Kijima, Masaaki and Yusuke Miyake, "On the Term Structure of Lending Interest Rates When a Fraction of Collateral is Recovered Upon Default", Japan Journal of Industrial and Applied Mathematics, Vol. 21, No. 1, (February 2004), pp. 35-56.
Abstract: This article provides an arbitrage-free model to evaluate the term structure of lending interest rates when a fraction of collateral is recovered upon default of the borrower. Unlike the previous literature, we assume that the value of the collateral asset fluctuates over time with certain correlation to the risk-free interest rate as well as the default hazard rate of the borrower. It is shown that the bank loan is a sum of holding a coupon-bearing bond and selling a put option, both being callable upon default. A Gaussian model is considered, as a special case, to derive an analytic expression of the appropriate lending interest rates, and some numerical example is given to demonstrate that bank loans exhibit different properties from corporate bonds.
Keywords: risk-neutral valuation, Cox process, stochastic recovery rate, Gaussian model, collateral, prepayment risk.