The Distribution of the Value of the Firm and Stochastic Interest Rates

The time evolution of the value of a firm is commonly modeled by a linear, scalar stochastic differential equation (SDE) of the type   d d d t t t v t V rV t t V W      v t  d t r v is where the coefficient in the drift term denotes the (exogenous) stochastic short term interest rate and is the given volatility of the value process. In turn, the dynamics of the short term interest rate, , are modeled by a scalar SDE. It is shown that exhibits a lognormal distribution when is a normal/Gaussian process defined by a common variety of narrow sense linear SDEs. The results can be applied to different financial situations where modeling value of the firm is critical. For example, with the context of the structural models, using this result one can readily compute the probability of default of a firm. t r