P&L, not about any single day. How large the noise around it is depends on how often the hedge is rebalanced, which is the subject of does hedging monthly instead of daily lose money.
Two errors account for most wrong break-evens. The first is using raw theta for a delta-hedged position, which ignores that the hedge itself earns or pays carry; here it inflates the answer from 1.2562 to 1.5042. The second is mixing day counts. Theta in most systems is per calendar day, and then the matching move is σS/√365. If the desk thinks in trading days the break-even is σS/√252, or 1.5119; applying a calendar-day theta to a trading-day move, or the reverse, misstates the position by about a fifth. Weekends are the practical test: a calendar-day theta charges three days of decay over a weekend in which the stock moves, on average, by less than three days' worth.
The implied volatility the break-even depends on is solved in how to calculate implied volatility with Newton's method.
All five Greeks, each with a finite-difference check, and the break-even against both thetas are live in the free workbook for this case.
The daily move in the underlying at which a delta-hedged option's gamma gain, half of gamma times the move squared, exactly pays for one day of time decay. On a one-year at-the-money call with gamma 0.015953 and net daily decay of 0.012588 it is 1.2562, or 1.26 per cent of a spot of 100.
Because the Black-Scholes equation sets half of sigma squared, S squared and gamma equal to theta net of financing. Solve for the move and the gamma and theta cancel out, leaving sigma times S over the square root of the day count: 24 per cent of 100 over 19.1050 is 1.2562.
Use the same convention for theta and the move. With theta per calendar day the break-even is the volatility over the square root of 365, 1.2562 here; with trading days it is over the root of 252, 1.5119. Mixing a calendar-day theta with a trading-day move misreads the position by about a fifth.
The Greeks and the break-even move are worked in the sensitivities workbook for chapter 8 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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