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Alternative models for stock price dynamics. (English) Zbl 1043.62087

Summary: This paper evaluates the role of various volatility specifications, such as multiple stochastic volatility (SV) factors and jump components, in appropriate modeling of equity return distributions. We use estimation technology that facilitates nonnested model comparisons and use a long data set which provides rich information about the conditional and unconditional distribution of returns. We consider two broad families of models: (1) the multifactor loglinear family, and (2) the affine-jump family. Both classes of models have attracted much attention in the derivatives and econometrics literature.
There are various tradeoffs in considering such diverse specifications. If pure diffusion SV models are chosen over jump diffusions, it has important implications for hedging strategies. If logarithmic models are chosen over affine ones, it may seriously complicate option pricing. Comparing many different specifications of pure diffusion multifactor models and jump diffusion models, we find that (1) log linear models have to be extended to two factors with feedback in the mean reverting factor, (2) affine models have to have a jump in returns, stochastic volatility or probably both. Models (1) and (2) are observationally equivalent on the data set at hand. In either (1) or (2) the key is that the volatility can move violently.
As we obtain models with comparable empirical fit, one must make a choice based on arguments other than statistical goodness-of-fit criteria. The considerations include facility to price options, to hedge and parsimony. The affine specification with jumps in volatility might therefore be preferred because of the closed-form derivatives prices.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
91B28 Finance etc. (MSC2000)
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[1] Aı̈t-Sahalia, Y., Maximum likelihood estimation of discretely sampled diffusionsa closed-form approximation approach, Econometrica, 70, 223-262, (2002) · Zbl 1104.62323
[2] Aı̈t-Sahalia, Y.; Brandt, M., Variable selection for portfolio choice, Journal of finance, 56, 1297-1351, (2001)
[3] Alizadeh, S.; Brandt, M.W.; Diebold, F.X., Range-based estimation of stochastic volatility models, Journal of finance, 57, 1047-1091, (2002)
[4] Andersen, T.; Benzoni, L.; Lund, J., An empirical investigation of continuous-time equity return models, Journal of finance, 57, 1239-1284, (2002)
[5] Anderson, E., Hansen, L.P., Sargent, T., 2002. A quartet of semi-groups for model specification, detection, robustness, and the price of risk. Working Paper, University of North Carolina.
[6] Bakshi, G.; Cao, C.; Chen, Z., Empirical performance of alternative option pricing models, Journal of finance, 52, 2003-2049, (1997)
[7] Bates, D., Post-’87 crash fears in the S&P 500 futures option market, Journal of econometrics, 94, 181-238, (2000) · Zbl 0942.62118
[8] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of political economy, 81, 637-654, (1973) · Zbl 1092.91524
[9] Brace, A.; Gatarek, D.; Musiela, M., The market model of interest rate dynamics, Mathematical finance, 7, 127-155, (1997) · Zbl 0884.90008
[10] Chacko, G., Viceira, L., 1999. Spectral GMM estimation of continuous-time processes. Discussion Paper, Harvard Business School. · Zbl 1026.62085
[11] Chen, X., Hansen, L.P., Carrasco, M., 2001. Nonlinearity and temporal dependence. Discussion Paper, University of Chicago. · Zbl 1431.62600
[12] Chernov, M.; Ghysels, E., A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation, Journal of financial economics, 56, 407-458, (2000)
[13] Dai, Q.; Singleton, K., Specification analysis of affine term structure models, Journal of finance, 55, 1943-1978, (2000)
[14] Das, S.; Sundaram, R., Of smiles and smirksa term-structure perspective, Journal of financial and quantitative analysis, 34, 211-240, (1999)
[15] Detemple, J., Garcia, R., Rindisbacher, M., 2002. Asymptotic properties of Monte Carlo estimators of diffusion processes. Working Paper, CIRANO. · Zbl 1418.62287
[16] Duffie, D.; Pan, J.; Singleton, K., Transform analysis and option pricing for affine jump-diffusions, Econometrica, 68, 1343-1377, (2000) · Zbl 1055.91524
[17] Durham, G., Gallant, A.R., 2000. Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. Working Paper, University of North Carolina.
[18] Elerian, O.; Chib, S.; Shephard, N., Likelihood inference for discretely observed non-linear diffusions, Econometrica, 69, 959-993, (2001) · Zbl 1017.62068
[19] Engle, R.F.; Gonzales-Rivera, G., Semi-parametric ARCH models, Journal of business and economic statistics, 9, 345-359, (1991)
[20] Engle, R.F., Lee, G., 1999. A long-run and short-run component model of stock return volatility. In: Engle, R.F., White, H. (Eds.) Cointegration, Causality and Forecasting—A Festschrift in Honour of Clive W.J. Granger, Oxford University Press, Oxford.
[21] Eraker, B., Johannes, M., Polson, N., 2003. The impact of jumps in equity index volatility and returns. Journal of Finance, forthcoming.
[22] Feller, W., 1971. An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York. · Zbl 0219.60003
[23] Gallant, A.R.; Hsieh, D.; Tauchen, G., Estimation of stochastic volatility models with diagnostics, Journal of econometrics, 81, 159-192, (1997) · Zbl 0904.62134
[24] Gallant, A.R., Hsu, C.-T., Tauchen, G., 1999. Using daily range data to calibrate volatility diffusions and extract the forward integrated variance. Review of Economics and Statistics, forthcoming.
[25] Gallant, A.R.; Long, J.R., Estimating stochastic differential equations efficiently by minimum chi-square, Biometrika, 84, 125-141, (1997) · Zbl 0953.62084
[26] Gallant, A.R.; Tauchen, G., Seminonparametric estimation of conditionally constrained heterogeneous processesasset pricing applications, Econometrica, 57, 1091-1120, (1989) · Zbl 0679.62096
[27] Gallant, A., Tauchen, G., 1992. A nonparametric approach to nonlinear time series analysis: estimation and simulation, in: Parzen, E., Brillinger, D., Rosenblatt, M., Taqqu, M., Geweke, J., Caines, P. (eds.), New Dimensions in Time Series Analysis. Springer-Verlag, New York.
[28] Gallant, A.R., Tauchen, G., 1993. SNP: a program for nonparametric time series analysis, version 8.3, User’s Guide. Discussion Paper, University of North Carolina at Chapel Hill.
[29] Gallant, A.R.; Tauchen, G., Which moments to match?, Econometric theory, 12, 657-681, (1996)
[30] Gallant, A.R., Tauchen, G., 1997. EMM: a program for efficient method of moments estimation, Version 1.4, User’s guide. Discussion Paper, University of North Carolina at Chapel Hill.
[31] Gallant, A.R.; Tauchen, G., Reprojecting partially observed systems with application to interest rate diffusions, Journal of American statistical association, 93, 10-24, (1998) · Zbl 0920.62132
[32] Gallant, A.R., Tauchen, G., 2003. SNP: a program for nonparametric time series analysis, a user’s guide. Manuscript, University of North Carolina. (Available along with code via http://www.econ.duke.edu/ get./snp.html).
[33] Ghysels, E.; Ng, S., A semiparametric factor model of interest rates and tests of the affine term structure, Review of economics and statistics, 80, 535-548, (1998)
[34] Heston, S.L., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Review of financial studies, 6, 327-343, (1993) · Zbl 1384.35131
[35] Jones, C., 2003. The dynamics of stochastic volatility. Journal of Econometrics, this issue.
[36] Kloeden, P.E.; Platen, E., Numerical solution of stochastic differential equations, (1995), Springer Berlin · Zbl 0858.65148
[37] Meddahi, N., 2001. An eigenfunction approach for volatility modeling. Working Paper, University of Montreal.
[38] Nelson, D., ARCH models as diffusion approximations, Journal of econometrics, 45, 7-38, (1990) · Zbl 0719.60089
[39] Pan, J., The jump-risk premia implicit in options: evidence from an integrated time-series study, Journal of financial economics, 63, 3-50, (2002)
[40] Platen, E.; Rebolledo, R., Weak convergence of semimartingales and discretisation methods, Stochastic processes and their applications, 20, 41-58, (1985) · Zbl 0584.60060
[41] Powell, J.; Stock, J.; Stoker, T., Semiparametric estimation of index models, Econometrica, 57, 1403-1430, (1989) · Zbl 0683.62070
[42] Scott, L., Option pricing when the variance changes randomlytheory, estimation and an application, Journal of financial and quantitative analysis, 22, 419-438, (1987)
[43] Stoker, T., 1993. Lectures on Semiparametric Econometrics. CORE Lecture Series.
[44] van der Vaart, A., Asymptotic statistics, (2000), Cambridge University Press Cambridge, MA · Zbl 0910.62001
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