Pranhita Capital is short a one-month at-the-money call on a Rs 1,000 stock at 25% volatility. Compare the hedging error from rehedging daily and weekly, and the cost at 0.05% of the traded value per rehedge. Which frequency would you choose?

Hedge daily: the error standard deviation falls from about Rs 12.7 a share weekly to Rs 5.6 daily, while the extra cost is about Rs 0.41 a share. On 50,000 shares that is Rs 6.4 lakh against Rs 2.8 lakh of uncertainty, for Rs 0.21 lakh of extra cost. Error falls and cost rises with the square root of the number of rehedges, so the balance sits where they are equal, which here is well above daily; at ten times the cost it moves down to roughly every day and a half.

Think of steering a car by looking at the road once a second against once every five seconds. Either way you stay roughly on the road, but the second driver wanders further between corrections. A delta hedge set at the start of a day is right only at that instant; as the stock moves, the option's delta changes by its gammaThe rate at which an option delta changes as the stock moves. Here about 0.0055 per rupee per share, so a Rs 16 move shifts delta by about 0.09. and the hedge drifts out of line until the next rebalance, so each interval leaves a small gain or loss that depends on how far the stock moved. These errors average to about zero when the option is fairly priced, but they do not cancel exactly, and their spread is the hedging error. The option here is worth about Rs 28.8 a share, with a delta of 0.51 and a vega of Rs 1.15 per volatility point.

There is a clean approximation, due to Derman and Kamal: the standard deviation of the hedging error is about the square root of pi over 4, times volatility, times the option's vega, divided by the square root of the number of rehedges. With vega of Rs 115.1 per unit of volatility and volatility of 25%, the numerator is Rs 25.5; divided by the square root of 21 it is Rs 5.56 a share for daily hedging, and divided by the square root of 4 it is Rs 12.75 for weekly. On 50,000 shares, one standard deviation is Rs 2.78 lakh daily against Rs 6.37 lakh weekly, on a premium of Rs 14.4 lakh. Weekly hedging means a one-in-six month can wipe out almost half the premium for reasons that have nothing to do with whether 25% was the right volatility.

Each rebalance trades the change in delta. Over one day the stock moves on average about Rs 12.6, and gamma of 0.0055 turns that into about 0.069 shares traded per share of option, worth about Rs 69; at 0.05% that is about Rs 0.035. Over the month daily rehedging costs about Rs 0.73 a share and weekly about Rs 0.32, because a weekly move is bigger but there are fewer of them, so cost grows with the square root of the number of rehedges, not in proportion. Both frequencies also pay about Rs 0.26 to put the first hedge on, which does not affect the choice. On 50,000 shares the gap between daily and weekly is about Rs 0.21 lakh, less than a tenth of the reduction in one standard deviation of error.

Put a price on risk: if one rupee of standard deviation is treated as costing one rupee, the total is cost times the square root of n plus error divided by the square root of n, and that is smallest where the two are equal. At 0.05% the balance is near 160 rehedges in the month, several a day, so daily hedging is the conservative choice and weekly is plainly too few; at 0.5%, a thin stock with wide spreads, the balance falls to about 16, roughly every day and a half. The answer is therefore daily for Kodachadri, and the reason is the cost, not a rule of thumb. In practice desks use a band rather than a clock: rebalance when the hedge has drifted more than a set number of shares, which spends transaction cost only when the stock has actually moved, and widen the band as expiry approaches and gamma rises.

State the limits. The error formula assumes the volatility the stock actually delivers equals the 25% used to price and hedge, and treats gamma as roughly constant; near expiry an at-the-money option's gamma rises sharply, so the last few days carry more of the error than the formula shows, and a simulation of this option lands within about a tenth of these figures. The trade-off also depends on how much the desk is willing to pay to reduce variance, which is a choice about risk appetite rather than arithmetic.