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.
- 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.