The gamma diagrams in Figures 3 and 5 have a frown shape. The gammas are the highest when the convertibles are at-the-money. It is intuitive that when the stock prices rise or fall, profits increase because of favorably changing deltas. For this reason, convertible bonds are very good candidates for delta neutral hedging. Relatively large positive gammas of convertibles could be one of the main drivers of profitability in convertible arbitrage.
  1. Conclusion
This paper aims to value hybrid financial instruments (e.g., convertible bonds) whose values may simultaneously depend on different assets subject to credit risk in a proper and consistent way. The motivation for our model is that if a company goes bankrupt, all the securities (including the equity) of the company default. The recovery is realized in accordance with the priority established by the Bankruptcy Code. In other words, different securities have the same probability of default, but different recovery rates.
Our study shows that risky asset pricing is quite different from risk-free asset pricing. In fact, the expectation of a defaultable asset actually grows at a risky rate rather than the risk-free rate. This conclusion is very important for risky valuation.
We propose a hybrid framework to value risky equities and debts in a unified way. The model relies on the probability distribution of the default jump rather than the default jump itself. The model is quite accurate for pricing convertible bonds.
Empirically, we do not find evidence supporting a systematic underpricing hypothesis. We also find that convertible bonds have relatively large positive gammas, implying that convertible arbitrage can make a profit on a large upside and downside movement in the underlying stock price.