Option prices contain forward looking information about stock price volatility and, potentially, the probability of bankruptcy. We develop a risk-neutral density (RND) model consisting of a mixture of two lognormal densities with a probability of bankruptcy. We calibrate this model to daily...

In this paper, we derive a closed-form explicit model-free formula for the (Black-Scholes) implied volatility. The method is based on the novel use of the Dirac Delta function, corresponding delta families, and the change of variable technique. The formula is expressed through either a limit or...