Implied Volatility in the Variance Gamma Model
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
This Demonstration explores the implied volatility "smile" and "skews" for the variance gamma model. "Implied volatility" means the value of the volatility parameter in the Black–Scholes model (assumed to be constant in the model), which gives the same option value as the one quoted on an options exchange. It is well known that implied volatility for market option prices when plotted against the strike price shows "skews" and "smiles". This Demonstration shows that skews and smiles are also obtained if, instead of market option prices, you use option prices generated by the three-parameter variance gamma model.
Contributed by: Andrzej Kozlowski (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
The variance gamma process was first used in option pricing models by Madan and Seneta [1] and generalized by Madan, Carr, and Chang [2]. Explicit formulas for European-style options can be given, generalizing the Black–Scholes formulas. The model has been shown to perform better than the Black–Scholes model under the "historical approach" [3].
[1] D. B. Madan and E. Seneta, "The Variance Gamma Process (V.G.) Model for Share Market Returns," Journal of Business, 63(4), 1990 pp. 511–524.
[2] D. B. Madan, P. P. Carr, and E. C. Chang, "The Variance Gamma Process and Option Pricing," European Finance Review, 2(1), 1998 pp. 79–105.
[3] K. Lam, E. Chang, and M. C. Lee, "An Empirical Test of the Variance Gamma Option Pricing Model," Pacific Basin Finance Journal, 10(3), 2002 pp. 267–285.
Permanent Citation
"Implied Volatility in the Variance Gamma Model"
http://demonstrations.wolfram.com/ImpliedVolatilityInTheVarianceGammaModel/
Wolfram Demonstrations Project
Published: March 7 2011
Correlated Gamma Variance Processes with Common Subordinator
Andrzej Kozlowski
The Return Distribution of the Variance Gamma Process
Andrzej Kozlowski
Merton's Jump Diffusion Model
Andrzej Kozlowski
Option Prices in the Kou Jump Diffusion Model
Andrzej Kozlowski
Option Prices under the Fractional Black-Scholes Model
Andrzej Kozlowski
The Difference between European Option Prices in the Black-Scholes and NIG Models Computed with the DFT
Andrzej Kozlowski
Implied Volatility in Merton's Jump Diffusion Model
Peter Falloon
Markov Volatility Random Walks
Jason Cawley
Monopoly Model
David Youngberg
Competitive Model
David Youngberg
-
The Black-Scholes European Call Option Formula Corrected Using the Gram-Charlier Expansion
Andrzej Kozlowski -
Option Prices in the Kou Jump Diffusion Model
Andrzej Kozlowski -
The Polar and Bipolar of a Convex Polytope
Andrzej Kozlowski -
Two Jump Diffusion Processes
Andrzej Kozlowski -
The Vieta Mapping for the Coxeter Group A_2
Andrzej Kozlowski -
The Bifurcation Set of the Space of Smooth Real Functions
Andrzej Kozlowski -
The Swallowtail Singularity
Andrzej Kozlowski -
Singularities of an Ellipsoidal Wave Front
Andrzej Kozlowski -
Nonuniqueness of Option Pricing Under the Meixner Model
Andrzej Kozlowski -
Coin Machine
Andrzej Kozlowski -
A Mean-Reverting Jump Diffusion Process
Andrzej Kozlowski -
Standard American and European Options
Andrzej Kozlowski -
Density of the Kou Jump Diffusion Process
Andrzej Kozlowski -
The Meixner Process
Andrzej Kozlowski -
Fitting the Meixner Distribution to S&P 500 Returns
Andrzej Kozlowski -
Simple Graphs and Their Binomial Edge Ideals
Andrzej Kozlowski -
The Eneström-Kakeya Bounds for Roots of a Polynomial with Positive Coefficients
Andrzej Kozlowski -
Newton Polygon and Branching of Algebraic Curves
Andrzej Kozlowski -
Counting the Number of Roots of Transcendental Functions in Bounded Regions Using Winding Numbers
Andrzej Kozlowski -
The Space of Inner Products
Andrzej Kozlowski