Effective Stochastic Volatility: Applications to ZABR-type Models

27 Pages Posted: 5 Feb 2020 Last revised: 20 Jan 2021

See all articles by Mike Felpel

Mike Felpel

University of Wuppertal

Joerg Kienitz

University of Wuppertal - Applied Mathematics; University of Cape Town (UCT); Quaternion Risk Management

Thomas McWalter

University of Cape Town (UCT); University of Johannesburg

Date Written: January 12, 2020

Abstract

There are numerous models for specifying the uncertainty of future instantaneous volatility or variance, including the Heston, SABR and ZABR models. Often it is observed that a specific stochastic volatility model is chosen not for particular dynamical features, relevant for exotic payoff structures, but instead for convenience and ease of implementation. The SABR model, with its semi-closed form approximate solution for the prices of vanilla options, is a well-known example. In this article, we consider a general approach that includes all practically relevant stochastic volatility models and introduces new variants of the ZABR model. In particular, we consider the mean-reverting ZABR and free ZABR models. We use the method of deriving an effective partial differential equation for the density. This approach leads to the known approximation formula for the SABR model, but also provides expressions for arbitrage-free models. Numerical experiments illustrate our approach.

Keywords: Stochastic volatility, SABR, ZABR, Free boundary ZABR, Mean-reverting ZABR, Effective PDE, Approximation formula

Suggested Citation

Felpel, Mike and Kienitz, Joerg and McWalter, Thomas, Effective Stochastic Volatility: Applications to ZABR-type Models (January 12, 2020). Available at SSRN: https://ssrn.com/abstract=3518141 or http://dx.doi.org/10.2139/ssrn.3518141

Mike Felpel

University of Wuppertal ( email )

Gaußstraße 20
42097 Wuppertal
Germany

Joerg Kienitz (Contact Author)

University of Wuppertal - Applied Mathematics ( email )

Gaußstraße 20
42097 Wuppertal
Germany

University of Cape Town (UCT) ( email )

Private Bag X3
Rondebosch, Western Cape 7701
South Africa

Quaternion Risk Management ( email )

54 Fitzwilliam Square North
Dublin, D02X308
Ireland

Thomas McWalter

University of Cape Town (UCT) ( email )

Private Bag X3
Rondebosch, Western Cape 7701
South Africa

University of Johannesburg ( email )

PO Box 524
Auckland Park
Johannesburg, Gauteng 2006
South Africa

Do you have a job opening that you would like to promote on SSRN?

Paper statistics

Downloads
141
Abstract Views
782
rank
239,589
PlumX Metrics