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Vulnerable Options, Risky Corporate Bond and Credit Spread

The focus of credit risk analysis has been either on the valuation of risky corporate bonds and credit spreads, or on the valuation of vulnerable options, but never on both in the same context. Risky corporate bonds and credit spreads have been modelled by Black and Scholes (1973) and Merton (1974), and extended by Black and Cox (1976), Longstaff and Schwartz (1995), Briys and de Varenne (1997), and others. Studies of vulnerable options are pioneered by Johnson and Stulz (1987), and subsequently advanced by Hull andWhite (1995), and Jarrow and Turnbull (1995). The existing studies can be improved in several aspects.

First, corporate bonds and credit spreads are generally analyzed in a context where corporate debt is the only liability of the firm and firm's value follows a continuous stochastic process. The obvious implication of this set-up is a zero short term credit spread, which is strongly rejected by empirical observations (see Fons 1994, Jones, Mason and Rosenfeld 1984, Sarig and Warga 1989). Since a corporation generally has more than one type of liabilities and one maturing liability may cause a sharp decrease in firm's value, modelling multiple liabilities may help to incorporate discontinuity in firm's value and hence lead to non-zero short term credit spreads.

Second, vulnerable options are priced under either the assumption that the option is the only liability of the firm (e.g., Johnson and Stulz (1987)), or the assumption that the option is fully paid off if the firm's value is above the default barrier at option's maturity (e.g. Hull and White (1995)). Both assumptions are questionable. On the one hand, most firms have debts in their capital structure and contingent liabilities (in the form of derivatives) are only part of the total liability. On the other hand, it is not appropriate to evaluate an option's vulnerability by focusing only on technical solvency since a corporation can find itself in a solvent position at option's maturity but with assets insu±cient to payoff the option.

To illustrate, suppose the value of the firm's assets until option's maturity has always been above the default barrier which is $40. Further suppose that the firm's value is $50 at option's maturity. If the option is $20 in the money, the firm is "threshold-solvent" but unable to pay $20 in full to the option holder. The downfall of Barings Bank serves as a convincing illustration the default on debentures was purely due to the large loss on derivatives positions.

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Vulnerable Options, Risky Corporate Bond and Credit Spread