In this paper, a new differential equation, driven by aleatory and epistemic forms of uncertainty, is introduced and applied to describe the dynamics of a stock price process. This novel class of differential equations is called uncertain stochastic differential equations(USDES) with uncertain jumps. The existence and uniqueness theorem for this class of differential equations is proposed and proved. An appropriate version of the chain rule is derived and applied to solve some examples of USDES with uncertain jumps. The differential equation discussed is applied in an American call option pricing problem. In this problem, it is assumed that the evolution of the stock price is driven by a Brownian motion, the Liu canonical process and an uncertain renewal process. MATLAB is employed for implementing the derived option pricing model. Results show that option prices from the proposed call option pricing formula increase as the jump size increases. As compared to the proposed call option pricing formula, the Black-Scholes overprices options for a certain range of strike prices and under-prices the same options for another range of exercise prices when the jump size is zero.

In practice, financial decisions are made in the context of indeterminacy. Randomness, uncertainty, and fuzziness are three basic types of indeterminacy. A multiplicity of differential equations have been designed to depict various processes powered by different kinds of indeterminacy. Among others, these differential equations include uncertain differential equations, stochastic differential equations, and fuzzy differential equations. In this study, we propose that the value of a firm can be described by an uncertain differential equation powered by a geometric canonical Liu process. Uncertain differential equations describe processes driven by uncertainty. Implementing the uncertain Liu option pricing theory, we develop and analyse a framework for valuing debt and equity for a levered firm in uncertain markets. Numerical calculations are demonstrated.

No abstract

No abstract

Uncertain stochastic calculus is a relatively new sub discipline of mathematics. This branch of mathematical sciences seeks to develop models that capture aleatory and epistemic features of generic uncertainty in dynamical systems. The growth of uncertain stochastic theory has given birth to a novel class of differential equations called uncertain stochastic differential equations (USDEs). Exact and analytic solutions to this family of differential equations are not always available. In such cases, numerical analysis provides a gateway to approximate solutions. This paper examines a Runge-Kutta method for solving USDEs. Before examining and applying the Runge-Kutta method, the paper states and proves the existence and uniqueness theorem. The Runge-Kutta method is then applied to solve an American call option pricing problem. This numerical algorithm proves to be effective and efficient because it produces almost the same results as compared to Chen's analytic formula and the classical Black-Scholes model.