Prudent upper and lower valuations from the literature on arbitrage free two price economies provide risk characteristics driving required returns. The risk characteristics assess the risk of price fluctuations. The difference between the upper and lower prudent valuations can be viewed as a...

A local volatility model is enhanced by the possibility of a single jump to default. The jump has a hazard rate that is the product of the stock price raised to a prespecified negative power and a deterministic function of time. The empirical work uses a power of -1.5. It is shown how one may...