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Filtering and forecasting with misspecified ARCH models. II: Making the right forecast with the wrong model. (English) Zbl 0820.62098
Summary: In part I of this paper, ibid. 52, No. 1/2, 61-90 (1992; Zbl 0761.62169), the first author showed that in data observed at high frequencies, an ARCH model may perform well in estimating the conditional variance of a process, even when the ARCH model is severely misspecified. While such models may perform reasonably well at filtering (i.e., at estimating unobserved instantaneous conditional variances), they may perform disastrously at medium- and long-term forecasting of the process and its volatility.
In this paper, we develop conditions under which a misspecified ARCH model sucessfully performs both tasks, filtering and forecasting. The key requirement (in addition to the conditions for consistent filtering) is that the ARCH model correctly specifies the functional form of the first two conditional moments of all state variables. We apply these results to a diffusion model employed in the options pricing literature, the stochastic volatility model of J. Hull and A. White [J. Finance 42, 281-300 (1987)], L. Scott [J. Financial Quant. Anal. 22, 419-437 (1987)], and J. Wiggins [J. Financial Econ. 19, 351-372 (1987)].

62P20 Applications of statistics to economics
62M20 Inference from stochastic processes and prediction
91B84 Economic time series analysis
Full Text: DOI
[1] Kaushik, I. Amin; Ng, Victor K.: ARCH processes and option valuation. (1993)
[2] Brian, D. O. Anderson; Moore, John B.: Optimal filtering. (1979) · Zbl 0688.93058
[3] Arnold, Ludwig: Stochastic differential equations: theory and applications. (1973) · Zbl 0278.60039
[4] Billingsley, Patrick: Probability and measure. (1986) · Zbl 0649.60001
[5] Bollerslev, Tim: Generalized autoregressive conditional heteroskedasticity. Journal of econometrics 31, 307-327 (1986) · Zbl 0616.62119
[6] Bollerslev, Tim; Chou, Ray Y.; Kroner, Kenneth F.: ARCH modeling in finance: A review of the theory and empirical evidence. Journal of econometrics 52, 5-60 (1992) · Zbl 0825.90057
[7] Bollerslev, Tim; Engle, Robert F.; Nelson, Daniel B.: ARCH models, forthcoming. Handbook of econometrics 4 (1994)
[8] Bollerslev, Tim; Engle, Robert F.; Wooldridge, Jeffrey M.: A capital asset pricing model with time varying covariances. Journal of political economy 96, 116-131 (1988)
[9] Charles, Q. Cao: Pricing options with stochastic volatility: A general equilibrium approach. (1992)
[10] Richard, M. Dudley: Real analysis and probability. (1989) · Zbl 0686.60001
[11] Robert, F. Engle: Autoregressive conditional heteroskedasticity with estimates of the variance of united kingdom inflation. Econometrica 50, 987-1008 (1982) · Zbl 0491.62099
[12] Engle, R. F.; Ng, V.; Rothschild, M.: Asset pricing with a factor ARCH covariance structure: empirical estimates for treasury bills. Journal of econometrics 45, 213-238 (1990)
[13] Stewart, N. Ethier; Kurtz, Thomas G.: Markov processes: characterization and convergence. (1986) · Zbl 0592.60049
[14] Stewart, N. Ethier; Nagylaki, Thomas: Diffusion approximations of Markov chains with two time scales and applications to population genetics, II. Advances in applied probability 20, 525-545 (1988) · Zbl 0669.60067
[15] Dean, P. Foster; Nelson, Daniel B.: Continuous record asymptotics for rolling sample variance estimators. (1994) · Zbl 0860.62101
[16] Andrew, C. Harvey; Shephard, Neil: The econometrics of stochastic volatility. Working paper (1993)
[17] Hull, John; White, Allan: The pricing of options on assets with stochastic volatilities. Journal of finance 42, 281-300 (1987)
[18] Jacquier, Eric; Polson, Nicholas G.; Rossi, Peter E.: Bayesian analysis of stochastic volatility models. Journal of business and economic statistics 12, 371-389 (1994) · Zbl 1082.62103
[19] Karatzas, Ioannis; Shreve, Steven E.: Brownian motion and stochastic calculus. (1988) · Zbl 0638.60065
[20] Kuwahara, Hiroto; Marsh, Terry A.: The pricing of Japanese equity warrants. Management science 38, 1610-1641 (1992)
[21] Harold, J. Kushner: Approximation and weak convergence methods for random processes, with applications to stochastic systems theory. (1984) · Zbl 0551.60056
[22] Melino, Angelo; Turnbull, Stuart M.: Pricing foreign currency options with stochastic volatility. Journal of econometrics 45, 239-267 (1990) · Zbl 1126.91374
[23] Daniel, B. Nelson: The time series behavior of stock market volatility and returns. (1988)
[24] Daniel, B. Nelson: ARCH models as diffusion approximations. Journal of econometrics 45, 7-38 (1990) · Zbl 0719.60089
[25] Daniel, B. Nelson: Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347-370 (1991) · Zbl 0722.62069
[26] Daniel, B. Nelson: Filtering and forecasting with misspecified ARCH models I: Getting the right variance with the wrong model. Journal of econometrics 52, 61-90 (1992) · Zbl 0761.62169
[27] Daniel, B. Nelson; Foster, Dean P.: Asymptotic filtering theory for univariate ARCH models. Econometrica 62, 1-41 (1994) · Zbl 0804.62085
[28] Boaz, A. Schwartz; Nelson, Daniel B.; Foster, Dean P.: Variance filtering with ARCH models: A Monte Carlo investigation. (1993)
[29] Louis, O. Scott: Option pricing when the variance changes randomly: theory, estimation, and an application. Journal of financial and quantitative analysis 22, 419-437 (1987)
[30] Daniel, W. Stroock; Varadhan, S. R. S.: Multidimensional diffusion processes. (1979) · Zbl 0426.60069
[31] James, B. Wiggins: Option values under stochastic volatility: theory and empirical estimates. Journal of financial economics 19, 351-372 (1987)
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