Exponential Lévy Models with Stochastic Volatility and Stochastic Jump-Intensity

We introduce a class of stock models that interpolates between exponential Lévy models based on Brownian subordination and certain stochastic volatility models with Lévy-driven volatility, such as the Barndorff-Nielsen–Shephard model. The driving process in our model is a Brownian motion subordinated to a business time which is obtained by convolution of a Lévy subordinator with a deterministic kernel. We motivate several choices of the kernel that lead to volatility clusters while maintaining the sudden extreme movements of the stock. Moreover, we discuss some statistical and path properties of the models, prove absence of arbitrage and incompleteness, and explain how to price vanilla options by simulation and fast Fourier transform methods.

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Highly Efficient Option Valuation Under the Double Jump Framework with Stochastic Volatility and Jump Intensity Based on Shannon Wavelet Inverse Fourier Technique

SSRN Electronic Journal ◽

10.2139/ssrn.3087866 ◽

2017 ◽

Author(s):

Chun-Sung Huang ◽

John G O'Hara ◽

Sure Mataramvura

Keyword(s):

Stochastic Volatility ◽

Option Valuation ◽

Fourier Technique ◽

Highly Efficient ◽

Jump Intensity ◽

Shannon Wavelet ◽

Efficient Option

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Time-Varying Skew in VIX Derivatives Pricing

Management Science ◽

10.1287/mnsc.2021.4168 ◽

2021 ◽

Author(s):

Peixuan Yuan

Keyword(s):

Reduced Form ◽

Time Varying ◽

Derivatives Pricing ◽

Nested Models ◽

Stochastic Jump ◽

Out Of Sample ◽

Jump Intensity ◽

Form Model ◽

Reduced Form Model ◽

Vix Derivatives

This paper proposes a new reduced-form model for the pricing of VIX derivatives that includes an independent stochastic jump intensity factor and cojumps in the level and variance of VIX, while allowing the mean of VIX variance to be time varying. I fit the model to daily prices of futures and European options from April 2007 through December 2017. The empirical results indicate that the model significantly outperforms all other nested models and improves on benchmark by 21.6% in sample and 31.2% out of sample. The model more accurately portrays the tail behavior of VIX risk-neutral distribution for both short and long maturities, as it better captures the time-varying skew found to be largely independent of the level of the VIX smile. This paper was accepted by Kay Giesecke, finance.

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Integrating Volatility Clustering Into Exponential Lévy Models

Journal of Applied Probability ◽

10.1239/jap/1253279842 ◽

2009 ◽

Vol 46 (3) ◽

pp. 609-628 ◽

Cited By ~ 10

Author(s):

Christian Bender ◽

Tina Marquardt

Keyword(s):

Fourier Transform ◽

Brownian Motion ◽

Stochastic Volatility ◽

Stochastic Volatility Models ◽

Volatility Clustering ◽

Volatility Models ◽

Fourier Transform Methods ◽

Exponential Lévy Models ◽

Absence Of Arbitrage ◽

Path Properties

We introduce a class of stock models that interpolates between exponential Lévy models based on Brownian subordination and certain stochastic volatility models with Lévy-driven volatility, such as the Barndorff-Nielsen–Shephard model. The driving process in our model is a Brownian motion subordinated to a business time which is obtained by convolution of a Lévy subordinator with a deterministic kernel. We motivate several choices of the kernel that lead to volatility clusters while maintaining the sudden extreme movements of the stock. Moreover, we discuss some statistical and path properties of the models, prove absence of arbitrage and incompleteness, and explain how to price vanilla options by simulation and fast Fourier transform methods.

Download Full-text