A General Valuation Framework for SABR and Stochastic Local Volatility Models

44 Pages Posted: 9 Mar 2019

See all articles by Zhenyu Cui

Zhenyu Cui

Stevens Institute of Technology - School of Business

Justin Kirkby

Georgia Institute of Technology - The H. Milton Stewart School of Industrial & Systems Engineering (ISyE)

Duy Nguyen

Marist College - Department of Mathematics

Date Written: April 24, 2018

Abstract

In this paper, we propose a general framework for the valuation of options in stochas-tic local volatility (SLV) models with a general correlation structure, which includes the Stochastic Alpha Beta Rho (SABR) model and the quadratic SLV model as special cases. Standard stochastic volatility models, such as Heston, Hull-White, Scott, Stein-Stein, α-Hypergeometric, 3/2, 4/2, mean-reverting, and Jacobi stochastic volatility models, also fall within this general framework. We propose a novel double-layer continuous-time Markov chain (CTMC) approximation respectively for the variance process and the underlying asset price process. The resulting regime-switching continuous-time Markov chain is further reduced to a single CTMC on an enlarged state space. Closed-form matrix expressions for European options are derived. We also propose a recursive risk-neutral valuation technique for pricing discretely monitored path-dependent options, and use it to price Bermudan and barrier options. In addition, we provide single Laplace transform formulae for arithmetic Asian options as well as occupation time derivatives. Numerical examples demonstrate the accuracy and efficiency of the method using several popular SLV models, and reference prices are provided for SABR, Heston-SABR, quadratic SLV, and the Jacobi model.

Keywords: SABR, stochastic local volatility, quadratic local volatility, continuous-time Markov Chains, exotic options, American options, Asian options, option pricing, CTMC, occupation time derivatives, barrier options

JEL Classification: G12, G13

Suggested Citation

Cui, Zhenyu and Kirkby, Justin and Nguyen, Duy, A General Valuation Framework for SABR and Stochastic Local Volatility Models (April 24, 2018). Available at SSRN: https://ssrn.com/abstract=3168155

Zhenyu Cui (Contact Author)

Stevens Institute of Technology - School of Business ( email )

Hoboken, NJ 07030
United States

HOME PAGE: http://sites.google.com/site/zhenyucui86/publications

Justin Kirkby

Georgia Institute of Technology - The H. Milton Stewart School of Industrial & Systems Engineering (ISyE) ( email )

765 Ferst Drive
Atlanta, GA 30332-0205
United States

Duy Nguyen

Marist College - Department of Mathematics ( email )

NY
United States

HOME PAGE: http://sites.google.com/site/nducduy/

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