One-month implied volatility on an index is 30% and your desk forecasts 22% realised. You sell a delta-hedged at-the-money straddle on Rs 10 crore notional. Estimate the expected profit from the gamma-theta relationship, the result if realised volatility is 35% instead, and what can go wrong between hedges.

Expect about Rs 18 lakh if the month realises 22%, and lose about Rs 12 lakh if it realises 35%. The straddle sells for about Rs 69 lakh, roughly Rs 2.3 lakh per volatility point. Day by day it collects Rs 1.64 lakh of theta and pays gamma on each move, breaking even on a 1.89% day. Between hedges a gap, a trend, a rise in implied volatility and the noise in one month's realised volatility can each overturn the edge.

An insurer that charges premiums for a 3% claim rate and sees 2% of claims arrive earns the gap, not the premium; the premium mostly pays for claims that do happen. A delta-hedged short straddle is the same business: it earns the difference between the volatility it was sold at and the volatility the index actually realises, because the hedging gives back the part of the premium that pays for real moves. At the money with zero rates, each option is worth about 0.4 times spot times volatility times the square root of time, so the straddle is about 0.8 x 30% x sqrt(21/252) = 6.91% of notional, Rs 69.1 lakh. At 22% it would be worth Rs 50.7 lakh. Hedged at the realised volatility, the profit is exactly the difference: Rs 18.4 lakh, which is also the straddle's vegaThe change in an option position’s value for a one-point change in implied volatility. of about Rs 2.3 lakh per volatility point times the 8-point gap.

Each day the short position collects thetaThe value an option loses per day as time passes, which a short option position collects. and pays for the index's move through gammaThe rate at which an option’s delta changes as the underlying moves; a short gamma position loses on moves in either direction.: the day's profit is theta minus half gamma times the move squared. On the first day theta is about Rs 1.64 lakh, and the position breaks even on a move of 30% / sqrt(252) = 1.89%, the daily move implied volatility is charging for. A day at the forecast 22%, a 1.39% move, earns about Rs 0.76 lakh; a day at 35%, a 2.20% move, loses about Rs 0.59 lakh. Twenty-one days of the first kind give roughly Rs 16 lakh, in line with the Rs 18 lakh from the price difference; gamma grows as expiry approaches, which is why the two do not match exactly.

The straddle would have been worth about Rs 80.6 lakh at 35%, so the position loses about Rs 11.5 lakh: five points wrong costs more than half of what eight points right would have earned, and the loss has no ceiling. Before trusting the forecast, ask how precisely one month can even measure volatility: with 21 daily returns, the standard error of realised volatility is about 22% / sqrt(42) = 3.4 points, so a month that realises 26% or 27% is ordinary luck even when the forecast is right. The 8-point edge is a little over two standard errors of that noise, which is a good trade to repeat, not a trade to size as if the Rs 18 lakh were in hand.

First, gaps. The parabola has no floor: a single 5% day costs about Rs 9.9 lakh, roughly the whole expected profit of the month, and an overnight gap cannot be hedged at all. Second, the mark. The desk is paid at expiry, but it is marked every day on implied volatility, and if implied rises from 30% to 35% in a sell-off the position shows a loss of about Rs 12 lakh before realised volatility has had its say, which a risk limit may force it to close at the worst moment. Third, the hedge itself: hedging once a day at the close means a trending day that opens down 1% and closes down 2% is hedged only at the close, and the path between hedges is the gamma loss. Fourth, pin risk in the last days, when gamma near the strike is many times what it was at the start, so the final 2% move costs more than the first. The limitation of the whole estimate is that it treats volatility as a single number for the month; the desk should size the trade to the gap day and to the mark, not to the average.