Case 025Greeks and managing an options bookHard

An options desk is short gamma of 8,000 shares per rupee on a Rs 400 stock, delta hedged, when news gaps it 8% to Rs 432. Implied volatility jumps from 30% to 38% and vega is minus Rs 1.5 lakh a point. Estimate the loss, the new delta, and what you do in the next ten minutes.

The situation

Noyyal Trading's options desk makes markets in Subarnarekha Energy, which trades at Rs 400. After a week of selling options to clients, the book is short gamma of 8,000 shares per rupee and short vega of Rs 1.5 lakh per volatility point, and it is delta hedged at 400. Implied volatility is 30%.

A headline about a government contract hits the screens and the stock gaps 8% to Rs 432 before the desk can trade. Implied volatility jumps to 38%. You are the trader on the book.

Your task

Estimate the P&L from the gap and from the volatility move, compute the book's delta at 432, and lay out what you do in the next ten minutes and why in that order.

Quick check

Before computing: if the gap had been 4% instead of 8%, how would the gamma loss compare?

Worked solution

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

30-second answerThe answer to give first

The gap costs about Rs 40.96 lakh from gamma, one half times 8,000 times 32 squared, and Rs 12 lakh from the 8 point rise in volatility, about Rs 52.96 lakh in all. The book is now short 2.56 lakh shares of delta, 8,000 times 32, worth about Rs 11.1 crore. The first job is to buy back that delta, because every further rupee of rally costs another Rs 2.56 lakh; then re-mark, check limits, and only then decide whether to cut the short gamma.

Step 1What does the gap cost, and why the square?

A car that is braking harder the faster it goes loses distance on a long skid far faster than on a short one. A short gamma book that was hedged at 400 picks up short delta as the stock rises, 8,000 shares for every rupee, so across a 32 rupee gap its delta goes from zero to minus 2.56 lakh shares, and the loss is the area under that line: one half times 8,000 times 32 squared, Rs 40.96 lakh. The square is why gaps are the worry: the book earns thetaThe value an options position gains or loses each day from the passage of time alone. A short gamma book is paid theta in exchange for losing on large moves. for every quiet day and gives it back in a single large move. On top of that, the 8 point jump in implied volatility costs 8 times Rs 1.5 lakh, Rs 12 lakh, because the book is short options that are now worth more.

The relationship
P&L≈12 Γ (ΔS)2+ν Δσ=12(−8,000)(32)2+(−1.5 lakh)(8)=−52.96 lakh\text{P\&L} \approx \tfrac{1}{2}\,\Gamma\,(\Delta S)^2 + \nu\,\Delta\sigma = \tfrac{1}{2}(-8{,}000)(32)^2 + (-1.5\text{ lakh})(8) = -52.96\text{ lakh}
\Gammagamma, minus 8,000 shares of delta per rupee
\Delta Sthe gap, 8% of 400, Rs 32
\nuvega, minus Rs 1.5 lakh per volatility point
\Delta\sigmathe rise in implied volatility, 8 points
What it says in wordsThe gamma loss grows with the square of the gap and the vega loss with the rise in volatility; together they come to about Rs 53 lakh before any rehedging.
The loss from the gap, piece by piece: gamma first, then vegaRs lakh lost, from zero downwards-40.96Gamma0.5 x 8,000 x 32 squared-12.00Vega8 points x 1.5 lakh-52.96Totalbefore any rehedgeIn contextTheta earned a dayabout Rs 1.58 lakhThe gap costabout 34 days of itA normal day's moveabout Rs 6.3; thisgap was 5.1 of those
The 32 rupee gap costs Rs 40.96 lakh through gamma and the 8 point volatility jump Rs 12 lakh through vega, Rs 52.96 lakh in all, roughly 34 days of the Rs 1.58 lakh a day the book earns in theta.
Step 2What is the delta now?

Gamma is the change in delta for a rupee move. The book started flat at 400; after 32 rupees at minus 8,000 shares a rupee, it is short 2.56 lakh shares, about Rs 11.1 crore of stock at 432. The figure shows it as the slope of the P&L curve at 432. That number is the live risk: with the book short 2.56 lakh shares, a further rally of Rs 5 costs another Rs 12.8 lakh plus the extra gamma, and the headline has not finished moving the stock. Two second-order effects make the true delta a little different, and a trader would say so: gamma itself changes as the stock moves and volatility rises, and the jump in volatility shifts the deltas of out-of-the-money options. The risk system's number should be used once it refreshes; 2.56 lakh shares is the right first estimate.

Short gamma: the loss grows with the square of the gap, and the delta grows with it-20-40-600Rs lakh360380400420440hedged at 400: delta 0432: -40.96 lakhslope = delta = -2.56 lakh shares-0.5 x 8,000 x (move) squared
Hedged at 400, the short gamma book's P&L is an upside-down parabola: at 432 it is down Rs 40.96 lakh from gamma, and the slope of the curve there, the new delta, is short 2.56 lakh shares, which is the position to buy back first.
Step 3What do you do in the next ten minutes?

Order matters, and the interviewer is listening for it. First, buy back the delta: about 2.56 lakh shares or the futures equivalent, in pieces rather than one order into a moving market, because the delta is the only risk that is losing money every second. Second, tell the desk head and risk what happened and the estimated Rs 53 lakh loss, before they find it on a screen. Third, re-mark the book at the new spot and volatility and check it against the gamma, vega and loss limits. Fourth, read the headline: a contract award that changes the company's value is a one-off repricing, and the volatility jump may fade; a story still developing means more gaps. Only then decide whether to buy options to cut the short gamma. Buying protection now means paying 38% volatility after the event, which locks in the loss; holding the position means betting the next move is smaller. A trader who rehedges and then waits for information is not being passive.

Say the limits. The gamma and vega figures are local estimates, accurate for small moves and approximate for an 8% gap; a full revaluation of the book at 432 and 38% would give the true number. The theta comparison uses the textbook relation between gamma and theta for a delta-hedged book at 30% volatility, about Rs 1.58 lakh a day, so the gap took roughly 34 trading days of income. And the daily one standard deviation move at 30% volatility is about Rs 6.3, so this gap was a 5 standard deviation day: rare under a normal distribution, routine for single stocks on news.

Where candidates lose it

Candidates compute the gamma loss as 8,000 times 32, Rs 2.56 lakh, treating gamma as if it were delta. The loss is one half times gamma times the move squared, Rs 40.96 lakh, and the 2.56 lakh is the new delta in shares.

The second loss is jumping to the strategic question, whether to buy back options, before rehedging. The open delta is losing money on every tick while the candidate debates volatility; the first ten seconds belong to the delta.

What the interviewer asks next

  • The stock reverses to 410 after you rehedged at 432. What is the P&L on the rehedge and on the gamma?
  • How would the loss have differed if the book had been short the same gamma in options expiring tomorrow rather than in three months?
  • The desk head wants the book flat gamma by the close. What would you buy, and what does it cost at 38% volatility?
  • How would you set a gamma limit so that a 10% gap cannot cost more than one month of theta?

Company names and figures are illustrative.