Chapter 7 Conclusion
Estimation of probability of defaults is an important task in credit analysis and risk management of bond portfolios. An important way to monitor risk-neutral probabilities of default is to observe an appropriate spread on the market. The problem is much of the information that markets provide is the spread of traded securities. Therefore the probabilities of default directly available are the probabilities of default of bonds maturing at one specific date.
From the theory of arbitrage pricing we know that there exists a probability, equivalent to the original historical probability, that is risk-neutral for nonde- faultable bonds. In the default context we show that there exists an equivalent probability that is risk-neutral for defaultable bonds.
Foqueet al [33] presented a mathematical framework for default event mod- elling that is flexible enough to reproduce the real features of credit events in financial world. The basic idea is to work with small and large intervals sep- arately, where we assume that the mean reversion is slow or fast, and then the constant volatility model is a better approximation. They concluded that approximation methods are very efficient in capturing the effects of stochastic volatility to the first-passage model developed by Black and Cox [6] in mod- elling defaultable bonds. By using models incorporating fast and slow stochas- tic volatility factors and a combination of singular and regular perturbations techniques they obtain reasonable fits to defaultable bonds data.
Chebbi [17] in his recent paper using the first-passage model concluded that the credit spreads observed in the market are largely explained by the risk of default. The basic ideas lead to an expression of the option price which may be used to correct the Black-Scholes formula.
In spite of all these approaches and methods we still experience the market crash, which means there is a lot of research to be done in finance.
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