Pricing vulnerable European options when the option’s payoff can increase the risk of financial distress

Abstract

We extend the results of Johnson and Stulz (Johnson, H., Stulz, R., 1987. Journal of Finance 42, 267–280) and Klein (Klein, P.C., 1996. Journal of Banking and Finance 20, 1211–1229) for valuing European options subject to the risk of financial distress on the part of the option writer. Our model incorporates a default boundary which depends on the potential liability of the written option as in Johnson and Stulz (1987), and also on the option writer’s other liabilities as in Klein (1996). As in both of these papers, the pay-out ratio in the event of financial distress is linked to the assets of the option writer, and the correlation between the assets of the option writer and the asset underlying the option is explicitly modeled. Although no analytical solution is available, we illustrate the importance of this approach through examples, which are evaluated numerically. We also develop an approximate analytical solution, which works well in most situations.