Rebalancing less often does not drain a hedged option book; it widens the range of results the book can deliver.
No. Across many paths the mean profit of a fairly priced, correctly hedged short option is zero at every frequency tested — 0.0092 at twelve rebalances a year, 0.0004 at 252, on a premium of 11.454068. What monthly hedging costs is certainty. One standard deviation of outcome rises from 0.5265 at daily rebalancing to 2.3031 at monthly, which is 20.1 per cent of the premium against 4.6 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 case is a one-year call: spot 100.00, strike 100.00, a risk-free rate of 4.0 per cent, sold at an implied volatility of 24 per cent. The closed form prices it at 11.454068, and the lot of 10,000 contracts is sold for 114,540.68. To measure what rebalancing frequency is worth on its own, everything else is removed: the realised volatility of the underlying is set exactly equal to the 24 per cent that was sold, so the volatility gap is zero and the expected profit is zero by construction. Only the number of hedge adjustments across the year changes.
| Frequency | Rebalances | Mean | Standard deviation | Share of premium |
|---|---|---|---|---|
| Monthly | 12 | 0.0092 | 2.3031 | 20.1 per cent |
| Weekly | 52 | 0.0044 | 1.1512 | 10.1 per cent |
| Daily | 252 | 0.0004 | 0.5265 | 4.6 per cent |
| Four times a day | 1008 | 0.0018 | 0.2632 | 2.3 per cent |
The mean column is the control and it behaves: 0.0092, 0.0044, 0.0004, 0.0018, on a premium of 11.454068. That is zero at every frequency, to within the noise of the experiment. Coarse hedging is not a slow leak. The errors it makes are symmetric and they cancel over many paths.
What hedging less often costs is not money but certainty. At monthly rebalancing the option was sold at a fair price, the volatility forecast was exactly right, the hedge was computed with the correct model and executed without slippage — and one standard deviation of outcome is still a fifth of the premium, 23,031.00 on the lot. A two-standard-deviation year, which is not a rare event, moves 40 per cent of the premium in either direction.
Set the daily figure beside the quantity desks spend their meetings on instead. The full spread between three independently written pricing engines on this same option is 0.029701. The dispersion left over by hedging once a business day is 0.5265 — nearly eighteen times as large. A desk that has argued for a month about which model to run, and has never asked how often the hedge is adjusted, has its attention in the wrong place by an order of magnitude.
Each rebalancing interval contributes an error, and the errors are roughly independent, so dispersion ought to scale with the square root of the number of intervals: quadruple the frequency, halve the dispersion. The table permits a check rather than an assumption.
| Step in frequency | Reduction achieved | Rule predicts |
|---|---|---|
| Monthly to weekly | 2.0006 | 2.0817 |
| Weekly to daily | 2.1865 | 2.2014 |
| Daily to four times a day | 2.0004 | 2.0000 |
Across the whole range, from 12 rebalances to 252, the dispersion falls by a factor of 4.37 where the rule promises 4.58. The rule over-promises by about five per cent of the reduction it advertises, and the shortfall should be reported that way rather than rounded into agreement. It is also not spread evenly. At the fine end the rule is essentially exact: going from 252 to 1008 halves the dispersion to four significant figures. The entire failure sits in the single step from monthly to weekly.
The reason is gamma. The rule assumes each interval contributes an equal, independent error, and equal contributions require gamma to be the same on every interval. It is nothing of the sort. Gamma on this option is 0.015953 at inception, largest near the strike and near expiry, negligible once the underlying has wandered far away. Over a short interval it is approximately constant, the assumption holds and the rule works. Over a month the underlying can travel a long way, and the position can pass straight through the region where gamma is largest without any adjustment happening at all. The approximation is worst precisely on the paths that dominate the answer, which means any extrapolation of hedging risk from a fine frequency down to a coarse one will misstate it, and not in a predictably conservative direction.
Hedging more often shrinks the dispersion and costs money, and the two effects run in opposite directions, so an optimum exists and is not at either extreme. Calibrate the constant straight from the table: 0.5265 multiplied by the square root of 252 gives 8.3579, which reproduces 0.2632 at 1008 rebalances. Minimise the sum of what is spent on adjustments and the price the desk puts on the dispersion that remains, and the optimal count rises with the two-thirds power of that ratio. Two things follow from the exponent. The optimum is insensitive — halving the all-in cost of an adjustment raises the optimal frequency by a factor of only 1.5874, and a tenfold reduction by 4.6416 — and the answer belongs to the desk rather than to the model, because two desks with different spreads and different appetites have two different correct frequencies, neither of them wrong.
Then report it. The residue belongs beside the premium in the same breath: sold 114,540.68 of calls, hedged daily, one standard deviation of hedging residue 5,265.00. That second figure is not a stress scenario or an adverse case; it is what the position does in the ordinary course of events with every forecast correct, and anyone signing for the book should see it in the same size of type as the premium. It also pre-empts a standing reporting failure. When a hedged book prints a result far from zero in a quiet quarter, the instinct is to look for a broken model, a bad mark or poor execution. A book hedged weekly carries a dispersion of 10.1 per cent of premium, and a result two standard deviations from the mean will arrive about as often as chance says it should.
Read down the standard deviation column once more: 2.3031, 1.1512, 0.5265, 0.2632. The numbers fall, they will keep falling if the experiment is extended, and they never arrive at zero. There is no rebalancing frequency at which a sold option becomes a riskless annuity. The choice is not whether to carry the residue. It is how much of it to buy back, and at what price.
There is no universal answer. Dispersion falls as a constant divided by the square root of the number of rebalances, the constant being 8.3579 on this option, while trading cost rises in proportion to that number. The optimum is finite and insensitive: halving the all-in cost of an adjustment raises the optimal frequency by only 1.5874.
Only at fine frequencies. From 252 rebalances to 1008 the dispersion halves to four significant figures, 2.0004 achieved against 2.0000 predicted. Across the whole range, from 12 to 252, it falls by 4.37 where the rule promises 4.58. The failure concentrates at the coarse end, because gamma is far from constant over a month.
Because delta hedging removes the first-order exposure only at the moment of each adjustment. Between adjustments the position carries an unhedged exposure that grows with gamma and with the size of the move. At daily rebalancing that residue has a standard deviation of 0.5265, nearly eighteen times the 0.029701 separating three pricing engines.
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.
Get the book on Amazon →Free companion files
Also on Amazon UK · Amazon Germany · Amazon France · Amazon Canada
Reading guide: corporate finance, valuation and markets → · All 1706 articles →
If this book helped — or didn’t — a few lines on Amazon are worth more than they look: they are what the next reader goes on. Write a review. The workbook stays free either way.