Case 012Margin, clearing and risk limitsHard

A book has vega of Rs 62 lakh against a Rs 50 lakh limit and gamma of minus 3.5 against minus 3.0. With a one-month and a three-month option to trade, find a combination that brings both inside and explain the trade-off.

The situation

After a large client trade, Lingana Capital's index options book shows vega of Rs 62 lakh per volatility point against a desk limit of Rs 50 lakh, and gamma of minus 3.5 against a limit of minus 3.0, both measured in the desk's units. Risk wants both inside limits before the close.

Two liquid at-the-money options are available. Buying one lot of the one-month option adds vega of Rs 12,000 and gamma of 0.008; buying one lot of the three-month option adds vega of Rs 25,000 and gamma of 0.003. Selling a lot subtracts the same amounts. Delta is hedged separately and can be ignored.

Your task

Why does neither option on its own solve the problem, what combination of the two brings both Greeks inside limits, and what does the fix cost the desk?

Quick check

Before solving: the book is long too much vega and short too much gamma. What does that tell you about the hedge?

Worked solution

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

30-second answerThe answer to give first

Buy about 105 lots of the one-month option and sell about 100 lots of the three-month: vega falls to Rs 49.6 lakh and gamma rises to -2.96, both inside. Selling the three-month alone fixes vega but takes gamma to -3.64; buying the one-month alone fixes gamma but lifts vega to Rs 69.5 lakh. The exact corner is buy 98 and sell 95. The cost is a calendar position: long short-dated theta bleed against short long-dated vega, which must be rolled every month.

Step 1Why does one instrument fail?

The book needs less vega, which means selling options, and less negative gamma, which means buying them. One tool cannot do both. Selling three-month options to cut vega by Rs 12 lakh takes 48 lots and pushes gamma from minus 3.5 to -3.64, further outside; buying one-month options to add 0.5 of gamma takes 62 lots and pushes vega to Rs 69.5 lakh, further outside. A kitchen with a tap that is both too hot and too weak needs the hot tap turned down and the cold tap turned up, not one knob. The way out is that the two options carry the Greeks in different proportions: per lot, the one-month has 0.067 units of gamma per lakh of vega and the three-month only 0.012.

One instrument fixes one Greek and breaks the other; the mix fixes bothSell 48 three-month onlyVega, Rs lakh per point50.0 against limit 50Gamma, short-3.64 against limit -3.0one limit still brokenBuy 62 one-month onlyVega, Rs lakh per point69.5 against limit 50Gamma, short-3.00 against limit -3.0one limit still brokenBuy 105 1m, sell 100 3mVega, Rs lakh per point49.6 against limit 50Gamma, short-2.96 against limit -3.0both inside
Selling 48 three-month lots brings vega to the Rs 50 lakh limit but takes gamma to -3.64; buying 62 one-month lots brings gamma to minus 3.0 but lifts vega to Rs 69.5 lakh; only the mix of buying 105 one-month and selling 100 three-month lots lands both inside, at Rs 49.6 lakh and -2.96.
Step 2How do you find the combination?
The relationship
62+0.12x−0.25y≤50−3.5+0.008x−0.003y≥−3.062 + 0.12x - 0.25y \le 50 \qquad -3.5 + 0.008x - 0.003y \ge -3.0
xone-month lots bought
ythree-month lots sold
0.12, 0.25vega per lot in Rs lakh
0.008, 0.003gamma per lot
What it says in wordsBuying x one-month lots and selling y three-month lots must take vega down by at least 12 lakh and gamma up by at least 0.5. Solving both as equalities gives the corner of the feasible region.

Treat both as equalities and solve: the corner is x of about 98 one-month lots bought and y of about 95 three-month lots sold. Rounding up to a margin inside both limits, buy 105 one-month lots and sell 100 three-month lots: vega is 62 plus 12.6 less 25, Rs 49.6 lakh; gamma is minus 3.5 plus 0.84 less 0.30, -2.96. Any point in the wedge above the vega line and below the gamma line works; the corner is the cheapest in lots traded, and a desk would pick a point slightly inside it so that a small move in spot or volatility does not push the book back out.

Both limits hold only in the wedge: buy short-dated for gamma, sell long-dated for vega050100150200050100150200250one-month lots boughtthree-month lots soldvega limit: feasible abovegamma limit: feasible to the rightcorner: buy 98, sell 95chosen: buy 105, sell 100sell 48 three-month only: gamma -3.64buy 62 one-month only: vega 69.5Per lotOne-month optionvega Rs 12,000, gamma 0.008gamma per unit of vega: highThree-month optionvega Rs 25,000, gamma 0.003vega per unit of gamma: highso each tool fixes one Greek
In the space of one-month lots bought against three-month lots sold, the vega limit is satisfied above one line and the gamma limit below a steeper one, so both hold only in the wedge to the upper right of their crossing at about 98 and 95; buy 105 and sell 100 sits just inside the corner.
Step 3What does the fix cost, and what would you say to risk?

The hedge is a calendar spread laid over the book: long one-month options that decay quickly, short three-month options that decay slowly. The desk now pays theta every day on the short-dated leg, and in a month the one-month options expire and the gamma hedge has to be bought again, so this is a running cost, not a one-off. The book also ends up long Rs 12.6 lakh of one-month vega against short Rs 25 lakh of three-month vega, which is a view that the volatility curve will steepen, whether the desk meant to take it or not. Say the limits: the Greeks per lot are at today's spot and will change as the index moves, so the fix needs to be rechecked tomorrow; the numbers assume the two options are at the money; and the clean answer to risk is that limits are met, the hedge costs roughly a month of theta, and the larger question is whether the client trade that caused the breach was priced for this cost.

Where candidates lose it

Candidates reach for one instrument, usually selling the longer option because vega is the larger breach in rupees, and report that vega is fixed. The gamma limit is then broken by more, and the interviewer asks what happened to it.

The second loss is solving the two equations and stopping at the corner without checking the sign of the trades. Buy the one-month, sell the three-month; reversing either leg makes both Greeks worse, and candidates who treat x and y as unsigned quantities often do.

What the interviewer asks next

  • The three-month option is only available at a wide bid-offer. How much of the vega cut could you get from a six-month option instead, and what does that do to gamma?
  • Spot rallies 3% overnight. Which of the two Greeks moves outside first, and why?
  • Risk asks for both Greeks at 80% of limit. Resize the trade.
  • Why might a desk prefer to run the vega breach for a day rather than pay for the hedge?

Company names and figures are illustrative.