Convertible Arbitrage
Convertible arbitrage is a hedge fund strategy that exploits perceived mispricings in convertible securities by typically buying the convertible bond (which contains embedded equity optionality) and selling short the underlying stock, profiting from the complexity premium and optionality embedded in convertibles that the market may mis-price relative to a theoretical fair value derived from option pricing models.
Key takeaways
- The core trade: long convertible bond (debt + embedded call option on equity) + short the underlying equity (to delta hedge the option component).
- Profit sources include: carry (coupon income minus short rebate), Vega (gains from rising implied volatility), Gamma (gains from rebalancing the hedge on large stock moves), and credit spread tightening.
- Convertible bonds are complex instruments — issuers are often small/mid-cap credits, with thin secondary market liquidity — creating persistent pricing inefficiencies.
- The strategy is short credit risk (convertible is a bond; default hurts the long) and long equity volatility through the embedded option.
- Convertible arb suffered severe losses in 2008 when both credit spreads widened and equity volatility spiked while liquidity dried up, causing forced deleveraging at the worst possible moment.
Explanation
A convertible bond is a hybrid security — a corporate bond with an embedded call option allowing the holder to convert the bond into a fixed number of shares at a predetermined conversion price. Its theoretical value is:
Convertible Bond Value = Straight Bond Value + Option Value (embedded call)
The embedded option's value can be modeled using a binomial tree or lattice model that accounts for credit risk (the bond floor can decline if the issuer's credit deteriorates), equity volatility (higher volatility increases option value), and any call provisions the issuer holds. In practice, convertibles are often mispriced because they fall between the domains of fixed income and equity analysts, with neither group having complete expertise in the other's dimension.
The mechanics of a convertible arbitrage trade begin with purchasing the convertible and computing its delta — the sensitivity of the convertible's price to changes in the underlying stock. To neutralize directional equity exposure, the trader shorts stock in proportion to the delta. If the convertible has a delta of 0.45 (each $1 increase in stock increases the convertible by $0.45), the trader shorts 0.45 shares for each $1 of par value in convertibles held.
The strategy generates carry from the net income: coupon received on the convertible minus the cost of borrowing shares to short (the short rebate in the prime brokerage account). Because convertible issuers tend to be speculative-grade or unrated companies paying higher coupons, the carry component can be meaningful (often 3–6% annually). Additionally, the Gamma component — periodically rebalancing the delta hedge as the stock price moves — generates systematic profits proportional to actual realized volatility minus implied volatility: if actual stock moves are larger than the implied volatility priced into the embedded option, rebalancing generates gains.
The key risks are credit risk (convertible bond defaults), liquidity risk (convertible bonds have wide bid-ask spreads and limited secondary market depth), forced deleveraging (many players use prime broker leverage of 3–5x; margin calls force simultaneous unwinding), and volatility regime changes (a crash in implied volatility while holding long Vega loses money despite market turmoil).
Formula
Convertible Value = Straight Bond Value + Embedded Call Option Value | Delta Hedge: Shares Short = Delta × (Par / Conversion Price)
Example
A convertible arb fund buys $10 million par of XYZ Corp's 3.50% convertible notes due 2027, convertible at $45/share (current stock price: $38). The bond is priced at 95 (95% of par = $9.5M market value), with a theoretical value of 92 (straight bond floor) + 6 (option value) = 98. The bond appears undervalued relative to model by ~3 points ($300,000). The embedded call option has a delta of 0.38, so the fund shorts 38,000 shares of XYZ (10,000,000/45 × 0.38 × $45/share ≈ 38,000 shares at $38) as the equity hedge. Annual carry: 3.50% coupon on $10M = $350,000 minus short borrow cost of 0.50% × $38 × 38,000 = $28,880, net carry = ~$321,000. Over the next quarter, XYZ stock rises 10% to $41.80. The fund captures Gamma income by rebalancing its delta hedge three times, netting approximately $85,000. The convertible also tightens from 95 to 98 (approaching theoretical value), generating $300,000 in mark-to-market appreciation.
Related terms
Arbitrage Bond Borrow Cost Call Option Convertible Bond Corporate Bond Credit Risk Dedicated Short Bias Deleveraging Delta Delta Hedge Equity