Convertible Arbitrage

A convertible bond is two things wearing one price tag: an ordinary corporate bond that pays coupons, plus an embedded option to swap the bond for a fixed number of the company's shares. Convertible arbitrage is the trade that pulls those two things apart. You buy the convertible, then short just enough of the stock to cancel the bond's sensitivity to the share price. What's left over is the option, and if the market sold it to you cheaply, you get paid for holding it.

The reason this works is that convertibles are often underpriced. They are sold in a hurry by companies raising money, held by investors who care about the bond, not the option, and are fiddly to value. So the embedded call frequently trades at a lower Implied Volatility than the same option would fetch on the listed market. The arbitrageur buys that cheap volatility and hedges away everything else.

The delta hedge

The convertible's price moves with the stock, but not one-for-one. Its delta is how many cents the convertible gains when the stock rises a dollar, scaled by the conversion ratio. To neutralise direction you short

$$\text{shares per bond} = \Delta \times (\text{conversion ratio}),$$

where $\Delta$ is the option delta (between 0 and 1) and the conversion ratio is how many shares one bond converts into. Hedged this way, small stock moves in either direction wash out of the profit and loss. What does not wash out is convexity: as the stock rises the convertible's delta rises too (positive gamma), so the hedge is always slightly wrong in your favour.

Worked example: the long-gamma engine

Take a convertible with face value $1,000, conversion ratio 20 (so it converts into 20 shares), while the stock trades at $40. Parity is $20 \times 40 = 800$ dollars, but the convertible trades at $1,020 — above both the bond floor and parity, because the option has value. Its delta is 0.45, so you short

$$0.45 \times 20 = 9 \text{ shares per bond},$$

worth $9 \times 40 = 360$ dollars.

• Stock rises 10% to $44. Parity climbs to $880, and because delta rose with it the convertible gains about $42, to $1,062. Your short loses $9 \times 4 = 36$ dollars. Net: $+42 - 36 = +6$ dollars.
• Stock falls 10% to $36. The bond floor cushions the drop, so the convertible loses only about $34, to $986. Your short gains $9 \times 4 = 36$ dollars. Net: $-34 + 36 = +2$ dollars.

Either direction makes money on the hedge — that is what "long gamma" means. Add the bond's coupon carry on top. You then re-hedge to the new delta (about 0.55 after the rise, so short two more shares) and wait for the next wiggle. The strategy profits whenever the stock's realised volatility beats the implied volatility you paid for the embedded option — exactly the logic of Gamma Scalping and Volatility Arbitrage.

Where it bites back

The hedge kills equity direction, but a convertible carries other risks that no share short can cancel.

• Credit risk. A convertible is still a bond. If the issuer's credit deteriorates, the bond floor itself sinks (a "busted convert"), and the equity hedge does nothing for that. Serious desks hedge credit separately with credit default swaps.
• Liquidity and funding. Convertibles are thinly traded and the strategy uses leverage. In 2008 convert-arb funds were forced to sell into a market with no bids, and the trade lost roughly a third of its value in weeks even though the "arbitrage" was sound.
• Financing the short. You must borrow the stock; hard-to-borrow names carry high fees that eat the edge, and a recall can force you to unwind at the worst moment.

Convertible arbitrage is the tidiest real-world example of separating an instrument into pieces and trading only the mispriced one. It sits alongside Merger Arbitrage and Statistical Arbitrage in the hedge-fund toolkit: a strategy that looks market-neutral on paper but earns its return by warehousing a specific, less-obvious risk.