Implied and Local Volatility Dynamics in the SABR Model
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
The stochastic
(SABR) model [1] provides a parsimonious parametrization of the implied volatility surface, based on a singular perturbation series of a particularly simple stochastic volatility model. The popularity of the SABR model is mainly due to its overwhelming empirical success, but also because it is extraordinary easy to calibrate. In this Demonstration, the asymptotic refinements of [3] are used. The local volatility surface is calculated exactly along the famous Dupire equation [2] in terms of implied volatility.
Contributed by: Thomas Mazzoni (August 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
The SABR model [1] is based on the simplest possible stochastic volatility model for forward prices
,
,
under the forward measure, where 
and 
are Brownian motions with 
and 
. In detail, 
is the initial volatility, 
is the CEV-exponent (CEV stands for constant elasticity of variance), 
is the volatility of volatility (volvol), and 
is the correlation between the two sources of random fluctuations.
Because the SABR model is the result of a perturbation series, there is a term of order 
, which may or may not be included. The corrections due to this term are usually about 1%.
References
[1] P. S. Hagan, D. Kumar, A. S. Lesniewski, and D. E. Woodward, "Managing Smile Risk," Wilmott Magazine, September 2002 pp. 84–108.
[2] B. Dupire, "Pricing with a Smile," Risk, 7, 1994 pp. 18–20.
[3] J. Oblój, "Fine-Tune Your Smile: Correction to Hagan et al.," Wilmott Magazine, May 2008.
Permanent Citation
Implied Volatility in the Variance Gamma Model
Andrzej Kozlowski
Implied Volatility in Merton's Jump Diffusion Model
Peter Falloon
Volatility Surface in the Heston Model
Slava Solganik
Dynamics in the Solow-Swan Growth Model
Richard Foltyn
Expected Dynamics of an Imitation Model in 2x2 Symmetric Games
Luis R. Izquierdo and Segismundo S. Izquierdo
A Spatial Dynamic Jury Model
Seth J. Chandler
Local Digit Values
Michael Schreiber
Expected Dynamics of an Imitation Model in the Hawk-Dove Game
Luis R. Izquierdo and Segismundo S. Izquierdo
A Model of Plasmodium Falciparum Population Dynamics in a Patient during Treatment with Artesunate
Sompob Saralamba
Dynamics of a Susceptible-Exposed-Infected-Recovered (SEIR) Epidemic Model with Time Delay
Clay Gruesbeck
