Real Options Assessment in the Time-Fractional Heston Model with Jump and Inertia

This research addresses the assessment of real options via the time-fractional Heston model, which takes into account jumps and inertia. First, we examine the existence and uniqueness of viscosity solutions leveraging the continuous model reformulated as a time-fractional Hamilton-Jacobi-Bellman (HJB) equation using Caputo fractional derivative and Rellich-Kondrachov compactness theorem. Secondly, we present a higher-order regularity result stylized with Sobolev embeddings and Caputo fractional derivative expressed using the Duhamel principle. Third, we investigate some well-known discrete approaches proving the backward martingale convergence theorem with the Caputo time-fractional derivative being approximated using the Grünwald-Letnikov scheme. Finally, we show the existence of Nash-equilibrium solution for the pricing of real options with both two players and two stopping times, which results in lengthy implications for risk management and strategic decision making.