Explain how the Black-Scholes-Merton model has been extended to overcome the assumption that default can only occur at maturity

What will be an ideal response?

The question is asking us how the BSM model is extended if we relax Assumption 2 (which supposes that the bond outstanding is a zero-coupon bond that matures in T years.). Thus, we want to extend the BSM model to the case where default can occur not only at maturity but at any time prior to maturity. The underlying legal principle to extend the BSM model for this situation is that there are typically covenants in a typical bond indenture granting the bondholders the right to restructure the corporation should the value of the corporate assets fall below a given amount, referred to as a default barrier. Thus what in needed are models that utilize this notion of a default barrier. Such models that do so are referred to as first-passage time models with the first such model being proposed by Black and Cox. In all of these models, a threshold is defined (default barrier) and default occurs when a corporation's asset value crosses that threshold. Default is viewed as a form of barrier option. A barrier option is a path dependent option. For such options, both the payoff of the option and the survival of the option to the stated expiration date depend on whether the price of the underlying asset reaches a specified level over the life of the option.