Case 094Options and volatility tradingWarm up

Bhavantar is long 100 at-the-money straddles with total gamma of 40 shares per rupee. The stock moves 1,000, 1,010, 1,000, 990, 1,000 during the day, theta costs Rs 3,000, and the desk re-hedges at each step. What is the net P&L?

The situation

Bhavantar Derivatives holds 100 at-the-money straddles, a call and a put at the same strike, on an invented stock trading at Rs 1,000. The position is delta-neutral at 1,000 and its total gamma is 40 shares per rupee: for every rupee the stock rises, the options' delta rises by 40 shares. Holding the options for a day costs Rs 3,000 of time decay.

During the day the stock trades 1,000, 1,010, 1,000, 990 and back to 1,000. At each step the desk trades shares to bring the delta back to zero. Ignore trading costs.

Your task

Compute the hedge trades, the gamma P&L at each step and the net P&L for the day, and say what the result means in volatility terms.

Quick check

The stock ends the day where it started. What is the net P&L?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Net P&L is plus Rs 5,000: four Rs 10 moves earn Rs 2,000 each through re-hedging, Rs 8,000, against Rs 3,000 of theta. The hedge sells 400 shares at 1,010, buys them back at 1,000, buys 400 at 990 and sells them at 1,000, locking in Rs 10 twice on 400 shares. The theta implies a breakeven daily move of about Rs 12.2; the day realised the equivalent of Rs 20, so realised volatility beat implied.

Step 1What does re-hedging do at each step?

Think of a shopkeeper who always keeps the same number of mangoes on the shelf: when a rush of buyers pushes the price up, he sells from stock; when the price falls, he restocks. Done mechanically, he sells high and buys low. A long-gamma position does this automatically: when the stock rises Rs 10, the straddles gain 40 x 10 = 400 shares of delta, so the desk sells 400 shares to get flat; when it falls back, the delta disappears and the desk buys them back. The sale at 1,010 and the purchase at 1,000 lock in Rs 10 on 400 shares, Rs 4,000. The same happens on the dip: buy 400 at 990, sell at 1,000.

The relationship
Gamma P&L per step=12Γ(ΔS)2=12×40×102=2,0004 steps=8,000,8,000−3,000=5,000\text{Gamma P\&L per step} = \tfrac{1}{2}\Gamma(\Delta S)^2 = \tfrac{1}{2}\times 40\times 10^2 = 2{,}000 \qquad 4 \text{ steps} = 8{,}000, \quad 8{,}000 - 3{,}000 = 5{,}000
Gammachange in delta per rupee, 40 shares
Delta Ssize of each move, Rs 10
thetathe day's time decay, Rs 3,000
What it says in wordsEach re-hedged move earns half of gamma times the move squared, whichever direction it goes; the day's total is set against the theta paid for holding the options.
StepMoveOptions P&LHedge P&LStep P&LHedge trade after
11,000 to 1,010+2,000+0+2,000sell 400
21,010 to 1,000-2,000+4,000+2,000buy 400
31,000 to 990+2,000+0+2,000buy 400
4990 to 1,000-2,000+4,000+2,000sell 400
Bhavantar's four steps. Each earns Rs 2,000 net: on the moves away from 1,000 the options' convexity pays, on the moves back the share hedge pays. The total is Rs 8,000 before theta.
Long gamma: the hedge sells rises and buys dips, automatically9901,0001,010sell 400 at 1,010buy 400 at 1,000buy 400 at 990sell 400 at 1,00004,0008,000gamma +8,000theta -3,000Running P and L, RsEach step: 0.5 x 40 x 10 squared = Rs 2,000. Net for the day: Rs 5,000.
Bhavantar's hedge sells 400 shares after the rise to 1,010 and buys after the dip to 990, so each Rs 10 move adds Rs 2,000 and the day's gamma gain of Rs 8,000 beats the Rs 3,000 theta, a net gain of Rs 5,000 on a stock that ended where it began.
Step 2Why does the stock ending flat still make money?

Because gamma P&L depends on the size of the moves squared, not on where the stock ends. The straddle holder is paid for movement and pays theta for time; the day's result is the sum of squared moves times half the gamma, minus theta. The squared moves sum to 400, so the gamma gain is 0.5 x 40 x 400 = Rs 8,000. A day that drifted quietly from 1,000 to 1,005 in small steps would have earned far less than its theta even though it moved further overall.

Step 3What does it say about volatility?

Theta is the price of the options' expected movement. Setting 0.5 x 40 x (move squared) equal to Rs 3,000 gives a breakeven sum of squared moves of 150, the same as one move of about Rs 12.2, or 1.22% of the price, which is an implied volatility of about 19% a year. The day's moves were equivalent to a single Rs 20 move, 2% of the price, a realised volatility of about 32%, so the long straddles won because realised beat implied. The limitation: real re-hedging pays bid-ask spread on every trade and cannot catch every wiggle, and on a quiet day the same hedging would lose the Rs 3,000 theta almost entirely.

Where candidates lose it

The common loss is saying the position lost Rs 3,000 because the stock went nowhere. That ignores the hedge trades, which turn every move into profit for a long-gamma book, whichever way it goes.

The second is computing the gamma P&L on the net move from 1,000 to 1,000, which is zero. Gamma P&L is path-dependent: it sums the squared moves between re-hedges.

What the interviewer asks next

  • How much would the day have earned if the desk re-hedged only at the close?
  • What would a short straddle book have lost on this path?
  • How would bid-ask spread of Rs 0.50 a share change the result?
  • Why does re-hedging more often not always increase the expected gamma P&L?

Company names and figures are illustrative.