Case 098Volatility, options and convertiblesHard

Prysmic Solar's convertible bond trades at 105 with a conversion ratio of 10 shares per 100 face, the stock at Rs 9, and a delta of 0.6. How many shares do you short per bond to hedge, and what do you make or lose if the stock moves 10% either way?

The situation

Prysmic Solar has a three-year convertible bond priced at Rs 105 per Rs 100 of face value. Each Rs 100 of face converts into 10 shares, a conversion price of Rs 10, and the shares trade at Rs 9. The desk's model gives the bond a delta of 0.6 to the share price.

To see the shape, value the bond as a bond floor plus 10 call options struck at Rs 10: at 4% rates and about 25% volatility, that reproduces the 105 price and the 0.6 delta, with a bond floor of about 88.8. Treat moves as instant, so time decay and coupons do not enter.

Your task

How many shares do you short per bond to hedge the delta, and what does the hedged position make or lose if the stock moves 10% up or down?

Quick check

The hedged position is long one bond and short 6 shares. The stock jumps 10% up or 10% down. What happens?

Worked solution

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

30-second answerThe answer to give first

Short 6 shares per bond, 0.6 times the 10-share conversion ratio, and the position makes a small gain on a 10% move either way. On the model, a 10% rise earns about Rs 0.37 per bond and a 10% fall about Rs 0.42, because the bond's delta climbs as the stock rises and shrinks as it falls. That convexity is paid for over time, and a sharp fall that widens Prysmic's credit spread can overwhelm it.

Step 1What is the bond made of, and where does 6 shares come from?

A convertible is a bond with a bundle of call options attached. Parity, what the bond is worth if converted now, is 10 x Rs 9, which is Rs 90; the bond trades at 105, a 16.7% premium, because it also has a bond floor under it and time for the shares to rise. The delta of 0.6 says the bond moves like 0.6 of its 10 shares, so the hedge is 0.6 x 10, which is 6 shares short per bond, Rs 54 of stock against Rs 105 of bond.

Convertible price against the share price: a curve above its own tangent80100120140bond floor about 89conversion value, 10 x sharehedge line, slope 6Rs 9: price 1055791113Prysmic share price, Rs (convertible price per 100 of face on the vertical axis)
Prysmic's convertible sits above both its bond floor of about 89 and its conversion value; at a share price of Rs 9 it trades at 105 with a slope of 6, and the curve bends above that tangent on both sides, which is the convexity a hedged holder earns.
Step 2What does the hedged position do on a 10% move?

Think of a bicycle that coasts faster downhill than the extra effort you put in, and slows less than you expect uphill. Up 10%, the bond rises to about 110.77 while the 6-share short loses Rs 5.40, a net gain of about Rs 0.37; down 10%, the bond falls to about 100.02 while the short gains Rs 5.40, a net gain of about Rs 0.42. Both gains come from gammaHow fast the delta of an option position changes as the underlying price moves. Positive gamma means a hedged position gains on large moves either way.: the bond's delta moves in the helpful direction as the stock moves, while the hedge stays fixed at 6 shares.

Stock moveShare price, RsBond priceBond P&LShort 6 sharesHedged P&L
-20%7.2095.93-9.07+10.80+1.73
-10%8.10100.02-4.98+5.40+0.42
+0%9.00105.00+0.00-0.00+0.00
+10%9.90110.77+5.77-5.40+0.37
+20%10.80117.21+12.21-10.80+1.41
Per Rs 100 of face, the hedged position earns about Rs 0.42 on a 10% fall and Rs 0.37 on a 10% rise, and more on 20% moves, because the bond's price curves above the fixed 6-share hedge.
Step 3If it gains both ways, where is the catch?

Two catches. First, time: the option part of the bond loses value every day the stock sits still, so the hedged holder is paid only if the stock moves more than the volatility built into the price; coupons and the interest on the short sale help carry the wait. Second, credit: the model holds the bond floor fixed, but when a solar company's shares fall 20%, lenders worry too and the floor drops. If a 20% fall also cuts the floor by 3 points, the hedged gain of Rs 1.73 turns into a loss of about Rs 1.27. That is why convertible arbitrage desks also hedge credit, with credit default swaps or by shorting extra shares on the downside.

Hedged P&L per bond (Rs per 100 of face): gains both ways, until credit bites+1.73-20%+0.42-10%+0.000%+0.37+10%+1.41+20%-1.27-20% and floor -3Stock move, bond long and 6 shares short
Long one bond and short 6 shares gains on every instant move, from Rs 0.42 on a 10% fall to Rs 1.41 on a 20% rise, but a 20% fall that also cuts the bond floor by 3 points turns the result into a loss of about Rs 1.27.

Close with how the hedge is run in practice: re-hedge as the delta changes, selling shares back after a fall and adding after a rise, which locks in the convexity gains. The trade is a bet that realised volatility beats the implied volatility in the bond's price, with credit as the risk that is easiest to forget.

Where candidates lose it

The frequent slip is shorting 0.6 shares, forgetting that one bond converts into 10 shares, or shorting all 10 as if the bond were stock. The hedge is delta times the conversion ratio.

The second is claiming the hedged position is riskless because it gains both ways. The gains are paid for through time decay, and the credit move on a sharp fall can wipe them out.

What the interviewer asks next

  • After a 10% rise the delta is higher. How many shares do you short now, and what does re-hedging lock in?
  • How would you hedge Prysmic's credit risk alongside the delta?
  • What happens to the hedged position if the stock does not move for six months?

Company names and figures are illustrative.