Paper: arXiv 2408.01642

Abstract

The additive process generalizes the Lévy process by relaxing its assumption of time-homogeneous increments and hence covers a larger family of stochastic processes. Recent research in option pricing shows that modeling the underlying log price with an additive process has advantages in easier construction of the risk-neural measure, an explicit option pricing formula and characteristic function, and more flexibility to fit the implied volatility surface. Still, the challenge of calibrating an additive model arises from its time-dependent parameterization, for which one has to prescribe parametric functions for the term structure. For this, we propose the neural term structure model to utilize feedforward neural networks to represent the term structure, which alleviates the difficulty of designing parametric functions and thus attenuates the misspecification risk. Numerical studies with S&P 500 option data are conducted to evaluate the performance of the neural term structure.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper is mathematically dense, featuring advanced stochastic calculus, Lévy processes, infinite divisibility, and specific derivations like Lemma 2.1 and Proposition 2.3, placing it at the high end of complexity. It demonstrates strong empirical rigor through calibration with S&P 500 option data, comparison to parametric models, and analysis of robustness across historical surfaces, making it backtest-ready despite focusing on theoretical construction.

Research Flowchart

  flowchart TD
  A["Research Goal: Assess Neural Term Structure for Additive Process Calibration"] --> B["Data: S&P 500 Options Data"]
  B --> C["Model: Neural Term Structure\nFeedforward Neural Networks"]
  C --> D["Computation: Calibrate Model to Market Data"]
  D --> E["Output: Improved Fit to Implied Volatility Surface"]
  E --> F["Outcome: Reduced Misspecification Risk\nExplicit Pricing Formula Achieved"]

Cited References (35)

  1. Financial modelling with jump processes (2004)
  2. Normal variance-mean mixtures and z distributions (1982)
  3. Handbook of the Logistic Distribution (1991)
  4. Approximation by superposition of sigmoidal functions (1989)
  5. Dynamics of implied volatility surfaces (2002)
  6. Recovering Volatility from Option Prices by Evolutionary Optimization (2005)
  7. Fast exact joint S&P 500/VIX smile calibration in discrete and continuous time (2024)
  8. Building arbitrage-free implied volatility: Sinkhorn’s algorithm and variants (2019) — link
and 27 more
  1. Bass construction with multi-marginals: Lightspeed computation in a new local volatility model (2021)
  2. Making Markov Martingales Meet Marginals: With Explicit Constructions (2002)
  3. Markov-komposition und eine anwendung auf martingale (1972)
  4. A note on sufficient conditions for no arbitrage (2005)
  5. The range of traded option prices (2007)
  6. The variance gamma process and option pricing (1998)
  7. Normal inverse Gaussian distributions and stochastic volatility modelling (1997)
  8. A jump-diffusion model for option pricing (2002)
  9. Option pricing when underlying stock returns are discontinuous (1976)
  10. Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options (1996)
  11. Managing smile risk (2002)
  12. Notes on option pricing I: Constant elasticity of diffusions, 1975 (1996)
  13. A closed-form solution for options with stochastic volatility with applications to bond and currency options (1993)
  14. Pricing with a smile (1994)
  15. The pricing of options and corporate liabilities (1973)
  16. Th 'e orie de la sp 'e culation (1900)
  17. An overview of gradient descent optimization algorithms (2016) — link
  18. Lecture 6.5-rmsprop: Divide the gradient by a running average of its recent magnitude (2012)
  19. Adam: A method for stochastic optimization (2014) — link
  20. TensorFlow: a system for large-scale machine learning (2016)
  21. Automatic differentiation in PyTorch (2017)
  22. L 'e vy processes and infinitely divisible distributions (1999)
  23. Characteristic Functions (1970)
  24. Families of distributions arising from distributions of order statistics (2004)
  25. A fast Monte Carlo scheme for additive processes and option pricing (2023)
  26. Explicit option pricing with additive processes (2023)
  27. Additive logistic processes in option pricing (2021)

0 of 35 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.