Linearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing

51 Pages Posted: 19 Mar 2011

See all articles by Peter Carr

Peter Carr

New York University Finance and Risk Engineering

Xavier Gabaix

Harvard University - Department of Economics; National Bureau of Economic Research (NBER); Centre for Economic Policy Research (CEPR); European Corporate Governance Institute (ECGI)

Liuren Wu

City University of New York, CUNY Baruch College - Zicklin School of Business

Date Written: March 18, 2011

Abstract

We propose to use the linearity-generating framework to accommodate the evidence of unspanned stochastic volatility: Variations in implied volatilities on interest-rate options such as caps and swaptions are independent of the variations on the interest rate term structure. Under this framework, bond valuation depends only on the transition dynamics of interest-rate factors, but not on their volatilities. Thus, interest-rate volatility is truly unspanned. Furthermore, this framework allows tractable pricing of options on any bond portfolios, including both caps and swaptions. This feat is not possible under existing exponential-affine or quadratic frameworks. Finally, the framework allows sequential estimation of the interest-rate term structure and the interest-rate option implied volatility surface, thus facilitating joint empirical analysis. Within this framework, we perform specification analysis on interest-rate factor transition dynamics and its relation to the interest-rate term structure; we also analyze the interest-rate volatility dynamics and its impact on interest-rate option pricing. We estimate several specifications for the transition dynamics to ten years worth of U.S. dollar LIBOR and swap rates across 15 maturities. We also estimate several interest-rate volatility dynamics specifications using ten years of swaption implied volatilities across a matrix of ten option maturities and seven swap tenors. The estimation results show that the volatility dynamics dictate the option implied volatility variation along the option maturity dimension, whereas the interest-rate transition dynamics dictate the implied volatility variation along the underlying swap maturity dimension.

Suggested Citation

Carr, Peter P. and Gabaix, Xavier and Wu, Liuren, Linearity-Generating Processes, Unspanned Stochastic Volatility, and Interest-Rate Option Pricing (March 18, 2011). Available at SSRN: https://ssrn.com/abstract=1789763 or http://dx.doi.org/10.2139/ssrn.1789763

Peter P. Carr

New York University Finance and Risk Engineering ( email )

6 MetroTech Center
Brooklyn, NY 11201
United States
9176217733 (Phone)

HOME PAGE: http://engineering.nyu.edu/people/peter-paul-carr

Xavier Gabaix

Harvard University - Department of Economics ( email )

Littauer Center
Cambridge, MA 02138
United States

National Bureau of Economic Research (NBER)

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

Centre for Economic Policy Research (CEPR)

London
United Kingdom

European Corporate Governance Institute (ECGI)

c/o ECARES ULB CP 114
B-1050 Brussels
Belgium

Liuren Wu (Contact Author)

City University of New York, CUNY Baruch College - Zicklin School of Business ( email )

One Bernard Baruch Way
Box B10-247
New York, NY 10010
United States
646-312-3509 (Phone)
646-312-3451 (Fax)

HOME PAGE: http://faculty.baruch.cuny.edu/lwu/

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