A Stochastic Delay Model For Pricing Debt And Loan Guarantees: Theoretical Results

TL;DR

This paper develops a theoretical framework using stochastic delay differential equations to model firm and claim values, deriving a random PDE for pricing debt and loan guarantees.

Contribution

It introduces a novel stochastic delay differential equation model and derives a corresponding random PDE for pricing corporate claims.

Findings

Derived a random PDE for claim valuation

Solved the PDE for debt and guarantees in a specific setting

Provides a theoretical basis for pricing with delay effects

Abstract

We consider that the price of a firm follows a non linear stochastic delay differential equation. We also assume that any claim value whose value depends on firm value and time follows a non linear stochastic delay differential equation. Using self-financed strategy and replication we are able to derive a Random Partial Differential Equation (RPDE) satisfied by any corporate claim whose value is a function of firm value and time. Under specific final and boundary conditions, we solve the RPDE for the debt value and loan guarantees within a single period and homogeneous class of debt.