Paper: arXiv 2405.16333

Abstract

Binomial trees are widely used in the financial sector for valuing securities with early exercise characteristics, such as American stock options. However, while effective in many scenarios, pricing options with CRR binomial trees are limited. Major limitations are volatility estimation, constant volatility assumption, subjectivity in parameter choices, and impracticality of instantaneous delta hedging. This paper presents a novel tree: Gaussian Recombining Split Tree (GRST), which is recombining and does not need log-normality or normality market assumption. GRST generates a discrete probability mass function of market data distribution, which approximates a Gaussian distribution with known parameters at any chosen time interval. GRST Mixture builds upon the GRST concept while being flexible to fit a large class of market distributions and when given a 1-D time series data and moments of distributions at each time interval, fits a Gaussian mixture with the same mixture component probabilities applied at each time interval. Gaussian Recombining Split Tre Mixture comprises several GRST tied using Gaussian mixture component probabilities at the first node. Our extensive empirical analysis shows that the option prices from the GRST align closely with the market.

Complexity vs Empirical Score

  • Math Complexity: 7.0/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces a novel mathematical framework with complex derivations and comparisons to established models like CRR, Tian, and Joshi’s Split Tree, indicating high mathematical density. It also includes extensive empirical analysis comparing option prices to market data and addresses practical implementation issues like volatility estimation, demonstrating strong empirical rigor.

Research Flowchart

  flowchart TD
  A["Research Goal: <br>Overcome Binomial Tree Limitations"] --> B["Method: <br>Gaussian Recombining Split Tree GRST"]
  B --> C["Input: <br>Market Data & Option Parameters"]
  C --> D{"Computational Process"}
  D --> E["Generate Discrete Probability <br>Mass Function"]
  D --> F["Fit Gaussian Mixture <br>Component Probabilities"]
  E & F --> G["Recombine Tree Nodes"]
  G --> H["Compute Option Prices <br>American & European"]
  H --> I["Outcome: <br>Market-Consistent Valuation"]
  I --> J["Findings: <br>Close Alignment with Market Data"]