Case 047Volatility tradingCore
Painganga Capital sold a one-month at-the-money straddle on Torna Logistics at 22% implied volatility, vega Rs 2 lakh per vol point, delta hedged daily. Realised volatility was 18%. Estimate the P&L, and describe a path that would have made the trade lose even so.
1The situation
Painganga Capital, a volatility fund, thought one-month options on Torna Logistics were expensive at 22% implied volatility when the stock had been moving at nearer 18%. It sold an at-the-money straddle, the 500 call and the 500 put, with Torna at Rs 500, on about 1.74 lakh shares, so that the position's vega was Rs 2 lakh per volatility point. It collected about Rs 44 lakh of premium and delta hedged at every close.
Over the month Torna realised exactly 18% volatility, as the fund expected. The risk manager asks for a first estimate of the P&L, and then for an example of a month with the same 18% that would have lost money.
2Your task
Estimate the P&L from vega and the volatility gap, refine it, and construct a path with 18% realised volatility on which the hedged short straddle loses.
Quick check
Two months each realise exactly 18% volatility. Could the delta-hedged short straddle make money in one and lose in the other?
Worked solution
Try it on paper, then open one step at a time.
30-second answerThe answer to give first
A first estimate is vega times the gap, Rs 2 lakh times 4 points, about Rs 8 lakh; the variance version gives Rs 7.3 lakh. But the result depends on the path. Even daily moves of 1.13% around the strike make about Rs 12.4 lakh; twenty quiet days and one 4.4% move on the last day, the same 18% overall, lose Rs 1.2 lakh, because that one move hits when gamma is at its largest.
Step 1What is the first estimate?
A delta-hedged short option earns when the stock moves less than the option price assumed, and loses when it moves more. The quick estimate is vega times the volatility gap: Rs 2 lakh times 22 less 18, Rs 8 lakh. A slightly better one works in variance, because that is what the hedging P&L actually accumulates: vega times 22 squared less 18 squared, over twice 22, is Rs 7.27 lakh. Either way the answer is a few lakh on Rs 44 lakh of premium, which is the right order of magnitude for a four-point edge over one month.
Step 2What does each day actually contribute?
Think of a shopkeeper who sells umbrellas cheaply on a promise to buy them back after each shower. Light drizzle all month costs her little; one cloudburst at the wrong moment, when every umbrella is out, can cost her the month. Each day the hedged short straddle earns its theta and pays a half times gamma times the day's move squared, so a day breaks even when the stock moves about 1.39%, Rs 6.93 on Rs 500, the implied volatility scaled to one day. Smaller moves earn, larger moves lose. The catch is the weight: gamma and theta are both largest when the stock sits at the strike and expiry is close, so a big move there counts for far more than the same move early in the month or away from the strike.
Step 3Which path loses, and why?
Path one moves 1.13% a day, alternately up and down, so it stays at the strike, realises exactly 18% and, revalued day by day, makes Rs 12.4 lakh, more than the estimate because it sits where gamma and theta are richest all month. Path two moves 0.6% a day for twenty days, collecting Rs 28.1 lakh of near-free theta, then jumps 4.45% on the last day, with the stock at the strike and gamma at its peak, and gives back Rs 29.3 lakh in one session, ending Rs 1.2 lakh down. Its realised volatility is also exactly 18%. The vega estimate is an average that assumes the moves are spread evenly through gamma; the P&L is the sum of each day's move weighted by that day's gamma, and the two differ whenever the big days are not typical.
Close with what the risk manager should take from it. The volatility view was right in both months, and the trade's average edge is still a few lakh, but the spread of outcomes around that average is wide, and it is widest for short-dated options near expiry. Funds that sell short-dated straddles control this with position size and by buying back the options in the last few days, giving up the remaining theta to avoid the peak of gamma. State the limits: the paths are constructed to make the point, the hedge is assumed to trade at the close with no cost, and implied volatility is held at 22% throughout; a rise in implied volatility mid-month would add a mark-to-market loss even on path one.
Where candidates lose it
The common loss is answering vega times four points, Rs 8 lakh, and stopping, as though a correct volatility forecast locked in the profit. The estimate is an average over paths, and the question asked for a path that breaks it.
The second is offering a path with higher realised volatility as the losing example. The question fixes realised at 18%. The interviewer wants the gamma-weighting argument: the same total of squared moves, concentrated where gamma is largest, near the strike close to expiry, turns a winner into a loser.
What the interviewer asks next
- How would you construct an 18% path that makes much more than the estimate rather than less?
- Why would a variance swap at 22 have paid exactly the same on both paths, and what does that say about the difference between the two instruments?
- Would rehedging more often on the last day have saved path two?
- How would the answer change if the fund had sold a three-month straddle instead of one-month?
Company names and figures are illustrative.