Exact Solution to CEV Model with Uncorrelated Stochastic Volatility

14 Pages Posted: 28 Jan 2014

See all articles by Alexandre Antonov

Alexandre Antonov

Abu Dhabi Investment Authority; ADIA

Michael Konikov

Numerix

David Rufino

Citigroup, Inc. - Citigroup Global Markets

Michael Spector

Numerix

Date Written: January 28, 2014

Abstract

Stochastic volatility models are widely used in interest rate modeling to match the option smiles -- the two most popular are the Heston model and the SABR one. These have been incorporated into arbitrage-free term structure frameworks, Heston-LMM and SABR-LMM respectively.

In this paper we consider the CEV model with a general stochastic volatility. Assuming that rate-volatility correlation is zero we are able to obtain an exact integral representation of the option price provided that we have a closed form for a Moment Generating Function of the cumulative stochastic variance or of its inverse. Using this result we derive explicit solutions in terms of two-dimensional integral for both CEV-CIR model (or power generalization of the Heston) and the SABR one. Moreover the results in this paper may be easily extended to any affine process (possibly multi-factor and including jumps) leading to numerous practical applications.

Keywords: CEV, CIR, Heston, affine, stochastic volatility, SABR, closed formula, volatility surface

JEL Classification: C1, C3, C5, C6

Suggested Citation

Antonov, Alexandre and Konikov, Michael and Rufino, David and Spector, Michael, Exact Solution to CEV Model with Uncorrelated Stochastic Volatility (January 28, 2014). Available at SSRN: https://ssrn.com/abstract=2386731 or http://dx.doi.org/10.2139/ssrn.2386731

Alexandre Antonov (Contact Author)

Abu Dhabi Investment Authority ( email )

211 Corniche Road
Abu Dhabi, Abu Dhabi PO Box3600
United Arab Emirates

ADIA ( email )

211 Corniche
abu Dhabi
United Arab Emirates

Michael Konikov

Numerix ( email )

99 Park Avenue, 5th Floor
New York, NY 10016
United States

David Rufino

Citigroup, Inc. - Citigroup Global Markets ( email )

United Kingdom

Michael Spector

Numerix ( email )

99 Park Avenue, 5th Floor
New York, NY 10016
United States

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