Understanding Hedge Fund Convertible Bond Arbitrage
Convertible bond arbitrage is a quantitative hedge fund strategy that seeks to profit from pricing discrepancies between a company's convertible debt and its underlying equity. Because convertible bonds can be mathematically decomposed into a straight corporate bond floor plus an embedded equity call option, the arbitrageur can isolate and monetize specific risk factors—chiefly implied volatility and credit spread mispricing.
1. Constructing the Delta-Neutral Hedge
The core of the strategy is immunizing the position against small directional moves in the stock. The sensitivity of the convertible bond to share price changes is defined by its hedge delta:
\(\text{Bond Delta} = \text{Conversion Ratio} \times \Delta_{\text{call}}\)
For instance, if a bond converts into 20 shares and the embedded call option has a delta of 0.50, the bond delta is 10. For every 1,000 bonds purchased, the arbitrageur sells short exactly 10,000 shares of common stock. If the stock ticks up or down by $1, the gain on the long bond roughly cancels out the loss on the short equity position.
2. Gamma Scalping & Volatility Monetization
Unlike static positions, a delta-hedged convertible portfolio possesses positive gamma (convexity). As the stock rallies, delta rises, making the position net long; the manager must short more shares at higher prices to return to delta neutrality. When the stock declines, delta drops, compelling the manager to buy back shares at lower prices.
This disciplined re-hedging process—systematically selling high and buying low—is known as gamma scalping. The expected annualized profit from gamma scalping is approximately proportional to the spread between the stock's realized volatility and the implied volatility priced into the bond:
\(\text{Annual Gamma P&L} \approx \frac{1}{2} \cdot \Gamma_{\text{portfolio}} \cdot S^2 \cdot \left(\sigma_{\text{realized}}^2 - \sigma_{\text{implied}}^2\right)\)
3. Carry Economics: Repo, Borrow & Dividends
A successful arbitrage trade requires careful management of carry frictions:
- Bond Coupon Income: The steady cash yield received semi-annually from the issuer.
- Repo Financing Expense: Prime brokers finance the long bond position via repurchase agreements (repo), charging SOFR plus a financing spread.
- Stock Borrow Fees: To maintain the short hedge, the manager pays stock borrow fees to securities lenders. For hard-to-borrow or heavily shorted growth stocks, borrow fees can spike dramatically.
- Dividend Pass-Through: The short seller must pay any cash dividends declared by the issuer to the stock lender.