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gateway/knowledge/trading/strategies/convertibles/convertible-arbitrage.md

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+---
+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

gateway/knowledge/trading/strategies/convertibles/convertible-oas.md

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+---
+description: "Simultaneously buy and sell two convertible bonds from the same issuer, long the higher option-adjusted spread and short the lower, profiting when the spreads converge."
+tags: [convertibles, arbitrage, fixed-income]
+---
+
+# Convertible Option-Adjusted Spread (OAS) Arbitrage
+
+**Section**: 12.2 | **Asset Class**: Convertibles (Hybrid: Fixed Income + Equity) | **Type**: Relative Value / Arbitrage
+
+## Overview
+This strategy simultaneously buys and sells two different convertible bonds issued by the same company. The long position is in the bond with the higher option-adjusted spread (OAS) and the short position is in the bond with the lower OAS. The trade is profitable when the two spreads converge toward each other.
+
+## Construction / Mechanics
+The price of a convertible bond is decomposed as:
+
+```
+P_C = P_B + V                (494)
+```
+
+- `P_C` = convertible bond price
+- `P_B` = straight bond price (the bond without the embedded option), computed via standard discounting of future cash flows
+- `V` = value of the conversion option (a call option on the issuer's stock)
+
+**OAS Calculation Procedure:**
+1. At the initial iteration, compute `V^(0)` using a call option pricing model with the zero-coupon government Treasury curve as the risk-free rate
+2. Check if `V^(0)` matches the market-implied option value `P_C^mkt - P_B`
+3. If not, iteratively parallel-shift the Treasury curve (e.g., using bisection) until the computed `V` equals `P_C^mkt - P_B`
+4. The parallel shift obtained is the OAS
+
+## Return Profile
+Profits when the OAS of the long bond decreases (price rises) and/or the OAS of the short bond increases (price falls), i.e., when the two spreads converge. Returns are driven by relative mispricing between the two convertibles of the same issuer, not by the absolute level of spreads or interest rates.
+
+## Key Parameters / Signals
+- **OAS differential**: the spread between the two bonds' OAS values; wider differential implies larger potential profit but also higher risk if divergence continues
+- **Same-issuer requirement**: both bonds must be from the same issuer to neutralize credit risk
+- **Convergence horizon**: the expected time for OAS convergence to occur
+
+## Variations
+- **Multi-bond basket**: extend to a basket of convertibles from the same issuer, weighting by OAS rank
+- **Cross-issuer OAS**: relax the same-issuer constraint and use credit hedges to neutralize issuer-level credit risk
+
+## Notes
+- The OAS computation requires an option pricing model for `V`; model risk affects both legs
+- The iterative parallel-shift procedure assumes the Treasury curve shape is fixed; actual curve shape changes can affect the OAS estimate
+- Liquidity mismatch between the two convertible bonds can create mark-to-market losses even when the fundamental trade thesis is correct
+- This strategy is distinct from straight convertible arbitrage (12.1): there is no stock short; both legs are bonds from the same issuer