A Solution Approach to the Valuation of Contingent Claim by Utility Indifference with Unpredictable Risks

Abstract

In this article, we consider a variant of the Merton problem, where we choose a stochastic process (a risky tradable asset or a risky commodity asset) with random drift and volatility. It is well known in the literature that this leads to a stochastic optimal control problem, which allows for the derivation of a parabolic partial differential equation, known as the Hamilton-Jacobi-Bellman (HJB) equation. This equation is commonly used to evaluate certain credit instruments, such as corporate bonds and credit default swaps (CDS), which is the case in our present work. The construction or definition of the value function involves a power transformation based on the solution of a linear or semi linear parabolic equation. We will use reduced solutions of these equations to determine prices of corporate bonds and credit default swap spreads under utility indifference. This approach will not only allow us to obtain analytical results, but also provide better insights into the dynamics of credit markets. The derived formulas for the valuation of corporate bonds and credit default swaps (CDS) can be used to develop trading strategies in credit markets. Furthermore, the prediction of credit spread dynamics and default risk will enable investors to construct complex trading portfolios, such as arbitrage and hedging strategies.