Paper: arXiv 2507.11868

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.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 4.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper is mathematically dense, featuring advanced stochastic calculus, PDEs, and analytic characteristic function derivations for new models like OUOU, but its empirical testing is limited to a single currency pair (EURUSD) with no access to code, live data pipelines, or robust backtesting frameworks for trading.

Research Flowchart

  flowchart TD
  A["Research Goal:<br>Analytic Estimation of<br>Stochastic Volatility Parameters"] --> B["Methodology:<br>Implied Central Moments &<br>Exponential-Affine Characteristic Functions"]
  B --> C["Data:<br>EURUSD Options Market<br>Sample: 2007-2009"]
  C --> D["Computational Process:<br>Calibrate Heston, Schöbel-Zhu, Bates, & OUOU Models"]
  D --> E{"Analysis & Validation"}
  E --> F["Key Findings:<br>1. Novel analytic formulas for parameters k, ρ, σ<br>2. Superior calibration accuracy vs. standard methods<br>3. Introduced OUOU model with semi-analytical formula"]
  E --> G["Outcome:<br>Enhanced pricing accuracy & faster<br>calibration for currency options"]

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0 of 133 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.