Abstract
In this paper we provide a stock price model that explicitly incorporates credit risk, under a stochastic optimal control system. The stock price model also incorporates the managerial control of credit risk through a control policy in the stochastic system. We provide explicit conditions on the existence of optimal feedback controls for the stock price model with credit risk. We prove the continuity of the value function, and then prove the dynamic programming principle for our system. Finally, we prove the Viscosity Solution of the Hamilton–Jacobi–Bellman equation. This paper is particularly relevant to industry, as the impact of credit risk upon stock prices has been prominent since the commencement of the Global Financial Crisis.
Highlights
The Global Financial Crisis demonstrated the importance of credit risk in stock price models: many firms with high credit risk experienced highly volatile price moves, with some firms declaring bankruptcy
In this paper we provide a stock price model that explicitly incorporates credit risk, under a stochastic optimal control system
We provide explicit conditions on the existence of optimal feedback controls for the stock price model with credit risk
Summary
The Global Financial Crisis demonstrated the importance of credit risk in stock price models: many firms with high credit risk experienced highly volatile price moves, with some firms declaring bankruptcy (see for instance Haas & Horen, 2012; Ouenniche, 2017; du Jardin, 2019; Affes & Hentati-Kaffel, 2019). Given the importance of credit risk in stock price processes, especially since the start of the Global Financial Crisis, we would like to have a stock price model that directly incorporates credit risk dynamics. Whilst many stochastic differential equation models exist for stock prices, the incorporation of credit risk is limited and so does not provide a comprehensive model for credit risk (such as jump-diffusion models). We propose a new stock price model that incorporates credit risk dynamics by formulating a SDE (stochastic differential equation) that has a discrete process t , which is a continuous time, random jump process on a finite state space. In addition to modelling the credit risk dynamics in our stock price model, another distinguishing feature of our model is the incorporation of a decision or control variable which reflects management decisions to manage credit risk. The proceeding section details the proofs of all our Theorems and Proposition, and we end with a conclusion
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