A client buys Rs 25 crore notional of three-month 105% calls on a stock at 400. Mid volatility is 28% and the desk charges 1.5 points. Price the charge, set up the hedge, and work the P&L if realised volatility turns out to be 32%.

The charge is worth about Rs 7.4 lakh, the hedge is about 272,177 shares bought, and at 32% realised the desk loses about Rs 12.4 lakh. At 28% the call is worth Rs 16.69 a share; at 29.5% the client pays Rs 17.87, Rs 1.12 crore in all. The delta at mid is 0.44, so the desk buys that share of the notional. Realising 32% means the stock moved more than the volatility sold, and the hedging losses exceed the premium collected.

Price the call twice. At the 28% mid it is worth Rs 16.69 a share; at 29.5% it is Rs 17.87. The difference, Rs 1.18 a share, times 625,000 shares, is the desk's charge of about Rs 7.39 lakh, and it is roughly the option's vegaHow much an option price changes for a one point change in implied volatility. of Rs 0.79 a share times 1.5 points. It works like a dealer's margin on a used car: the screen price is the mid, the client pays a little above it, and the gap is what pays the dealer for holding the car until it is sold on. On the desk, the car is the risk of hedging the option for three months.

A 105% call, three months out, moves about 44 paise for each rupee of the stock at today's price: the delta from the mid model is 0.44. The desk is short the call, so it is short that delta, and it buys 0.44 times 625,000, about 272,177 shares, roughly Rs 10.9 crore of stock. The hedge is right only at today's price: as the stock rises the delta rises and the desk must buy more, and as it falls the desk sells. That buy-high, sell-low rebalancing is the cost of being short gamma, and the premium is what pays for it.

The hedging cost depends on how much the stock actually moves. If it moves exactly as 29.5% volatility implies, rebalancing losses use up the whole premium and the desk breaks even. If it realises 32%, the rebalancing losses are as if the desk had sold the option for its value at 32%, Rs 19.84 a share, so the desk is down Rs 1.98 a share, about Rs 12.4 lakh on the trade. Against mid the volatility miss is 4 points; the 1.5 point charge absorbs part of it and leaves 2.5 points of loss. The approximation assumes smooth daily moves and constant volatility; a single gap day would make the loss larger.

Close with what the desk does about it. It does not have to wait three months to find out: it can buy back some of the volatility in the listed market, or offset the gamma with another client's trade on the other side. A charge is a cushion against being wrong on volatility, not a profit, until the option has expired.