The Pricing of Options with Default Risk

This paper considers the pricing of options with default risk. The comparative statics of such options can differ from those of ordinary options, and early exercise of such American call options can be optimal. Several examples of options with default risk are considered. TRADITIONALLY, IT HAS BEEN assumed that options have no default risk. Such an assumption is reasonable when one studies options traded on an organized exchange in the U.S. Default on such options would require, as Cox and Rubinstein [5] put it, "a very sudden and strong movement in stock prices, one considerably more extreme than any on record" (p. 71). However, many options and financial assets containing option-like payoffs are sold by firms that have limited assets. For such options, default is often a possibility that must be taken seriously. In this paper, we study how options subject to default risk, which we hereinafter call vulnerable options, are priced. It turns out that many of the well-established results of the option-pricing literature (see, e.g., Smith [12] and references therein) do not hold for vulnerable options. In particular, the value of a vulnerable European option can fall with time to maturity, with the interest rate, and with the variance of the underlying asset. Furthermore, it may pay to exercise early a vulnerable American option on a non-dividend-paying asset.