Case 017Hedging a bookHard

A bank has sold one-year index puts to a client and delta-hedged them. The index falls 5% and the put delta moves from minus 0.35 to minus 0.48. What hedge trade does the desk make, what does the move cost, and why does this hedge bleed in falling markets?

The situation

Corvanta Bank sells a corporate client one-year put options on Rs 200 crore of an equity index, collecting a premium of about Rs 7.9 crore. The put delta is -0.35, meaning the put gains 35 paise for each rupee the index falls, for small moves.

The equity derivatives desk delta-hedges the position with index futures. The next day the index falls 5% and the put delta moves to -0.48. Ignore interest, dividends and the one day of time decay for the arithmetic.

Your task

What hedge does the desk hold before and after the move, what does the move cost the hedged book, and why is this a structural feature of being short options?

Quick check

The index has just fallen 5%. What does the desk now have to do with its hedge?

Worked solution

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

30-second answerThe answer to give first

The desk starts short Rs 70 crore of index futures and must sell Rs 26 crore more after the 5% fall. The short put loses about Rs 4.15 crore while the hedge gains Rs 3.50 crore, so the book is down about Rs 0.65 crore even though it was hedged. Short options are short gamma: the hedge always trades after the move, selling low and buying high, and the premium is what pays for that.

Step 1What does being short a put make the bank?

Selling a put is like selling flood insurance on houses near a river: you collect a premium, and you lose when the water rises, faster the higher it gets. Corvanta has promised to buy the index at the strike if the client wants to sell, so it gains when the index rises and loses when it falls, like an owner of the index. For small moves that exposure is the delta: 0.35 times Rs 200 crore, Rs 70 crore long. The desk hedges it by selling Rs 70 crore of index futures.

Step 2What hedge trade follows the fall?

After a 5% fall the put is closer to being exercised, and its delta has moved to minus 0.48. Corvanta's exposure is now 0.48 times Rs 200 crore, Rs 96 crore long, so the desk sells Rs 26 crore more of futures to stay hedged. That is the uncomfortable part: the desk is forced to sell after the market has fallen, and if the index bounces it must buy back after the market has risen.

Short an option and the hedge sells into every fallStart: index at 10070 crore: sell 70After a 5% fall96 crore: sell 26 moreIndex futures sold to stay hedged, Rs croreRebound and the desk must buy back what it sold: it sells low and buys high, again and again.
Corvanta starts by selling Rs 70 crore of index futures against the short put, and after a 5% fall it must sell Rs 26 crore more, taking the hedge to Rs 96 crore, always trading after the market has moved.
Step 3What does the move cost a hedged book?

Value the two legs over the move. The index falls 5% of Rs 200 crore, Rs 10 crore. The put gains roughly the average delta times the move, 0.415 times Rs 10 crore, Rs 4.15 crore, which the bank loses; the hedge, short Rs 70 crore, gains only Rs 3.50 crore. The hedged book is down about Rs 0.65 crore. A full option model priced on the same deltas gives Rs 0.64 crore; the average-delta shortcut is close enough to say out loud.

The relationship
Loss≈12 Δδ ΔS=12×0.13×10=0.65\text{Loss} \approx \tfrac{1}{2}\,\Delta\delta\,\Delta S = \tfrac{1}{2}\times 0.13 \times 10 = 0.65
\Delta\deltahow far the delta moved over the fall, 0.48 less 0.35
\Delta Sthe index move in rupees of notional, 5% of Rs 200 crore
What it says in wordsThe hedged loss is half the change in delta times the size of the move, because the hedge was sized to the starting delta while the true exposure grew steadily through the fall.
The hedge is a straight line; the short put bends away from it-6%-4%-2%+2%-4-2+2hedge covers -3.50short put -4.14Gap at -5%: -0.64 crorethe curvature the hedge missesdelta hedge lineshort putIndex moveChange in value, Rs crore
The delta hedge offsets the short put along a straight line, but the put's value bends below it in both directions; at a 5% fall the short put loses Rs 4.14 crore against a hedge gain of Rs 3.50 crore, a gap of about Rs 0.64 crore.
Step 4Why does this happen every time, and what pays for it?

Because the bank is short gammaHolding options sold to others, so that the position loses from large moves in either direction even when delta-hedged.. The curve bends away from the hedge line on both sides, so any large move, up or down, costs the hedged book money. What pays for it is the premium: the Rs 7.9 crore collected is, in effect, the price of the volatility the desk expects to hedge through. If the index moves more than the volatility priced into the put, the hedging losses exceed the premium; if it moves less, the desk keeps the difference. Close with the risk view: limit the desk's gamma and its exposure to a large gap move, because a hedge that trades only after the move cannot protect against a market that jumps.

Where candidates lose it

Candidates say the position was delta-hedged, so the fall cost nothing. Delta hedging removes only the first-order exposure; the change in delta through the move is what costs money, and it always goes against the seller of options.

The second miss is getting the direction of the hedge trade wrong: after a fall, a desk short puts sells more, not less, which is why dealer hedging can add to a sell-off.

What the interviewer asks next

  • The index rebounds 5% the next day. What does the desk trade, and what is the two-day result?
  • How would buying a small amount of put options reduce the desk's gamma?
  • What happens to this hedge if the index gaps down 10% overnight?
  • Why might many dealers being short puts at once make a market fall faster?

Company names and figures are illustrative.