Abstract¶
This research explores the implementation of two algorithms for options pricing in Lévy models, specifically comparing the Normal Inverse Gaussian (NIG) process with the Inverse Transform Method. Our implementation builds upon the foundational work by Feng et al. on simulating Lévy processes from their characteristic functions.
Research Implementation and Analysis¶
This research proposal outlines our comprehensive study of options pricing methodologies within Lévy models, specifically focusing on the implementation and comparison of two distinct algorithmic approaches.
Lévy Process Diagram - Illustration of Lévy process paths and their application in options pricing
Key Research Components¶
Normal Inverse Gaussian (NIG) Process¶
Implementation of advanced stochastic processes for more accurate market modeling. The NIG process captures important stylized facts of financial markets including:
- Heavy tails in return distributions
- Volatility clustering
- Asymmetric returns
Inverse Transform Method¶
Comparative analysis of computational efficiency and pricing accuracy through:
- Direct simulation from characteristic functions
- Optimized numerical integration techniques
- Performance benchmarking against traditional methods
Theoretical Foundation¶
Building upon the seminal work by Feng et al. on simulating Lévy processes from characteristic functions, we extend their methodology to:
- Discrete-time option pricing models
- Efficient calibration procedures
- Risk-neutral measure transformations
Practical Applications¶
Real-world options pricing scenarios and performance benchmarking including:
- European option pricing
- American option approximations
- Exotic derivatives valuation
- Greeks calculation and hedging strategies
Research Objectives¶
- Develop robust implementations of both pricing algorithms in R
- Conduct comprehensive performance analysis comparing computational efficiency
- Validate pricing accuracy against market data and theoretical benchmarks
- Provide practical guidance for practitioners in quantitative finance
Methodology¶
Our research methodology combines theoretical analysis with empirical validation:
Expected Contributions¶
This research aims to contribute to the quantitative finance literature by:
- Providing open-source R implementations of advanced Lévy process pricing models
- Establishing performance benchmarks for different algorithmic approaches
- Offering practical insights for model selection in real-world applications
- Creating educational resources for students and practitioners
Research Timeline¶
| Phase | Duration | Deliverables |
|---|---|---|
| Literature Review | 2 months | Comprehensive survey of Lévy models |
| Implementation | 3 months | R package with core algorithms |
| Testing & Validation | 2 months | Performance benchmarks and accuracy metrics |
| Documentation | 1 month | Research paper and user guides |
Download Research Proposal¶
Implementation Preview¶
Here’s a preview of our R implementation from the slides:
R code implementation of the Lévy process simulation algorithms
Collaborators¶
This research is conducted at the Illinois Institute of Technology by:
Yuchen Duan - Co-Researcher
Mathematical Modeling, Theoretical Analysis
Daniel Liberman - Co-Researcher
Empirical Validation, Market Data Analysis
References¶
- Feng, L., Linetsky, V., & Morales, J. L. (2016). Options Pricing in Lévy Models. Academic Paper.
- Barndorff-Nielsen, O. E. (1997). Normal inverse Gaussian distributions and stochastic volatility modelling. Scandinavian Journal of Statistics, 24(1), 1-13.
- Cont, R., & Tankov, P. (2004). Financial modelling with jump processes. Chapman and Hall/CRC.