Case 094Greeks and managing an options bookCore
A fund is long an at-the-money straddle position with gamma of 1,500 shares per rupee and theta of Rs 45,000 a day, rehedged at each close. Closes run 800, 812, 805, 790, 798, 801. Estimate each day's P&L and the week's.
1The situation
Chaliyar Capital is long a one-month at-the-money straddle position on Betwa Motors, struck at Rs 800. The book's gamma is 1,500 shares per rupee and its theta is Rs 45,000 a day. The desk delta-hedges back to flat at each close, so each day starts with zero delta.
Over five days the stock closes at 800, 812, 805, 790, 798, 801. Treat gamma and theta as constant for the week. The head of trading wants each day's P&L, the week's total, and the move the stock needs to make each day for the position to break even.
2Your task
What does the position make or lose each day, what is the week's total, and what is the daily breakeven move?
Quick check
On day four the stock moves Rs 8. Does the long straddle make or lose money that day?
Worked solution
Try it on paper, then open one step at a time.
30-second answerThe answer to give first
The week nets about Rs 1.43 lakh: big moves on days one and three pay for the quiet days. Each day earns half of 1,500 times the move squared and pays Rs 45,000 of theta. Moves of 12, 7, 15, 8 and 3 give daily nets of +63,000, -8,250, +123,750, +3,000, -38,250 rupees. The breakeven move is the square root of 45,000 over 750, about Rs 7.75, or 0.97% a day.
Step 1Why does a hedged straddle make money when the stock moves either way?
Think of a stall that buys mangoes more cheaply when prices fall and sells more when prices rise, and pays a fixed daily rent for the spot. Any swing in either direction is good for business; a flat day just costs the rent. A long gammaHow much delta changes for a one rupee move in the stock. Long gamma means delta rises as the stock rises and falls as it falls. position makes half of gamma times the move squared each day after rehedging, whatever the direction, and pays theta every day regardless. When the stock rises, the position gets longer and the hedge sells into strength; when it falls, the hedge buys into weakness. The square in the formula is why direction does not matter.
| \Gamma | gamma, 1,500 shares per rupee |
| \Delta S | the day's move in the stock, Rs |
| \Theta | the day's time decay, Rs 45,000 |
Step 2What does each day come to?
Day one moves 12: 750 times 144 is Rs 108,000, net Rs +63,000. Day two moves 7: Rs 36,750, net Rs -8,250, just short of breakeven. Day three moves 15: Rs 168,750, net Rs +123,750. Day four moves 8: net Rs +3,000. Day five moves 3: only Rs 6,750 against the theta, net Rs -38,250. The week totals Rs +143,250, about Rs 1.43 lakh, and two days, the 12 and the 15, earn more than the whole. That is the usual shape of long gamma: steady small losses, occasional large wins.
Step 3What is the breakeven, and what does it say about volatility?
Set 750 times m squared equal to 45,000: m is the square root of 60, about Rs 7.75, or 0.97% of the price. Multiply a daily breakeven of 0.97% by the square root of 252 and you get about 15.4%, which is, to a close approximation, the implied volatility the straddle was bought at. This week the moves had a root mean square of about 9.91 rupees a day, a realised volatility near 19.7%. Realised above implied is the condition for a long gamma position to make money, and the P&L is that condition written in rupees.
State the limits of the estimate. Gamma is not constant: it is highest at the strike and falls as the stock moves away, and it rises as expiry approaches, as does theta. A move of 15 is large enough that the true gamma P&L differs a little from the formula. Rehedging only at the close also misses intraday swings, which can add to or subtract from the result depending on the path. The constant-Greek calculation is the one to give out loud; the risk system's full revaluation is the one to book.
Where candidates lose it
The common loss is multiplying gamma by the move instead of half gamma by the move squared. That turns day one into Rs 18,000 instead of Rs 108,000 and misreads every day.
The second is netting the moves first: the stock went from 800 to 801 over the week, so a candidate says the straddle lost five days of theta. Gamma pays on each day's move, and the path, not the endpoint, is what earns.
What the interviewer asks next
- If the desk rehedged only every two days, how would the week's gamma P&L change?
- What realised volatility would make this week exactly break even?
- Why does theta rise as the straddle approaches expiry, and what does that do to the breakeven move?
Company names and figures are illustrative.