Linetsky, Vadim, "Pricing Equity Derivatives Subject To Bankruptcy", Mathematical Finance, Vol. 16, No. 2, (April 2006), pp. 255-282.
Abstract: We solve in closed form a parsimonious extension of the Black-Scholes-Merton model with bankruptcy where the hazard rate of bankruptcy is a negative power of the stock price. Combining a scale change and a measure change, the model dynamics is reduced to a linear stochastic differential equation whose solution is a diffusion process that plays a central role in the pricing of Asian options. The solution is in the form of a spectral expansion associated with the diffusion infinitesimal generator. The latter is closely related to the Schrödinger operator with Morse potential. Pricing formulas for both corporate bonds and stock options are obtained in closed form. Term credit spreads on corporate bonds and implied volatility skews of stock options are closely linked in this model, with parameters of the hazard rate specification controlling both the shape of the term structure of credit spreads and the slope of the implied volatility skew. Our analytical formulas are easy to implement and should prove useful to researchers and practitioners in corporate debt and equity derivatives markets.
Keywords: bankruptcy, credit risk, hazard rate, credit spread, stock options, implied volatility skew, Asian options, Brownian exponential functionals, Schrödinger operator with Morse potential, spectral expansions.