A Generative Model-based Estimation of Option Prices and Greeks, and Forecasting VIX using Interpretable Neural Networks = 생성 모형 기반의 옵션 가격 및 그릭스 추정과 해석 가능한 신경망을 활용한 VIX 예측

다국어 초록 (Multilingual Abstract)

The accurate and efficient estimation of option prices and Greeks is crucial for effectively managing and hedging risks in financial derivatives.
Traditional approaches often struggle with the analytical and numerical complexities involved, especially in sophisticated option pricing models.
For example, pricing exotic options under non-Markovian models may not even be formulated as PDEs, and thus relies on simulation schemes.
This dissertation introduces a deep learning-based framework utilizing generative neural networks, specifically conditional normalizing flows, to estimate option prices and Greeks. Using these generative networks, two approaches are proposed: simulation-based and integration-based approaches. These methods are universally applicable across a variety of models-from the simple Black-Scholes model to complicated non-Markovian models with rough volatility- and a wide range of option types$-$including vanilla options, cliquet-style options, and multi-asset options.
Experimental results demonstrate that our proposed methods provide more accurate and reliable results compared to a feed-forward neural network method.
Furthermore, our approach can compute option prices and Greeks thousands to hundreds of thousands of times faster than traditional Monte-Carlo methods with similar accuracy.
This significant acceleration, while maintaining high accuracy, highlights the considerable potential of our methods to enhance both the efficiency and scalability of financial derivatives computation.

Next topic presents the use of Kolmogorov-Arnold Networks (KANs) for forecasting the CBOE Volatility Index (VIX).
Unlike traditional MLP-based neural networks that are often criticized for their black-box nature, KAN offers an interpretable approach via learnable spline-based activation functions and symbolification.
Based on a parsimonious architecture with symbolic functions, KAN expresses a forecast of the VIX as a closed-form in terms of explanatory variables, and provide interpretable insights into key characteristics of the VIX, including mean reversion and the leverage effect.
Through in-depth empirical analysis across multiple datasets and periods, we show that KANs achieve competitive forecasting performance while requiring significantly fewer parameters compared to MLP-based neural network models.
Our findings demonstrate the capacity and potential of KAN as an interpretable financial time-series forecasting method.

Together, these two studies, while addressing distinct financial modeling tasks, contribute to the development of AI-based models in finance.
The first study highlights the potential of generative neural networks for the fast and accurate estimation of option prices and Greeks across a variety of model settings.
The second study demonstrates the effectiveness of Kolmogorov–Arnold Networks in forecasting volatility with interpretability, revealing structural insights such as mean reversion and the leverage effect.
By tackling two challenges in financial modeling—derivatives pricing and volatility forecasting—this dissertation offers a cohesive framework that achieves both practical performance and conceptual clarity.

목차 (Table of Contents)

• 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
• 2 A generative neural network-based approach for efficient es-
• timation of option prices and Greeks . . . . . . . . . . . . . . . . . . . . . . . . . 14
• 2.1 Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
• 2.1.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
• 2.1.2 Option types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
• 2.1.3 Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
• 2.2 Conditional normalizing flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
• 2.2.1 Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
• 2.2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
• 2.2.3 Training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
• 2.2.4 Batch-normalization pretraining . . . . . . . . . . . . . . . . . . . . . 23
• 2.3 Goodness-of-fit test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
• 2.4 Option prices and Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
• 2.4.1 Option price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
• 2.4.2 Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
• 2.4.3 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
• 3 Forecasting VIX using interpretable Kolmogorov-Arnold net-
• works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
• 3.1 Kolmogorov-Arnold networks: representation, structure, and
• training . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
• 3.1.1 Kolmogorov-Arnold representation theorem . . . . . . . . . . 46
• 3.1.2 Kolmogorov-Arnold networks . . . . . . . . . . . . . . . . . . . . . . . 47
• 3.2 VIX forecasting using KAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
• 3.2.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
• 3.2.2 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
• 4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
• Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
• 4.1 The Greeks for a vanilla put option under the BS model . . . . . 80
• Appendix for Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
• 4.2 Training data and hyperparameters used for experiments . . . . . 80
• 4.2.1 Training data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
• Appendix for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
• 4.3 Analyses on hyperparameter configurations . . . . . . . . . . . . . . . . 82
• 4.3.1 Impact of depth and width in KAN . . . . . . . . . . . . . . . . . . 82
• 4.3.2 Impact of grid size on activation functions in KAN . . . . 83
• 4.3.3 Impact of node and edge pruning thresholds on KAN
• structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
• 4.3.4 Impact of regularization parameters on learned acti-
• vation functions in KAN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
• 4.3.5 Hyperparameter configurations explored for the bench-
• mark neural networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
• Appendix for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
• 4.4 KAN training result before and after symbolification in Pe-
• riod 1 and Period 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
• References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
• Abstract in Korean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106