自 Black-Scholes 模型提出後，有關描述資產報酬之模型及相關問題即為一引人注目的研究課題。然而 Black-Scholes 模型中關於常數波動性及常態誤差的假設與實証結果並不吻合。實務模型中其分配函數尾端通常較常態模型厚，因此本文考慮隨機波動模型。我們的目標為計算在該模型下之風險值，並以自助法評價其估計量。
本文證明風險值估計量具有漸進常態性以及自助法的適當性。模擬結果顯示即使在誤差分配是自由度為 1 之下的 t 分配，對於參數估計與風險值估計均有不錯的結果。此外，我們應用參數自助法在隨機波動模型下建構預測區間，並以美國標準普爾 500 指數為例。結果顯示預測區間與實際資料的走勢一致。最後我們考慮在隨機波動模型下對於稀少事件的機率估計問題。本文提出以條件重點抽樣的概念估計此機率問題。模擬結果顯示條件重點抽樣估計量比蒙地卡羅估計量有較高的有效性，
也就是條件重點抽樣達到變異數降低的效果。

Abramovitz, M. and N. Stegun. (1970). Handbook of Mathematical Func-
tions with Formulas, Graphs, and Mathematical Tables.
Asmussen, S. and Rubinstein, R. Y. (1995). Complexity properties of
steady-state rare events simulation in queueing models. Advances in
Queueing: Theory, Methods and Open Problems, CRC Press, 429-462.
Bucklew, J. A. (2004). Introduction to Rare Event Simulation. Springer-
Verlag: New York.
Bollerslev, T. (1986). Generalized autoregressive conditional heteroscedas-
ticity. Journal of Econometrics, 31, 307-327.
Bollerslev, T., Chou, R. Y., and Kroner, K. F. (1992). ARCH modeling
in nance: a selective review of the theory and empirical evidence.
Econometrics, 52, 5-59.
Box, G. E. P. and Jenkins, G. M. (1976). Time Series Analysis Forecasting
and Control. Holden-Day: San Francisco.
Burmeister, E., and Wall, K. D. (1982). Kalman ltering estimation of
unobserved rational expections with an application to the German
hyperin ation. Journal of Econometrics, 4,147-160.
Chang, Y. P., Hung, M. C. and Wu, Y. F. (2003). Nonparametric estima-
tion for risk in Value-at-Risk estimator. Communications In Statistics:
Simulation and Computation, 32, 1041-1064.
Duffie, D. and Pan, J. (1997). An Overview of Value at Risk. Journal of
Derivative, 7, 7-49.
Dunsmuir, W. (1979). A central limit theorem for parameter estimation
in stationary time series and its applications to models for a signal
observed white noise. Annals of Statistics, 7, 490-506.
Efron, B. (1979). Bootstrap methods: Another look at the jackknife.
Annals of Statistics, 7,23-55.
Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with
estimates of the variance of United Kingdom in ation. Econometrica,
50, 987-1008.
Fuh, C. D. (2004). E cient likelihood estimation in state space models.
Technical Report, No. C-4. Institute of Statistical Science, Academia Sinica.
Glasserman, P., Heidelberger, P., and Shahabuddin, P. (2000). Variance
reduction techniques for estimating Value-at-Risk. Management Science,46,1349- 1364.
Glasserman, P., Heidelberger, P., and Shahabuddin, P. (2002). Portfolio
Value-at-Risk with heavy tailed risk factors. Mathematical Finance,
12, 239-270.
Glynn, P. W. and Iglehart, D. L. (1989). Importance sampling for stochastic
simulations. Management Science, 35, 1367-1391.
Hall, P. and Yao Q. (2003). Inference in ARCH and GARCH models with
heavy-tailed errors. Econometrica, 71, 285-317.
Harvey, A. C. (1989). Forecasting, Structural Models and The Kalman
Filter. Cambridge University Press, Cambridge.
Harvey, A. C., Ruiz, E., and Shephard. N. G. (1992). Multivariate
stochastic variance models. Review of Economic Studies, 61, 247-264.
Hendricks, D. (1996). Evaluation of Value-at-Risk models using historical
data. Federal Reserve Bank of New York Economic Policy Review, April, 39-69.
Hull, J. and White, A. (1998). Incorporating volatility updating into the
historical simulation method for Value-at-Risk. Journal of Risk, 1, 5-19.
Jacquier, E., Polson, N. G., and Rossi, P. E. (1994). Bayesian analysis
of stochastic volatility models. Journal of Business and Economic
Statistics, 12, 371-389.
Jorion, P. (2002). Value at Risk: The New Benchmark for Managing
Financial Risk. McGraw-Hill: New York.
Ljung, L., and Caines, P. E. (1979). Asymptotic normality of prediction
error estimators for approximate system models. Stochastics, 3, 29-46.
Miguel, J. A. and Olave P. (1999a). Bootstrapping forecast intervals in
ARCH models. Test, 8, 345-364.
Migue, J. A. and Olave, P. (1999b). Forecast intervals in ARCH models:
bootstrap versus parametric methods. Applied Economics Letters, 6,
323-327.
Morgan, J. P. (1996). RiskMetrics Technical Document, Forth edition,
New York.
Ridder, T. (1997). Basic of statistical VaR-estimation. In: Bol, D.,
Nakhaeizadeh, G., Vollmer, K. H., eds. Risk Measurement, Econo-
metrics and Neural Networks.Heidelberg: Physica-Verlag. 161-187.
Ross, S. M. (2002). Simulation. Academic Press: San Diego.
Ruiz, E. (1994). Quasi-maximum likelihood estimation of stochastic volatility
models. Journal of Econometrics, 63, 289-306.
Sen, P. K. and Singer, J. M. (1993). Large Sample Methods in Statistics:
An Introduction with Applications. Chapman & Hall: New York.
Shephard N. (1993). Fitting nonlinear time-series models with applications to
stochastic variance models. Journal of Applied Econometrics, 8, 135-152.
So, M. K. P., Li, W. K., and Lam, K. (1997). Multivariate modelling of
the autoregressive random variance process. Journal of Time Series
Analysis, 18, 429-446.
Stoffer, D. S. and Wall, K. D. (1991). Bootstrapping state-space models:
Gaussian maximum likelihood estimation and the Kalman lter.
Journal of the American Statistical Association, 86, 1024-1033.
Taylor, S. J. (1982). Financial returns modelled by the product of two
stochastic process, a study of daily sugar price 1961-79. In Time Series
Analysis: Theory and Practice 1 (ed. O. D. Anderson). Amsterdam:
North-Holland. 203-226.
Taylor, S. J. (1994). Modelling stochastic volatility. Mathematical Finance,
4, 183-204.
Thombs, L. A. and Schucany, W. R. (1990). Bootstrap prediction intervals
for autoregression. Journal of the American Statistical Association, 85,
486-492.
Tsay, R. S. (2002). Analysis of Financial Time Series. John Wiley.
Wall, K. D., and Sto er, D. S. (2002). A state space model to bootstrap-
ping conditional forecasts in ARMA models. Journal of Time Series
Analysis, 23, 733-751.
Wong, C. M. and So, M. K. P. (2003). On conditional moments of GARCH
models, with applications to multiple period value at risk estimation.
Statistica Sinica, 13, 1015-1044.