---
title: "Convertible Arbitrage"
book: "Strategies II: Volatility, Relative Value, Macro and the Bank Desks"
subject: quant
language: en
chapter: 8
exercises: 8
source: https://one-course.com/books/quant/9/en/chapter/8-convertible-arbitrage
---

# Chapter 8 — Convertible Arbitrage

Buy the convertible, short the stock, and the trade is long volatility and credit and short the borrow. In the first quarter of 2005, investors withdrew more than 20% of the capital of convertible arbitrage funds. The funds sold convertibles to pay them, and prices fell below theoretical values, by a median of 2.7% in mid-May. The funds lost 7.2% from January to May, about what a 2.7% cheapening costs at the typical leverage of three to one. In 2008 the prime brokers who lent to hedge funds cut the leverage they offered. On this chapter’s synthetic market, a delta-hedged book of thirty convertibles, levered three to one, earns 9.8% a year on its capital in ordinary times. It loses 11.6% in a planted three-month episode of forced selling, of which 12.6 points come from cheapening and 6.9 from credit, and it recovers 22.0% in the six months after. The build is `firm.convarb`.

## 8.1 What a convertible holder owns

A convertible bond (Book 5, chapter 21) is a corporate bond the holder may exchange for a fixed number of shares. It is a straight bond, plus an option on the shares, minus whatever the issuer’s call removes. The whole is exposed to the issuer’s default. Book 5’s pricer values it on a grid in the share price, with a default hazard that rises as the share falls: an equity-to-credit link.

The chapter’s bond is five years long, with a 2% coupon, and converts into 2.5 shares (a conversion price of 40). It is callable after two years if the share is 30% above the conversion price. The hazard is 3% a year at a share price of 40, rising in proportion to $40/S$, with 40% recovery. At issue, with the share at 40, the pricer values it at 117.7. Its conversion value is 100, and its delta is 1.80 shares per bond ([Figure 8.1](#fig-s2-convertible-arbitrage-profile)). At a share price of 80 it trades close to its conversion value (204.8 against 200) with a delta of 2.40.

**Definition 8.1 (Busted convertible).**

A *busted convertible* is a convertible whose share price has fallen far below its conversion price, so that the conversion option is nearly worthless and the bond trades on its credit; its remaining equity sensitivity comes mostly from the link between the share price and the issuer’s default risk.

At a share price of 15 the bond is busted. It is worth 81.2, below the straight bond’s 87.34 at the issue’s credit spread, because the hazard has risen with the fall. Its delta is still 1.27 shares, an equity elasticity of only 0.23, and almost all of that delta is credit.

![The chapter’s five-year convertible at issue, by share price: model value from Book 5’s pricer with an equity-to-credit hazard, conversion value (2.5 shares), and the straight bond discounted at the issue’s credit spread. At a share price of 19 and below the bond is worth less than the straight bond, as the hazard rises with the fall. Data: s2_convarb.profile.](https://one-course.com/images/onecourse/chapters/quant-9/s2-convertible-arbitrage/fig-83b9002433f0.svg)

***Figure 8.1.** The chapter’s five-year convertible at issue, by share price: model value from Book 5’s pricer with an equity-to-credit hazard, conversion value (2.5 shares), and the straight bond discounted at the issue’s credit spread. At a share price of 19 and below the bond is worth less than the straight bond, as the hazard rises with the fall. Data: `s2_convarb.profile`.*

## 8.2 The hedged trade and its Greeks

**Definition 8.2 (Convertible delta hedge).**

A *convertible delta hedge* is a short position in the issuer’s shares equal to the convertible’s delta, rebalanced as the delta changes, so that the position’s value does not move with small share-price changes and its P&L comes from gamma, carry, credit and the bond’s cheapness.

Agarwal, Fung, Loon and Naik found that a buy-and-hedge strategy, long convertibles with the equity risk hedged, explains much of convertible arbitrage funds’ returns. Market-wide extremes and the supply of new convertibles matter as well. They read the funds’ returns as payment for funding issuers and passing part of the equity risk on to the equity market.

Mitchell, Pedersen and Pulvino describe the market’s structure. Convertible arbitrage funds held up to 75% of the convertible market. Convertibles are often issued below their fundamental value and go on trading below it, because they are illiquid. The arbitrageur’s main risk is that the bond cheapens further and has to be sold at the wrong moment.

`firm.convarb` buys one bond on each of the synthetic market’s thirty stocks at issue, on day 1 000, with the share at the conversion price. The bonds cost 3% below model value, falling to 1% over six months. Each position is short its delta, rebalanced daily, at a borrow fee of 0.5% a year. The book is levered three to one: capital is a third of the bonds’ value, 114.19 per bond at purchase. Each day’s P&L is split into buckets ([Listing 8.1](#lst-s2-convertible-arbitrage-book)): the bond’s move at unchanged credit and the stock hedge (together, gamma), the change in the credit state, the change in cheapness, coupons and borrow.

## 8.3 Credit and borrow

**Definition 8.3 (Credit-hedged convertible).**

A *credit-hedged convertible* is a delta-hedged convertible to which the arbitrageur adds credit protection (credit default swaps, Book 2, chapter 23, or a short position in the issuer’s straight bonds) sized to the bond’s sensitivity to the issuer’s credit, so that the position keeps its option value and carry without the default risk.

Credit protection costs a premium. In the synthetic market it pays back the bond’s loss from any rise in the hazard multiplier. Each year it costs the bond’s value at risk from a doubled hazard divided by the protection’s annuity. The borrow is the other leg’s cost. A short position pays the lender a fee, and when shares are scarce, the fee rises or the loan is recalled.

## 8.4 2005 and 2008

The synthetic market plants an episode 500 days after purchase, starting with the first crash. For three months the bonds cheapen from 1% to 6% below model value, the hazard doubles and the borrow fee rises to 5%. Afterwards the cheapness returns to 1% over six months. On capital:

| thirty bonds, three to one | delta-hedged | delta- and credit-hedged |
| --- | --- | --- |
| carry outside the episode and recovery (%/yr) | 9.8 | 6.0 |
| three-month episode (%) | $-11.6$ | $-5.8$ |
| of which cheapness | $-12.6$ | $-12.6$ |
| credit, net of any hedge | $-6.9$ | 0.0 |
| gamma (bond and stock hedge) | 3.7 | 3.7 |
| six-month recovery (%) | 22.0 | 10.6 |
| five years (%) | 51.9 | 30.1 |

In the episode the shares fell with the crash. The bonds lost 17.0% of capital at unchanged credit, and the short stock gained 20.7%: the convexity paid 3.7 points. Cheapening cost 12.6 points and the doubled hazard 6.9. The annual coupon that fell in the window added 5.3, and the borrow cost 1.1. The credit hedge paid back its 6.9 points in the episode. It cost 21.8% of capital over five years, and it gave back its share of the recovery. The book is long everything at once ([Figure 8.2](#fig-s2-convertible-arbitrage-book)): volatility, credit and liquidity. The worst days for each are the same days.

![The synthetic delta-hedged convertible book, thirty bonds levered three to one: cumulative P&L as a share of capital and its buckets over five years. The episode starts at year 2.0 with the first planted crash. Data: s2_convarb.cumulative.](https://one-course.com/images/onecourse/chapters/quant-9/s2-convertible-arbitrage/fig-3d04862ce10e.svg)

***Figure 8.2.** The synthetic delta-hedged convertible book, thirty bonds levered three to one: cumulative P&L as a share of capital and its buckets over five years. The episode starts at year 2.0 with the first planted crash. Data: `s2_convarb.cumulative`.*

The real episodes had their own mechanics. In 2005 the pressure came from investors: redemptions of more than 20% of capital in one quarter, 28 specialised funds selling 35% of their convertibles by the end of the year, and a discount largest around the deadlines for redemption notices. In 2008, Mitchell and Pulvino found, the pressure came from the lenders. The imminent failure of prime brokers suddenly cut the leverage hedge funds could get, and what had looked like long-term financing became short-term. In both years the arbitrageurs who should have bought the cheap bonds were the ones selling them.

## 8.5 Strategy files

**Strategy file 8.1 — Delta-hedged convertible.**

**Who pays you, and why.** Issuers, who sell convertibles below fundamental value for quick, cheap capital; arbitrageurs fund them and pass the equity risk to the stock market.

**Instruments and venues.** Convertible bonds (mostly 144A issues); the issuers’ shares, borrowed.

**Signal.** Market price against model value; the new-issue discount.

**Sizing and execution.** Delta-hedged, rebalanced as the delta moves; leverage from prime brokers.

**Costs.** Bond spreads, borrow fees, financing.

**How it dies.** Cheapening in forced selling; leverage withdrawn; borrow recalled.

**Horizon, capacity, infrastructure.** Months to years; a convertible pricer, stock loan, prime broker.

**Backtest honestly.** Model values with realistic credit inputs; episodes like 2005 and 2008; financing that can vanish.

**Sources.** Agarwal, Fung, Loon and Naik (2011); Mitchell, Pedersen and Pulvino (2007): $-7.2\%$ for the funds from January to May 2005; this chapter: 9.8% a year on capital, $-11.6\%$ in the episode.

**Strategy file 8.2 — Credit-hedged convertible.**

**Who pays you, and why.** As above, less the credit premium paid away.

**Instruments and venues.** Convertibles, shares, and credit default swaps or straight bonds of the issuer.

**Signal.** Cheapness against the cost of protection.

**Sizing and execution.** Protection sized to the bond’s credit sensitivity, resized as the share moves.

**Costs.** Protection premiums; the credit default swap’s spread.

**How it dies.** Cheapening is not hedged: it cost 12.6 points in the chapter’s episode either way.

**Horizon, capacity, infrastructure.** As above; credit trading lines.

**Backtest honestly.** Charge the protection every day, not only when it pays.

**Sources.** This chapter: carry 6.0% instead of 9.8%, the episode $-5.8\%$ instead of $-11.6\%$.

**Strategy file 8.3 — Busted convertible credit trade.**

**Who pays you, and why.** Holders who must sell convertibles that no longer behave like equity, and credit buyers who cannot hold them.

**Instruments and venues.** [Busted convertibles](#def-s2-convertible-arbitrage-busted); the issuer’s straight debt or credit default swaps.

**Signal.** The convertible’s yield against the issuer’s straight credit.

**Sizing and execution.** Credit-sized; small equity hedges from the equity-to-credit delta.

**Costs.** Illiquid bonds.

**How it dies.** Default: the bond pays its recovery.

**Horizon, capacity, infrastructure.** Months; credit analysis.

**Backtest honestly.** Recoveries and restructurings as they happened.

**Sources.** No performance figure verified.

**Strategy file 8.4 — New-issue convertible.**

**Who pays you, and why.** Issuers who pay a discount for speed: convertibles are often issued below fundamental value.

**Instruments and venues.** New 144A issues; the issuers’ shares.

**Signal.** Issue price against model value.

**Sizing and execution.** Allocations in the book-build; hedged on day one.

**Costs.** Borrow on newly shorted shares.

**How it dies.** Crowded new issues; discounts that shrink.

**Horizon, capacity, infrastructure.** Weeks; relationships with underwriters.

**Backtest honestly.** Allocations were not guaranteed; assume partial fills.

**Sources.** Mitchell, Pedersen and Pulvino (2007); this chapter’s new-issue discount of 3% is an assumption.

## 8.6 Tutorial: long everything at once

**Goal.** Price convertibles with Book 5’s pricer, hedge them in delta and credit, and run the book through the planted episode. **End state:** the table and the two figures.

1. **The book**. `def run_book (S, cfg: ArbConfig | None = None , credit_hedge: bool = False ) -> dict : """One bond bought at the market price on day 0, short its delta daily, held to the end of S (at most maturity).""" cfg = cfg or ArbConfig() tables = value_tables(cfg) S = np.asarray(S, float ) * cfg.s0 / S[0 ] T = min (len (S) - 1 , int (cfg.maturity * YEAR) - 1 ) sc = scenario(T + 1 , cfg) keys = (" bond " , " hedge " , " cheapness " , " credit " , " credit_hedge " , " coupon " , " borrow " ) out = {k: np.zeros(T) for k in keys} coupon_days = {int (round ((cfg.maturity - k) * YEAR)) for k in range (int (cfg.maturity))} for t in range (T): tau0, tau1 = cfg.maturity - t / YEAR, cfg.maturity - (t + 1 ) / YEAR st0, st1 = bool (sc[" stressed " ][t]), bool (sc[" stressed " ][t + 1 ]) m0 = mark(tables, tau0, S[t], st0) m1 = mark(tables, tau1, S[t + 1 ], st0) # the day's move at yesterday's credit state m1s = mark(tables, tau1, S[t + 1 ], st1) # then the credit state changes d = delta(tables, tau0, S[t], st0) out[" bond " ][t] = m1 - m0 out[" credit " ][t] = m1s - m1 out[" cheapness " ][t] = -(m1s * sc[" cheap " ][t + 1 ] - m0 * sc[" cheap " ][t]) out[" hedge " ][t] = -d * (S[t + 1 ] - S[t]) out[" borrow " ][t] = -d * S[t] * sc[" fee " ][t] / YEAR if credit_hedge: # protection that pays the credit bucket back sens = mark(tables, tau0, S[t], False ) - mark(tables, tau0, S[t], True ) lam = cfg.lam0 * (S[t] / cfg.s0) ** -cfg.p annuity = (1 - math.exp(-(cfg.r + lam) * tau0)) / (cfg.r + lam) out[" credit_hedge " ][t] = -out[" credit " ][t] - sens / annuity / YEAR # its premium, a year's worth / annuity if (t + 1 ) in coupon_days: out[" coupon " ][t] = 2.0 out[" total " ] = sum (out[k] for k in keys) out[" price0 " ] = mark(tables, cfg.maturity, S[0 ]) * (1 - cfg.new_issue) return out` **Listing 8.1.** One bond, short its delta, with credit protection optional, attributed day by day. code/firm/convarb/firm_convarb.py
2. **Returns on capital**. `def summary (credit_hedge: bool = False ): """Returns on capital (bond price / leverage): annual carry outside the episode and its recovery, the episode's loss by bucket, the recovery's gain, and the whole five years.""" cfg, b = books(credit_hedge) capital = b[" price0 " ] / LEVERAGE a, e = cfg.stress_start - 1 , cfg.stress_start + cfg.stress_len - 1 rec = e + cfg.recovery_days days = len (b[" total " ]) calm = np.ones(days, bool ) calm[a:rec] = False ep = {k: float (b[k][a:e].sum() / capital) for k in BUCKETS + (" total " ,)} return {" carry " : float (b[" total " ][calm].sum() / capital / (calm.sum() / YEAR)), " episode " : ep, " recovery " : float (b[" total " ][e:rec].sum() / capital), " total " : float (b[" total " ].sum() / capital), " years " : days / YEAR, " price0 " : b[" price0 " ], " buckets " : {k: float (b[k].sum() / capital) for k in BUCKETS}}` **Listing 8.2.** Carry, the episode by bucket, the recovery. code/strategies-2/08-convertible-arbitrage/python/s2_convarb.py
3. **Run** `summary()` , `summary(True)` , `profile()` and `fig_convarb.py` .

**What to change next.** Make the borrow recall force a partial unwind of the hedge; lever to five to one and find the episode loss that triggers a margin call; start the book at a share price of 20 and see what a [busted convertible](#def-s2-convertible-arbitrage-busted)’s hedge is made of.

## 8.7 Build: convertible arbitrage

**Purpose.** A convertible book on Book 5’s pricer with delta and credit hedges, borrow, and a forced-selling scenario.

**Interface.** `ArbConfig(…)`, `value_tables(cfg)`, `mark(tables, tau, S, stressed)`, `delta(tables, tau, S, stressed)`, `scenario(T, cfg)`, `run_book(S, cfg, credit_hedge)`.

**Rules.** Values tabulated monthly in maturity for a normal and a stressed hazard; market price is model value less cheapness; buckets add up to the total.

**Acceptance tests.** `code/firm/convarb/tests/`: the scenario’s shape; marks equal the pricer at table nodes, stress lowers the value, the delta is below the conversion ratio; buckets add up on a calm path with no credit P&L.

**Stretch.** Borrow recalls; margin calls; several issuers’ credit correlated.

Sources and further reading

- M. Mitchell, L. H. Pedersen and T. Pulvino, “Slow moving capital”, *American Economic Review* 97(2), 2007 (NBER working paper 12877).
- M. Mitchell and T. Pulvino, “Arbitrage crashes and the speed of capital”, *Journal of Financial Economics* 104(3), 2012.
- V. Agarwal, W. H. Fung, Y. C. Loon and N. Y. Naik, “Risk and return in convertible arbitrage: evidence from the convertible bond market”, *Journal of Empirical Finance* 18(2), 2011.

## 8.8 Exercises

**Exercise 8.1 ★.**

A bond converts into 2.5 shares. What is its conversion price per 100 of face, and its conversion value at a share price of 60?

**Solution of Exercise 8.1.**

$100/2.5 = 40$ a share; at 60 the conversion value is $2.5 \times 60 = 150$.

**Exercise 8.2 ★.**

Bonds cheapen by 2.7%. What does a fund levered three to one lose on its capital?

**Solution of Exercise 8.2.**

Its bonds are three times its capital, so $3 \times 2.7 = 8.1\%$ of capital, close to the $-7.2\%$ the funds reported for January to May 2005.

**Exercise 8.3 ★.**

A bond worth 81.2 has a delta of 1.27 shares at a share price of 15. What is its equity elasticity?

**Solution of Exercise 8.3.**

$1.27 \times 15/81.2 = 0.23$: a 1% fall in the share takes 0.23% off the bond.

**Exercise 8.4 ★★.**

Why can a [busted convertible](#def-s2-convertible-arbitrage-busted) be worth less than the straight bond at the issue’s credit spread?

**Solution of Exercise 8.4.**

The straight bond is discounted at the spread for the issue’s hazard at a share price of 40. At 15 the equity-to-credit link has raised the hazard, so the bond is worth less than a bond at the old spread; the conversion option is too far out of the money to make up for it.

**Exercise 8.5 ★★.**

In the episode, the bonds lost 17.0% of capital and the stock hedge gained 20.7%. Why did the hedge gain more than the bonds lost?

**Solution of Exercise 8.5.**

Convexity: as the share fell, the bond’s delta fell, so it lost less per unit of fall than the short hedge, sized at the higher delta, gained. The book is long gamma, and the crash’s large moves paid 3.7 points net.

**Exercise 8.6 ★★.**

Why is cheapening the risk that no hedge removes?

**Solution of Exercise 8.6.**

Cheapness is the gap between market price and model value, driven by who must sell and who can buy. The share, the credit default swap and the option market do not carry it, so no traded hedge offsets it; only the ability to hold until it closes does.

**Exercise 8.7 ★★★.**

*Coding.* Compare `summary()` with `summary(True)`. What does the credit hedge cost over five years, and what does it buy?

**Solution of Exercise 8.7.**

Over five years the credit hedge costs 21.8% of capital in premiums and gives back the credit rebound; it lowers the carry from 9.8% to 6.0% a year and the five-year return from 51.9% to 30.1%. It buys the episode’s 6.9 points of credit loss, halving the episode from $-11.6\%$ to $-5.8\%$, and leaves the 12.6 points of cheapening untouched.

**Exercise 8.8 ★★★.**

*Find the flaw.* “Our convertible book is delta-hedged and credit-hedged, so it has no risk and we can lever it ten to one.”

**Solution of Exercise 8.8.**

Delta and credit hedges leave cheapening, borrow and financing risk: in the chapter’s episode cheapening alone cost 12.6% of capital at three to one, so at ten to one it would cost about 42%, before any margin call. In 2005 and 2008 the risk that mattered was exactly the one the hedges left.

## 8.9 Problem: Long Everything at Once

**Problem 8.1.**

Weekend problem — a convertible arbitrage book

The chapter’s synthetic book, Book 5’s pricer and the public record.

**Part I — The bond.**

1. What does a convertible holder own?
2. Give the chapter’s bond’s value, conversion value and delta at issue.
3. Define a [busted convertible](#def-s2-convertible-arbitrage-busted) and describe the bond at a share price of 15.
4. Why does the equity-to-credit link matter for the hedge?

**Part II — The trade.**

5. Define the [convertible delta hedge](#def-s2-convertible-arbitrage-delta) and the [credit-hedged convertible](#def-s2-convertible-arbitrage-credit) .
6. What did Agarwal and co-authors find?
7. What did Mitchell, Pedersen and Pulvino find about 2005?
8. What changed in 2008?

**Part III — The book.**

9. How is the synthetic book built, and how is its P&L attributed?
10. Give its carry, with and without the credit hedge.
11. Decompose the episode’s loss.
12. What happened in the recovery?

**Part IV — The verdict.**

13. State the *named result* : the hedged book’s carry and its loss in the planted forced-selling episode, and the share from credit.
14. What does the credit hedge cost and buy?
15. Why is leverage the book’s real risk?
16. Why were arbitrageurs sellers when bonds were cheapest?
17. How would you size the book?
18. How would you backtest it honestly?
19. Which strategy file depends most on the new-issue discount?
20. In one sentence: what is a convertible arbitrageur paid for?

**Solution of Problem 8.1.**

1. A straight bond, an option on the shares, less the issuer’s call, all exposed to default.
2. 117.7 against a conversion value of 100, a delta of 1.80 shares.
3. A convertible far below its conversion price that trades on credit; at 15 it is worth 81.2, below the straight bond’s 87.34, with a delta of 1.27 and an elasticity of 0.23.
4. The hedge’s delta includes the credit effect of the share price; a hedge that ignores it is too small when the share falls.
5. As defined in the chapter: a short stock position equal to the delta; plus credit protection sized to the credit sensitivity.
6. A buy-and-hedge strategy explains much of the funds’ returns; extreme events and convertible supply matter.
7. Redemptions of more than 20% in a quarter, forced sales, a 2.7% median discount in May, $-7.2\%$ fund returns.
8. The leverage offered by prime brokers fell suddenly as they faced failure.
9. Thirty bonds at issue, short their deltas daily, levered three to one; P&L in gamma, credit, cheapness, coupons and borrow.
10. 9.8% a year delta-hedged; 6.0% credit-hedged.
11. $-11.6\%$ : cheapness $-12.6$ , credit $-6.9$ , gamma $+3.7$ , coupon $+5.3$ , borrow $-1.1$ .
12. The bonds recovered their cheapness and the hazard fell back: $+22.0\%$ in six months.
13. **Named result.** The delta-hedged book carries 9.8% a year on capital and loses 11.6% in the planted three-month episode, 6.9 points from credit and 12.6 from cheapening; hedging credit halves the loss to 5.8% at the cost of carry (6.0% a year).
14. It costs 21.8% of capital over five years and buys the 6.9 points of credit loss.
15. Cheapening is multiplied by leverage and can force selling at the worst prices.
16. They faced redemptions (2005) or lost leverage (2008): capital, not prices, decided.
17. By the loss in a forced-selling episode at the chosen leverage, against the capital that could be lost without a forced unwind.
18. Model values with realistic inputs, episodes of cheapening, borrow and financing costs that change.
19. The new-issue convertible.
20. For funding issuers quickly and holding illiquid bonds through the times when others must sell.

## 8.10 Interview questions

**Interview question 8.1 ★ trader.**

Walk through a convertible arbitrage trade.

**Solution of Interview question 8.1.**

Buy the convertible, short the delta in shares, possibly buy credit protection; earn the coupon, the short rebate less borrow, the gamma from rehedging and the convergence of a cheap bond to its value; risks are cheapening, credit, borrow and financing.

**Interview question 8.2 ★★ researcher.**

How does the link between the share price and credit change a convertible’s delta?

**Solution of Interview question 8.2.**

When the share falls the hazard rises, so the bond falls more than an option model with constant credit says: the delta is larger at low share prices, and at very low prices most of the delta is credit.

**Interview question 8.3 ★★ risk.**

A convertible book is hedged in delta and credit. What risks remain, and how would you stress them?

**Solution of Interview question 8.3.**

Cheapening, borrow recall, financing withdrawal, gamma in gaps, issuer calls and takeovers, model error in the credit input; stress with a 2005-style cheapening, a doubled hazard and a borrow recall at once, at the book’s leverage.

**Interview question 8.4 ★★ trader.**

The borrow on one of your short hedges is recalled. What do you do?

**Solution of Interview question 8.4.**

Find borrow elsewhere, hedge with options or a correlated instrument, or reduce the position; accept a less precise hedge rather than an unhedged bond, and record the cost for the book’s borrow risk.

**Interview question 8.5 ★★ developer.**

Design the daily process that marks and hedges a book of 200 convertibles.

**Solution of Interview question 8.5.**

Each day: prices of bonds, shares, credit and vols; model values and deltas from the pricer (tabulated for speed); hedge adjustments net across issuers; borrow availability; cheapness monitoring and attribution; checks on corporate actions, calls and conversions.

**Interview question 8.6 ★★★ researcher.**

Show that a delta-hedged convertible earns about $\tfrac12\Gamma S^2(\sigma_r^2 - \sigma_i^2)\,\mathrm{d}t$ plus carry, and explain what a default does to the hedge.

**Solution of Interview question 8.6.**

As for any option, the hedged position’s value changes by $\Theta\,\mathrm{d}t + \tfrac12\Gamma(\mathrm{d}S)^2$, with the bond’s coupon and the short’s rebate less borrow added as carry. At default the share jumps to near zero: the short stock gains its value and the bond falls to its recovery; with an equity-to-credit model the delta hedge has been sized for part of that jump, and credit protection covers the rest.