Parameter Calibration

Analytic estimation of parameters of stochastic volatility diffusion models with exponential-affine characteristic function for currency option pricing ArXiv ID: 2507.11868 “View on arXiv” Authors: Mikołaj Łabędzki Abstract This dissertation develops and justifies a novel method for deriving approximate formulas to estimate two parameters in stochastic volatility diffusion models with exponentially-affine characteristic functions and single- or two-factor variance. These formulas aim to improve the accuracy of option pricing and enhance the calibration process by providing reliable initial values for local minimization algorithms. The parameters relate to the volatility of the stochastic factor in instantaneous variance dynamics and the correlation between stochastic factors and asset price dynamics. The study comprises five chapters. Chapter one outlines the currency option market, pricing methods, and the general structure of stochastic volatility models. Chapter two derives the replication strategy dynamics and introduces a new two-factor volatility model: the OUOU model. Chapter three analyzes the distribution and surface dynamics of implied volatilities using principal component and common factor analysis. Chapter four discusses calibration methods for stochastic volatility models, particularly the Heston model, and presents the new Implied Central Moments method to estimate parameters in the Heston and Schöbel-Zhu models. Extensions to two-factor models, Bates and OUOU, are also explored. Chapter five evaluates the performance of the proposed formulas on the EURUSD options market, demonstrating the superior accuracy of the new method. The dissertation successfully meets its research objectives, expanding tools for derivative pricing and risk assessment. Key contributions include faster and more precise parameter estimation formulas and the introduction of the OUOU model - an extension of the Schöbel-Zhu model with a semi-analytical valuation formula for European options, previously unexamined in the literature. ...

A Space Mapping approach for the calibration of financial models with the application to the Heston model ArXiv ID: 2501.14521 “View on arXiv” Authors: Unknown Abstract We present a novel approach for parameter calibration of the Heston model for pricing an Asian put option, namely space mapping. Since few parameters of the Heston model can be directly extracted from real market data, calibration to real market data is implicit and therefore a challenging task. In addition, some of the parameters in the model are non-linear, which makes it difficult to find the global minimum of the optimization problem within the calibration. Our approach is based on the idea of space mapping, exploiting the residuum of a coarse surrogate model that allows optimization and a fine model that needs to be calibrated. In our case, the pricing of an Asian option using the Heston model SDE is the fine model, and the surrogate is chosen to be the Heston model PDE pricing a European option. We formally derive a gradient descent algorithm for the PDE constrained calibration model using well-known techniques from optimization with PDEs. Our main goal is to provide evidence that the space mapping approach can be useful in financial calibration tasks. Numerical results underline the feasibility of our approach. ...