Callable Stock Loans

In a non-recourse collateralized loan agreement, the lender's recovery is only limited to the market value of the asset as a collateral. In practice, a loan lender can protect himself against any loss by introducing either a margin requirement or a callable feature in the loan contract. In this respect, it is natural to ask: (1) which one of the two mentioned features should be more preferable to the loan manager; (2) how the borrower reacts towards either one of these two features. In the present work, we address these issues by solving explicitly the respective valuation problems of collateralized lending with margin requirement and callable feature under the Black-Scholes model. For the callable feature, we also provide systematic discussions on (1) how to identify whether the smooth-fit condition holds or fails at the optimal stopping boundaries of the associated Dynkin game, and (2) how to solve it when the smooth-fit condition fails at one or both boundaries. We finally utilize these explicit solutions to conduct detailed comparative analysis.