Case 052Greeks and managing an options bookHard

You are short Rs 5 lakh of one-month vega and can only hedge with three-month options. Size the hedge, then show what happens when one-month vol rises 4 points and three-month vol rises 2.

The situation

Hemavati Capital's index options desk has sold a run of one-month options to clients. The book is short vegaThe change in an option position's value for a one-point change in implied volatility. Short vega loses money when volatility rises. of Rs 5 lakh per volatility point. The one-month options are illiquid to buy back, so the desk proposes to hedge with three-month options, which carry vega of Rs 2,500 per lot.

Overnight, a policy surprise lifts implied volatility across the curve. One-month implied volatility rises 4 points, from 14% to 18%; three-month rises 2 points, from 15% to 17%.

Your task

How many three-month lots flatten the book's vega, what is the P&L on the move, and what does the result say about hedging one expiry with another?

Quick check

The desk buys enough three-month lots to make total vega zero. On this move the book:

Worked solution

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

30-second answerThe answer to give first

200 lots flatten the vega, and the book still loses Rs 10 lakh. Rs 5 lakh of vega divided by Rs 2,500 a lot is 200 lots of three-month options to buy. When one-month vol rises 4 points the short leg loses Rs 20 lakh; the hedge's expiry rises only 2 points, so it earns Rs 10 lakh. A vega-flat book covered half the loss because short-dated volatility moves more than long-dated.

Step 1Why is vega sized lot by lot, and what does flat vega promise?

Vega adds across positions like weight on a scale. Rs 5,00,000 of short vega needs Rs 5,00,000 of long vega, and at Rs 2,500 a lot that is 200 lots of the three-month option. Buy them and the risk report shows vega of zero. The promise in that zero is narrower than it looks: it says the book is unmoved by a one-point rise in volatility at every expiry at once. It says nothing about a rise that is 4 points at the front and 2 points at three months.

The relationship

\text{lots} = \frac{5{,}00{,}000}{2{,}500} = 200 \qquad \text{P\&L} = -5{,}00{,}000 \times 4 + 200 \times 2{,}500 \times 2 = -10{,}00{,}000

5,00,000	rupee vega of the short one-month book, per vol point
2,500	rupee vega per lot of the three-month option
4 and 2	the volatility moves at one month and three months, in points

What it says in wordsThe hedge is the short vega divided by the vega per lot; the profit or loss is each leg's vega times the move at that leg's own expiry.

Step 2What happened on the night of the move?

A policy surprise is a short-dated event, and the market prices it where the event lives: in the front of the curve. One-month volatility went from 14 to 18 and three-month from 15 to 17, so the term structure inverted. The short one-month leg lost Rs 5 lakh times 4 points, Rs 20 lakh; the long three-month leg gained Rs 5 lakh times 2 points, Rs 10 lakh; net, Rs 10 lakh lost on a book whose vega was zero. This is like insuring a scooter with a policy on a car: both are vehicles, both get dented in a storm, but not by the same amount.

One-month implied volatility rose from 14 to 18 while three-month rose from 15 to 17, so the short one-month book lost Rs 20 lakh and the 200 lots of three-month options recovered only Rs 10 lakh.

The waterfall shows the Rs 20 lakh loss on the short one-month vega, the Rs 10 lakh gain on the three-month hedge, and the Rs 10 lakh net loss that flat vega did not prevent.

Step 3Could the hedge have been sized for the move that actually happens?

Yes, and the choice is the real content of the question. If history says three-month volatility moves about half as much as one-month, a desk can weight the hedge by that ratio and buy 400 lots instead of 200. On this move the hedge then earns Rs 20 lakh and the book breaks even. But the same 400 lots leave the book net long Rs 5 lakh of vega against a parallel shift, so a one-point fall across the curve costs about Rs 5 lakh. There is no number of three-month lots that hedges both a parallel move and a front-end move; that needs one-month options, which is where this began.

Hedge	3m lots	Total vega	This move, 1m +4, 3m +2	Parallel +1 at all expiries
Vega-flat	200	0	Rs -10 lakh	Rs 0
Ratio-weighted, 3m moves half as much	400	+Rs 5 lakh / point	Rs 0	Rs +5 lakh

Rs lakh. The vega-flat hedge is right for a parallel move and wrong for a front-end move; the ratio-weighted hedge is the reverse. The desk has to say which move it is hedging.

The report to the head of desk should say: vega is flat by expiry bucket only if we hold one-month options against one-month options; otherwise show vega by bucket and run a front-end stress alongside the parallel one. A single vega number is a parallel-shift number, and the moves that hurt option sellers are rarely parallel. The limitation is the ratio itself: the half-as-much relation is an average from calm periods, and in a panic the front can move three or four times the back.

Where candidates lose it

Candidates divide 5 lakh by 2,500, say 200 lots, and stop, as if the vega number were the risk. The interviewer then asks for the P&L on the move, and the candidate who has not thought about expiry buckets says zero.

The second miss, after seeing the Rs 10 lakh loss, is to declare the ratio-weighted hedge the right answer. It is right for this move and wrong for a parallel one; the honest answer names the trade-off.

What the interviewer asks next

How would you present vega by expiry bucket on the risk report, and what buckets would you use?
The three-month options you bought also carry gamma and theta the one-month book does not. What does that do to the daily P&L?
After the move the curve is inverted. Does the desk keep the 200 lots or sell them?

Company names and figures are illustrative.