Discontinuity in Continuous Time

It has been known for some time in the credit literature that many empirical results cannot be explained by diffusion processes alone. For an asset value that follows diffusion process, there is a minimum time required for the asset value to drop below the default threshold. Empirically observed credit spread and numerous financial crises show that default can take place at any time and instantly. Separately, the implied volatility surface observed in the financial markets also requires the possibility of a stock price jump in order to explain the steep skewness of the implied volatilities of short maturity options. In this chapter, we show how continuous time technique can be expanded to analyse and model jump processes needed to address many empirically observed phenomena.