Application of BSDE in Standard Inventory Financing Loan

Abstract

This paper examines the issue of loans obtained by the small and medium-sized enterprises (SMEs) from banks through the mortgage inventory of goods. And the loan-to-value (LTV) ratio which affects the loan business is a very critical factor. In this paper, we provide a general framework to determine a bank’s optimal loan-to-value (LTV) ratio when we consider the collateral value in the financial market with Knightian uncertainty. We assume that the short-term prices of the collateral follow a geometric Brownian motion. We use a set of equivalent martingale measures to build the models about a bank’s maximum and minimum levels of risk tolerance in an environment with Knightian uncertainty. The models about the LTV ratios are established with the bank’s maximum and minimum risk preferences. Applying backward stochastic differential equations (BSDEs), we get the explicit solutions of the models. Applying the explicit solutions, we can obtain an interval solution for the optimal LTV ratio. Our numerical analysis shows that the LTV ratio in the Knightian uncertainty-neutral environment belongs to the interval solutions derived from the models.