Heun Solutions to the SABR Model

17 Pages Posted: 18 Jan 2011

See all articles by Othmane Islah

Othmane Islah

Lloyds Banking Group; Quantuply Ltd

Date Written: January 18, 2011

Abstract

After investigation of the relative volatility of the spot price in the SABR model, we find that it is related to the general Fuschian differential equation with four singularities known as the Heun equation. The boundary behavior at zero and at infinity of the spot price in the SABR model, are identified for the first time through a classical Feller test. Expressions for the moment generating function of the log price are found analytically through solving an nonhomogeneous Heun equation. For the case of zero correlation between spot and vol, the differential equation to solve simplifies to an equation with three singularities for which the solutions can be expressed in terms of Hypergeometric functions.

Keywords: SABR Model, Stochastic Volatility Models, Fuschian Differential Equation, Heun Equation, Feller Test, Green Functions

Suggested Citation

Islah, Othmane, Heun Solutions to the SABR Model (January 18, 2011). Available at SSRN: https://ssrn.com/abstract=1742942 or http://dx.doi.org/10.2139/ssrn.1742942

Othmane Islah (Contact Author)

Lloyds Banking Group ( email )

10 Gresham Street
London, EC2V 7AE
United Kingdom

Quantuply Ltd ( email )

London
United Kingdom

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