Arbitrage-Based Pricing When Volatility is Stochastic

Caltech Social Science Working Paper 977

41 Pages Posted: 5 Sep 1996

See all articles by Peter Bossaerts

Peter Bossaerts

University of Melbourne - Department of Finance

Eric Ghysels

University of North Carolina Kenan-Flagler Business School; University of North Carolina (UNC) at Chapel Hill - Department of Economics

Christian Gourieroux

University of Toronto - Department of Economics; Center for Interuniversity Research and Analysis on Organization (CIRANO); Ecole Nationale de la Statistique et de l'Administration Economique (ENSAE); National Bureau of Economic Research (NBER)

Date Written: July 1996

Abstract

In one of the early attempts to model stochastic volatility, Clark [1973] conjectured that the size of asset price movements is tied to the rate at which transactions occur. To formally analyze the econometric implications, he distinguished between transaction time and calendar time. The present paper exploits Clark's strategy for a different purpose, namely, asset pricing. It studies arbitrage-based pricing in economies where: (i)trade takes place in transaction time, (ii) there is a single state variable whose transaction-time price path is binomial, (iii) there are riskfree bonds with calendar-time maturities, and (iv) the relation between transaction time and calendar time is stochastic. The state variable could be interpreted in various ways. E.g., it could be the price of a share of stock, as in Black and Scholes [1973], or a factor that summarizes changes in the investment opportunity set, as in Cox, Ingersoll and Ross [1985] or one that drives changes in the term structure of interest rates (Ho and Lee [1986], Heath, Jarrow and Morton [1992]). Property (iv) generally introduces stochastic volatility in the process of the state variable when recorded in calendar time.

The paper investigates the pricing of derivative securities with calendar-time maturities. The restrictions obtained in Merton [1973] using simple buy-and-hold arbitrage portfolio arguments do not necessarily obtain. Conditions are derived for all derivatives to be priced by dynamic arbitrage, i.e., for market completeness in the sense of Harrison and Pliska [1981]. A particular class of stationary economies where markets are indeed complete is characterized.

JEL Classification: D52, G13

Suggested Citation

Bossaerts, Peter L. and Ghysels, Eric and Gourieroux, Christian, Arbitrage-Based Pricing When Volatility is Stochastic (July 1996). Caltech Social Science Working Paper 977. Available at SSRN: https://ssrn.com/abstract=680 or http://dx.doi.org/10.2139/ssrn.680

Peter L. Bossaerts (Contact Author)

University of Melbourne - Department of Finance ( email )

Faculty of Economics and Commerce
Department of Finance
Carlton, Victoria 3010
Australia

HOME PAGE: http://bmmlab.org

Eric Ghysels

University of North Carolina Kenan-Flagler Business School ( email )

Kenan-Flagler Business School
Chapel Hill, NC 27599-3490
United States

University of North Carolina (UNC) at Chapel Hill - Department of Economics ( email )

Gardner Hall, CB 3305
Chapel Hill, NC 27599
United States
919-966-5325 (Phone)
919-966-4986 (Fax)

HOME PAGE: http://www.unc.edu/~eghysels/

Christian Gourieroux

University of Toronto - Department of Economics ( email )

150 St. George Street
Toronto, Ontario M5S 3G7
Canada
416-978-4349 (Phone)
416-978-6713 (Fax)

Center for Interuniversity Research and Analysis on Organization (CIRANO) ( email )

2020 rue University, 25th Floor
Montreal, Quebec H3C 3J7
Canada

Ecole Nationale de la Statistique et de l'Administration Economique (ENSAE) ( email )

15 Boulevard Gabriel Peri
92245 Malakoff Cedex
France
33.4117.7666 (Fax)

National Bureau of Economic Research (NBER)

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

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