Convertible Bond Valuation, Credit Spreads, and Embedded Equity Optionality
Summary
The document examines why a convertible bond model might show a higher fair value when the issuer’s CDS spread rises, even though wider spreads imply greater default risk. One explanation uses a delta hedged position: a convertible bond paired with a short equity hedge has positive gamma, so a higher modeled jump to default probability can increase the value associated with the hedge’s payoff in the jump scenario.
An alternative intuition holds the equity price fixed while credit risk changes. The bond captures much of the equity upside through conversion, while its downside may be partly cushioned by bond recovery relative to equity loss. The text also emphasizes that if worsening credit prospects lower the equity price, the combined effect can instead reduce the convertible’s value. A second answer attributes some marginal spread sensitivity to higher implied equity volatility and the embedded call, while cautioning that this may fail for large spread shocks as the option becomes less relevant and bond discounting dominates. These are model dependent explanations, not universal price rules.
Key ideas
- A convertible bond combined with a short equity hedge can have positive gamma.
- In a jump to default model, a higher jump probability may benefit the hedged position under the stated hedge logic.
- Holding equity price fixed, greater credit risk can coincide with greater equity upside potential, which the conversion feature captures.
- If worsening credit outlook also pushes the equity price down, the convertible bond’s total value may fall.
- A volatility based explanation may apply to marginal changes but can fail when credit spreads rise sharply.
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Full text
# Why does the price of a convertible bond go up if the CDS spread goes up? # Why does the price of a convertible bond go up if the CDS spread goes up? Looking at convertible bond prices in a commercial pricing tool, which is based on a model of Black-Scholes volatility plus a Poisson process of jump to default, I noticed that increasing the spread for CDS on the issuer causes the fair price of the convertible bond to go up. Since increased credit risk suggests that there is more chance that the bond will be defaulted on, why should it make the bond more valuable? ## Answer by jwg (score 3) https://quant.stackexchange.com/a/14780 A friend gave me the following reply in terms of dynamic hedging and portfolio management: ### Quantitative justification Pricing models for a CB are based on holding the CB hedged with a short equity position. The combined portfolio has zero delta. However, it has positive gamma. To see this, consider that delta increases when the probability of conversion increases. Of course, this means that it increases when spot of the underlying equity increases. Now, a dynamically delta-hedged portfolio should be gamma-neutral except in the case of jumps. In a CB pricing model based on diffusion and a jump to default, an increase in the probability of a jump (which would happen if the CDS spreads go up) means an increased chance of making profit from being long gamma. In the event of a jump, our short equity hedge will pay off much more than the CB will lose. This explanation, while correct, is problematic for someone who isn't used to arguments based on dynamic hedging. Like the question about why we would want to hold a hedged vanilla (which has negative theta, and so steadily loses money even if nothing happens!), it seems that it should be possible for a different hedging strategy to justify a different valuation. Of course, any other portfolio will not be delta-neutral, and so will preserve some risks which are not present here. However, I tried to come up with a different way of answering this question which is accessible without understanding dynamic delta hedging. ### Intuitive justification The price of a CB goes up if credit risk goes up ceteris paribus - all other things being equal. In particular if the equity price remains the same but the CDS spread goes up, the CB will go up. If the credit risk of a companay has gone up, but the equity price is unchanged, then the potential upside of the equity must have increased by enough to compensate the increased downside risk (from the point of view of the holder of the equity). But the CB is exposed to most of the upside of an equity (because if the equity rises enough the CB will be converted), but only a fraction of the downside (because in a credit event, the equity is likely to drop to zero, before any of the recovery value of the CB is impaired). So if the increased upside and increased downside are balanced for the equity, the upside should more than compensate the downside for the convertible bond. On the other hand, if the credit risk of the issuer increases in isolation of any other changes to its outlook, the CDS spread should rise and the equity price should fall. The combined effect on the price of the CB should be that it falls. ## Answer by Nicholas (score 0) https://quant.stackexchange.com/a/24709 Intuitive explanation: CDS spread goes up which results in an increased uncertainty in credit quality which itself results in the increase in equity volatility. CB is a debt instrument with an embedded call option, when you are long call option you are long gamma (volatility of underlying). Such relationship would generally work for marginal changes in CDS spread, and most likely will break for drastic increase in CDS spread as the CB becomes more out-of-the money and will act more as bond discounted at higher rates.
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