---
description: "Buy an undervalued convertible bond and short the underlying stock using a delta-based hedge ratio to capture the mispricing between the convertible's market price and its fair value."
tags: [convertibles, arbitrage]
---

# Convertible Arbitrage

**Section**: 12.1 | **Asset Class**: Convertibles (Hybrid: Fixed Income + Equity) | **Type**: Arbitrage

## Overview
A convertible bond is a hybrid security with an embedded option to convert the bond to a preset number of the issuer's shares (the conversion ratio) when the stock price reaches the conversion price. Empirically, convertibles at issuance tend to be undervalued relative to their fair value, creating arbitrage opportunities. The strategy buys the convertible bond and simultaneously shorts the underlying stock to hedge the equity exposure.

## Construction / Mechanics
The hedge ratio (number of shares to short) is:

```
h = Δ × C                    (492)
Δ = ∂V/∂S                   (493)
```

- `C` = conversion ratio (number of shares per bond)
- `V` = value of the conversion option (model-dependent)
- `S` = underlying stock price
- `Δ` = option delta (model-dependent)

The position is typically held for 6–12 months starting at the convertible's issuance date. The hedge ratio is updated daily as delta changes with the stock price.

## Return Profile
Profits when the market price of the convertible converges toward its theoretical fair value. The long convertible position captures the undervaluation premium. The short stock position hedges directional equity risk, leaving exposure primarily to the convergence of the mispricing.

## Key Parameters / Signals
- **Conversion ratio** `C`: fixed at issuance
- **Delta** `Δ = ∂V/∂S`: requires a model for the conversion option value `V`; changes daily with stock price `S`
- **Gamma hedging**: since delta itself changes with `S`, the option gamma can be used to refine dynamic hedging (see Section 7.4.1)
- **Entry timing**: position typically initiated at issuance when undervaluation is most pronounced

## Variations
- **Gamma hedging overlay**: use gamma to dynamically adjust the hedge ratio as the stock moves, capturing additional convexity profits

## Notes
- Hedge ratios are model-dependent; model risk is a key concern
- Nonparametric hedge estimation using historical data (constrained regression of MBS price P w.r.t. swap rate R) is an alternative to model-based delta
- Liquidity risk: convertible bonds are less liquid than the underlying stock
- Crowding risk: convertible arbitrage is a well-known strategy; forced unwinds by other funds can cause losses