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The paper aims to develop an algorithm to solve the inverse problem of identifying the volatility with given point measurements in a regime-switching economy. This value is of a particular importance since it is one of the main ingredients forming the option premia. After a review on the existing results for the classical Black–Scholes framework (equivalent to one regime), the direct and the inverse problems considering two regimes are defined. The forward problem is solved by applying the Simpson’s rule to an existing exact formula, while the inverse problem is tackled using a root-finding algorithm, namely Levenberg–Marquardt. What is more, a closed-form approximation formula for computing the implied volatility is proposed when at-the-money case is assumed. The numerical performance of the algorithms is studied and they are compared with a FDS approach. The paper is closed with a discussion and final remarks outlining the usefulness and applicability of the results. [ABSTRACT FROM AUTHOR]

We study the numerical approximation of the implied volatility for models in option pricing that generalize the Black–Scholes equation when the processes which generate the underlying stock returns may contain both continuous part and jumps. Here we propose weighted positive difference schemes with implicit-explicit iterations to solve the integral terms explicitly. We aim to identify the volatility from extra point measurement. First, we linearize in time the diffusion quadratic nonlinear term. Next, we employ a special decomposition of the approximate solution. Numerical tests are presented to show the computational efficiency of the algorithms. [ABSTRACT FROM AUTHOR]

This paper is devoted to develop a robust numerical algorithm to solve inverse problems of determining the right boundary conditions according to measurements inside a truncated domain for regime-switching models of European options. Difference schemes on Tavella-Randall grids are derived. We propose and discuss results of computational experiments for a several European options. [ABSTRACT FROM AUTHOR]