Set the various arbitrage strategies side by side and outline how convertible arbitrage works.

FRM® Part II · Risk Management and Investment Management · Hedge Fund Investment Strategies · FRM2.IM.8.b

The idea

Arbitrage hedge funds exploit relative mispricing between related instruments while hedging out broad market risk, but the trades are rarely riskless. Fixed income arbitrage bets on convergence of spreads such as swap spreads, yield curve shapes and mortgage securities against Treasuries. The spreads are small, so positions are heavily leveraged. Statistical arbitrage uses models across many short-horizon positions, and capital structure arbitrage trades different securities of one issuer. Convertible arbitrage buys an undervalued convertible bond and shorts shares against it, giving a delta-hedged position that is long volatility and long gamma. It earns the coupon and gamma profits when the stock moves sharply, and pays borrow and financing costs. Main risks are credit spread widening, falling implied volatility, and forced deleveraging in illiquid markets. Exam point: the hedge depends on the bonds held, the conversion ratio and the delta. Shares = bonds × conversion ratio × delta.

Formulas

Shares to short in a delta-neutral convertible arbitrage

Shares short = Δ × N × CR

Δ: the delta of the conversion option
N: the number of convertible bonds held
CR: the conversion ratio, shares per bond

Rebalance as delta changes. Shorting the full conversion shares ignores delta; leaving out the conversion ratio under-hedges.

Gamma profit on a delta-hedged convertible position

gamma P&L = 0.5 × Γ × ΔS² × shares

Γ: gamma per share per unit of stock price
ΔS: the change in the stock price
shares: the number of underlying shares covered

Second-order, so positive for a move in either direction for a long-gamma position. Ignores time decay, financing and credit; forgetting the half or the square is the slip.

A practice question

From the question bank, with the reason each wrong option is wrong.

In a classic convertible arbitrage position, what does the fund hold?

Short the convertible bond and long the issuer's stock.Why not: Reverses the position; the strategy buys the cheap convertible and shorts the stock.
Long the convertible bond and short a number of shares of the issuer's stock based on the option delta. - correct
Long the convertible bond and long the issuer's stock for extra upside.Why not: Doubles the equity exposure instead of hedging it; no hedge is created.
Long the convertible bond and short an equal par value of the issuer's straight bonds, ignoring the stock.Why not: Describes a credit-only trade that does not hedge or capture the equity option embedded in the convertible.

The convertible contains a bond plus an equity call option. Shorting delta times the conversion ratio shares neutralises the stock exposure and leaves a position that is long volatility and long gamma.

A harder one

A convertible arbitrage fund owns 10,000 convertible bonds, each convertible into 25 shares. The current option delta is 0.60. To be delta neutral, how many shares of the stock should the fund short?

250,000Why not: Shorts the full conversion ratio and ignores the delta.
100,000Why not: Uses 1 - delta (0.40) in place of delta.
150,000 - correct
6,000Why not: Multiplies delta by the number of bonds and omits the conversion ratio.

Equity exposure per bond is delta x conversion ratio = 0.60 x 25 = 15 shares. For 10,000 bonds the hedge is 150,000 shares.

The hardest tier

A convertible arbitrage fund is delta hedged. The issuer is downgraded, its credit spreads widen sharply, its stock falls modestly, and many other arbitrage funds sell convertibles at the same time. The fund loses money despite the gain on its short stock. Which explanation best fits the loss?

Delta hedging protects against all risks, so the loss must have come from the stock rising and hurting the short position.Why not: Assumes delta neutrality removes credit and liquidity risk; also contradicts the stock having fallen.
Falling stock prices always hurt convertible arbitrage because the fund is net long the equity market overall.Why not: The position is delta hedged; the fund is not net long the equity market.
The coupon on the bond stopped accruing as soon as the downgrade occurred, which turned the position into a loss.Why not: Coupons keep accruing until default; the loss comes from the price of the bond, not a stopped coupon.
The bond lost more from wider credit spreads and illiquid selling than the short stock gained; the equity hedge ignores credit. - correct

The convertible arbitrageur is long credit through the bond floor. A spread-widening shock and forced selling lower the convertible's price by more than the short stock earns, and borrow and funding conditions can worsen at the same time.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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