An Integral Equation Approach for the Valuation of Finite-maturity margin-call Stock Loans

TL;DR

This paper introduces an integral equation method using Fourier transforms to efficiently price finite-maturity margin-call stock loans, providing a reliable numerical scheme for valuation and optimal exit decision analysis.

Contribution

It develops a novel integral equation approach that simplifies the pricing of margin-call stock loans under the Black-Scholes-Merton model, enabling efficient computation of contract values and exit prices.

Findings

The proposed numerical scheme is reliable and efficient.

It accurately computes service fees and contract values.

The method quantifies the impact of margin-call features.

Abstract

This paper examines the pricing issue of margin-call stock loans with finite maturities under the Black-Scholes-Merton framework. In particular, using a Fourier Sine transform method, we reduce the partial differential equation governing the price of a margin-call stock loan into an ordinary differential equation, the solution of which can be easily found (in the Fourier Sine space) and analytically inverted into the original space. As a result, we obtain an integral representation of the value of the stock loan in terms of the unknown optimal exit prices, which are, in turn, governed by a Volterra integral equation. We thus can break the pricing problem of margin-call stock loans into two steps: 1) finding the optimal exit prices by solving numerically the governing Volterra integral equation and 2) calculating the values of margin-call stock loans based on the obtained optimal exit prices. By validating and comparing with other available numerical methods, we show that our proposed numerical scheme offers a reliable and efficient way to calculate the service fee of a margin-call stock loan contract, track the contract value over time, and compute the level of stock price above which it is optimal to exit the contract. The effects of the margin-call feature on the loan contract are also examined and quantified.