Higher order approximation of option prices in Barndorff-Nielsen and Shephard models

ArXiv ID: 2401.14390 “View on arXiv”

Authors: Unknown

Abstract

We present an approximation method based on the mixing formula (Hull & White 1987, Romano & Touzi 1997) for pricing European options in Barndorff-Nielsen and Shephard models. This approximation is based on a Taylor expansion of the option price. It is implemented using a recursive algorithm that allows us to obtain closed form approximations of the option price of any order (subject to technical conditions on the background driving Lévy process). This method can be used for any type of Barndorff-Nielsen and Shephard stochastic volatility model. Explicit results are presented in the case where the stationary distribution of the background driving Lévy process is inverse Gaussian or gamma. In both of these cases, the approximation compares favorably to option prices produced by the characteristic function. In particular, we also perform an error analysis of the approximation, which is partially based on the results of Das & Langrené (2022). We obtain asymptotic results for the error of the $N^{"\text{th"}}$ order approximation and error bounds when the variance process satisfies an inverse Gaussian Ornstein-Uhlenbeck process or a gamma Ornstein-Uhlenbeck process.

Keywords: Barndorff-Nielsen and Shephard models, stochastic volatility, Taylor expansion, Lévy process, characteristic function, Equity Derivatives (European Options)

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper is extremely heavy on advanced mathematics, featuring stochastic calculus, Lévy processes, characteristic functions, and high-order Taylor expansions with complex error bounds, placing it at the high end of math complexity. However, it lacks empirical backtesting, code implementation, or real-world data analysis, focusing instead on theoretical derivations and approximation bounds, resulting in low empirical rigor.

  flowchart TD
    A["Research Goal: Higher Order Option Pricing"] --> B{"Methodology: Taylor Expansion & Mixing Formula"}
    B --> C["Inputs: BNS Model Parameters"]
    C --> D{"Computational Process"}
    D --> E["Recursive Algorithm for N-th Order Approximation"]
    E --> F["Closed-Form Results for IG/Gamma Distributions"]
    F --> G["Error Analysis & Asymptotic Bounds"]
    G --> H["Outcome: Accurate European Option Prices"]