Convertible Bond Arbitrage for Quants

Convertible bond arbitrage is the classic hedge-fund relative-value trade: buy a convertible bond, short the underlying stock to hedge equity exposure, and earn the bond's yield and embedded optionality while neutralizing delta. In clean theory it is a volatility and credit package. In practice it is a multi-factor book with borrow, liquidity, and crowding risk. This article lays out the math, the Greeks that matter, and why 2008 and 2020 hurt convert arb more than textbooks predict.

The position

long convertible bond
short delta × shares of stock

The convertible is a corporate bond plus a call on the issuer's equity (roughly). Fair value depends on:

• Interest rates and credit spread of the issuer
• Equity price and implied/realized vol
• Conversion ratio, call features, put features, and dilution

convert_value ≈ bond_floor + option_value - credit_adjustment

When the package trades cheap to a model, buy convert / short stock. When rich, reverse (harder: shorting converts is often constrained).

Delta hedging and gamma

Delta of the convertible rises with stock price (more equity-like when deep in-the-money). Continuous hedging creates a gamma profile similar to long options:

hedge_shares = -delta(S, σ, r, credit, T)

Rehedging when delta moves:

• Rising stock → buy more stock? No: delta increases → short more shares
• Falling stock → delta falls → cover shorts (buy stock)

You are long gamma if you hedge dynamically — you buy low and sell high on the hedge if realized vol exceeds the vol implied in the convert price. That is the same economic story as gamma scalping.

def convert_delta_approx(stock: float, strike: float, vol: float,
                         ttm: float, rate: float) -> float:
    """Black-Scholes call delta as a crude convert proxy (ignores credit)."""
    import math
    if ttm <= 0 or vol <= 0:
        return 1.0 if stock >= strike else 0.0
    d1 = (math.log(stock / strike) + (rate + 0.5 * vol**2) * ttm) / (vol * math.sqrt(ttm))
    # Φ(d1) via erf
    return 0.5 * (1 + math.erf(d1 / math.sqrt(2)))

Real desks use convertible-specific models (tree/lattice with credit) — BS is a starting intuition only.

Credit and the bond floor

When equity crashes, the convert approaches bond floor (straight debt value). If credit spreads blow out at the same time — as in 2008 — the floor falls too. Long convert / short stock then loses on both legs: stock short profits, but convert drops more than delta predicts because credit widened.

Shock	Convert	Short stock	Net
Equity up, vol flat	Up	Loss on short	Near flat if delta-hedged
Equity down, credit stable	Down less than delta	Profit on short	Near flat / long gamma
Equity down + credit blowout	Down hard	Profit	Often large net loss
Vol crush	Option value down	Flat	Loss

Credit is not a residual — it is a first-class risk. Hedge with CDS when liquid, or size for tail credit scenarios.

Sources of edge

1. Cheap vol — convert implies lower vol than options / historical realized
2. New issue premium — primary market often offers cheapness vs secondary
3. Hard-to-borrow — stock borrow cost eats hedge returns (short selling)
4. Liquidity premium — converts trade wide; you get paid for providing capital
5. Forced selling — other convert funds delever into your bid

Edge #3 is why paper Sharpes die: borrow fees and buy-ins are not in naive backtests.

Implementation checklist

def convert_arb_pnl_day(bond_pnl: float, stock_pnl: float,
                        borrow_fee: float, financing: float,
                        hedge_error: float) -> float:
    """Daily P&L with the costs that matter."""
    return bond_pnl + stock_pnl - borrow_fee - financing - abs(hedge_error)

Operational realities:

• Corporate actions — conversion notices, calls, puts, dividends on short
• Asymmetric liquidity — selling the convert can take days
• Prime broker — margin, locate, and recall risk on the short
• Model risk — wrong credit curve → wrong delta → inventory risk

Capacity and crowding

Convert arb AUM is limited by the convertible universe size and borrow availability. When the strategy is crowded, cheapness compresses and crash correlation rises — everyone owns the same names, same hedges.

Stress test with:

• Equity -20%, credit +300 bps simultaneous
• Borrow recall on top 10 holdings
• Bid-ask widening 2–3× on converts

Relation to other arb

Convert arb sits between merger arb (event credit/equity) and vol arb (gamma/vega). It is not classical arb — residual risks remain.

Key takeaways

• Convert arb = long convert, short delta stock — earn carry + cheap vol
• Credit shocks break the hedge; model credit explicitly
• Borrow and liquidity costs dominate live P&L vs paper
• Crowding and deleveraging create correlated drawdowns
• Stress equity + credit + borrow jointly, not one factor at a time