Case 023Options and volatility tradingCore
An options book shows delta 2,000 shares, gamma 300 shares per rupee, vega Rs 2 lakh per vol point and theta minus Rs 1.5 lakh a day. The stock rises Rs 4 and implied volatility rises 1 point; reported P&L is Rs 1 lakh. Attribute the P&L and size the unexplained residual.
1The situation
Sarvikon Options runs a book of listed options on Pravalika Motors, a stock at about Rs 1,800 with implied volatility near 30%, most of it in one-month expiries. At the previous close the book's risk report showed delta 2,000 shares, gamma 300 shares per rupee of stock move, vega Rs 2 lakh per implied volatility point and theta minus Rs 1.5 lakh per calendar day.
Today the stock closed up Rs 4 and the at-the-money implied volatility rose 1 point. The P&L system reports a profit of Rs 1 lakh for the day. The desk head asks you to attribute it.
2Your task
Compute each Greek's contribution, the explained total and the residual, say whether the residual is large, and list what you would check to close it.
Quick check
Before computing: which Greek will be the largest contributor today?
Worked solution
Try it on paper, then open one step at a time.
30-second answerThe answer to give first
Explained P&L is Rs 60,400, so Rs 39,600 of the reported Rs 1,00,000 is unexplained. Delta contributes Rs 8,000, gamma Rs 2,400, vega Rs 2,00,000 and theta minus Rs 1,50,000. A residual of 40% of the day's P&L, larger than delta and gamma combined, is too big to sign off: the first suspects are a non-parallel move in the volatility surface, trades done during the day, and marks taken at different times.
Step 1How does each Greek turn a move into rupees?
Each Greek is a sensitivity; multiply it by what moved. Delta is 2,000 shares, so a Rs 4 rise earns 2,000 times 4, Rs 8,000. Gamma is the rate at which delta grows with the stock, so its P&L is one half of 300 times 4 squared, Rs 2,400. Vega is Rs 2 lakh per volatility point, and the surface rose a point, so Rs 2,00,000. Theta is the cost of one day passing, minus Rs 1,50,000. The four add to Rs 60,400, and the system says Rs 1,00,000, so Rs 39,600 is unexplained. A household budget works the same way: rent, food and travel explain most of the month, and if the bank balance moved by far more than their sum, something is missing from the list, not wrong in the arithmetic.
| \delta | delta, 2,000 shares |
| \gamma | gamma, 300 shares per rupee |
| \nu | vega, Rs 2 lakh per volatility point |
| \theta | theta, minus Rs 1.5 lakh per day |
| \Delta S, \Delta\sigma | the stock move, Rs 4, and the volatility move, 1 point |
Step 2Was Rs 4 a good day or a bad day for this book?
Bad, and the vega hid it. A book that is long gamma pays theta for the right to profit from movement, and it needs the stock to move enough to pay the rent. Gamma earns one half of 300 times the move squared, and theta costs Rs 1.5 lakh, so the breakeven move is the square root of 2 times 1.5 lakh over 300, about Rs 32; a Rs 4 day earns Rs 2,400 of gamma against Rs 1.5 lakh of theta. That is consistent with the book's own numbers: at 30% volatility a one standard deviation day on an Rs 1,800 stock is about Rs 34, on which gamma would earn about Rs 1,73,571, close to the theta. Today the stock barely moved, the gamma-theta trade lost about Rs 1.48 lakh, and the one-point rise in implied volatility earned Rs 2 lakh and turned the day positive. Reporting a profit without saying that is reporting the wrong story.
Step 3Is the residual large, and where would you look?
Judge it against the pieces. Rs 39,600 is 40% of the reported P&L, 66% of the explained P&L, and about 11% of the gross size of the four contributions; it is larger than delta and gamma together. A residual that would change the sign of the story if it reversed is not noise, and on a book whose two big Greeks are vega and theta the first place to look is the volatility surface. Vega of Rs 2 lakh per point assumes the whole surface moved by the same one point; if one-month implied rose two points while three-month rose half a point, the bucketed vegas explain a different number. Next come second-order cross terms, vanna and volga, which matter when volatility and spot move together; trades done during the day, whose P&L is not in the overnight Greeks at all; dividend or rate changes; and marks: a Rs 4 close against a risk report struck at a different time.
| Source | Rs | Share of reported |
|---|---|---|
| Delta, 2,000 x 4 | +8,000 | 8% |
| Gamma, 0.5 x 300 x 16 | +2,400 | 2% |
| Vega, 2 lakh x 1 | +2,00,000 | 200% |
| Theta, one day | -1,50,000 | -150% |
| Explained | +60,400 | 60% |
| Residual | +39,600 | 40% |
Close with the process. Rerun the attribution with vega by expiry bucket and the actual surface moves, add the intraday trade blotter, and check the mark times. If the residual falls under a few thousand rupees the model is fine and the surface moved unevenly; if it does not, the Greeks in the report are stale or the P&L system has a booking error, and either one is worth more to the desk than today's profit.
Where candidates lose it
The common loss is computing the four contributions, getting Rs 60,400, and stopping, as if the residual were a rounding error. The interviewer's real question is the Rs 39,600 and whether you can size it and say where it comes from.
The second is calling the day a profit from the stock move. Delta and gamma earned Rs 10,400 together; the stock Greeks lost money net of theta, and only the volatility rise made the day positive.
What the interviewer asks next
- Vega by bucket is Rs 1.5 lakh in one-month and Rs 0.5 lakh in three-month; one-month implied rose 1.8 points and three-month 0.4. What is the vega P&L now?
- How would you include vanna in the attribution, and when does it matter?
- The report was struck at 15:00 and the close was 15:30. How much residual could that explain?
- What residual as a share of gross Greeks would you accept without investigation?
Company names and figures are illustrative.