Models for S&P 500 Dynamics: Evidence from Realized Volatility, Daily Returns, and Option Prices

CREATES Research Paper No. 2007-37

39 Pages Posted: 24 Jun 2008

See all articles by Peter Christoffersen

Peter Christoffersen

University of Toronto - Rotman School of Management; Copenhagen Business School; Aarhus University - CREATES

Kris Jacobs

University of Houston - C.T. Bauer College of Business

Karim Mimouni

McGill University - Desautels Faculty of Management

Date Written: November 15, 2007

Abstract

Most recent empirical option valuation studies build on the affine square root (SQR) stochastic volatility model. The SQR model is a convenient choice, because it yields closed-form solutions for option prices. However, relatively little is known about the resulting biases. We investigate alternatives to the SQR model, by comparing its empirical performance with that of five different but equally parsimonious stochastic volatility models. We provide empirical evidence from three different sources. We first use realized volatilities to assess the properties of the SQR model and to guide us in the search for alternative specifications. We then estimate the models using maximum likelihood on S&P 500 returns. Finally, we employ nonlinear least squares on a panel of option data. In comparison with earlier studies that explicitly solve the filtering problem, we analyze a more comprehensive option data set. The scope of our analysis is feasible because of our use of the particle filter. The three sources of data we employ all point to the same conclusion: the SQR model is misspecified. Overall, the best of the alternative volatility specifications is a model with linear rather than square root diffusion for variance which we refer to as the VAR model. This model captures the stylized facts in realized volatilities, it performs well in fitting various samples of index returns, and it has the lowest option implied volatility mean squared errors in- and out-of-sample.

Keywords: Stochastic volatility, option valuation, particle filtering, skewness, kurtosis, mean

JEL Classification: G12

Suggested Citation

Christoffersen, Peter and Jacobs, Kris and Mimouni, Karim, Models for S&P 500 Dynamics: Evidence from Realized Volatility, Daily Returns, and Option Prices (November 15, 2007). CREATES Research Paper No. 2007-37. Available at SSRN: https://ssrn.com/abstract=1150644 or http://dx.doi.org/10.2139/ssrn.1150644

Peter Christoffersen (Contact Author)

University of Toronto - Rotman School of Management ( email )

105 St. George Street
Toronto, Ontario M5S 3E6 M5P 3C4
Canada
416-946-5511 (Phone)

HOME PAGE: http://www.christoffersen.com

Copenhagen Business School

Solbjerg Plads 3
Frederiksberg C, DK - 2000
Denmark

Aarhus University - CREATES

School of Economics and Management
Building 1322, Bartholins Alle 10
DK-8000 Aarhus C
Denmark

Kris Jacobs

University of Houston - C.T. Bauer College of Business ( email )

Houston, TX 77204-6021
United States

Karim Mimouni

McGill University - Desautels Faculty of Management ( email )

1001 Sherbrooke St. West
Montreal, Quebec H3A1G5 H3A 2M1
Canada

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