Valuation of the prepayment option of a perpetual corporate loan

We investigate in this paper a perpetual prepayment option related to a corporate loan. The default intensity of the firm is supposed to follow a CIR process. Two frameworks are discussed: first a constant interest rate and a secondly a multi-regime framework where the interest rate is augmented by a liquidity factor dependent on the regime. The prepayment option needs specific attention as the payoff itself is an implicit function of the parameters of the problem and of the dynamics. We establish in the unique regime case analytic formulas for the payoff of the option; in both cases we give a verification result that allows to compute the price of the option. Numerical results that implement the findings are also presented and are completely consistent with the theory; it is seen that when liquidity parameters are very different (i.e., when a liquidity crisis occur) in the high liquidity cost regime the exercise domain may entirely disappear meaning that it is not optimal for the borrower to prepay during such a liquidity crisis. The method allows to quantify and interpret these findings.