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Stochastic volatility

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Pricing Stock Options Under Stochastic Volatility And Interest Rates With Efficient Method Of Moments Estimati

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Second, while allowing for stochastic volatility can reduce the pricing errors and allowing for asymmetric volatility or “leverage effect” does help to explain the skewness of the volatility “smile”, allowing for stochastic interest rates has minimal impact on option prices in our case.

A simulation of the heston model with stochastic volatility using the finite difference method

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In the Heston model, the volatility satisfies. is the volatility of the variance process. S v t represents the price of volatility risk and is independent of the particular asset. Equations (2.4) and (2.5) are the governing equations for the Heston model. It is an extension of the Black-Scholes model in the sense that the behavior for the volatility process is assumed to be stochastic instead of being a constant.

Daily VaR forecasts with realized volatility and GARCH models

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X., Ebens, H., The Distribution of Realized Stock Return Volatility, “Journal of Financial Economics” 61, pp. G., Bollerslev, T., Diebold, F.X., Roughing It Up: Including Jump Components in the Measurement, Modeling, and Forecasting of Returns Volatility, “The Review of Economics and Statistics” 89(4), pp. E., Shephard, N., Econometric Analysis of Realised Volatility and Its Use in Estimating Stochastic Volatility Models, “Journal of the Royal Statistical Society”, B, 64, pp.

SAS/ETS 9.22 User's Guide 130

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Example 18.17: Simulated Method of Moments—Stochastic Volatility Model. This example illustrates how to use SMM to estimate a stochastic volatility model as in Andersen and Sorensen (1996):. This is called the stochastic volatility model because the volatility is stochastic as the random variable u t appears in the volatility equation. The following SAS statements use three moments: absolute value, the second-order moment, and absolute value of the first-order autoregressive moment.

Handbook of Economic Forecasting part 3

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Volatility and Correlation Forecasting. Uses of volatility forecasts 786. Further reading 797. Rolling regressions and RiskMetrics 798. Further reading 812. Stochastic volatility 814. Efficient method of simulated moments procedures for inference and forecasting 823. Markov Chain Monte Carlo (MCMC) procedures for inference and forecasting 826. Further reading 828. Realized volatility 830. Realized volatility modeling 834. Realized volatility forecasting 835. Further reading 837.

Handbook of Economic Forecasting part 73

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Another popular class of models is the so-called stochastic volatility models [see, e.g., Ghysels, Harvey and Renault (1996) for further discussion]. Models of stochastic volatility have been used extensively in the finance literature.

Handbook of Economic Forecasting part 90

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“Stochastic volatility”. Journal of Applied Econometrics 8, S85–S118.. Journal of Econometrics 40, 87–96.. Journal of Applied Econo- metrics . “Cross-security tests of the mixture of distributions hypothesis”. Journal of Financial and Quantitative Analysis 21, 39–46.. “Transactions data tests of the mixture of distributions hypothesis”. Journal of Financial and Quantitative Analysis

Charles J. Corrado_Fundamentals of Investments - Chapter 15

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The Black-Scholes-Merton option pricing model assumes a constant stock price volatility, and yields option prices that may differ from stochastic volatility option prices. Nevertheless, even when volatility is stochastic the Black-Scholes-Merton option pricing model yields accurate option prices for options with strike prices close to a current stock price.

Handbook of Economic Forecasting part 4

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Latent variables, like the volatilities h t in the stochastic volatility model of Section 2.1.2, are common in econometric modelling. Their treatment in Bayesian inference is no dif- ferent from the treatment of other unobservables, like parameters. For the stochastic volatility model Equations (4)–(5) provide the distribution of the latent variables (hyperparameters) con- ditional on the parameters, just as (12) provides the hyperparameter distribution in the illustration of shrinkage.

Handbook of Economic Forecasting part 81

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Sections 3–5 present a variety of alternative procedures for univariate volatility mod- eling and forecasting based on the GARCH, stochastic volatility and realized volatility paradigms, respectively. Section 6 extends the discussion to the multivariate problem of forecasting conditional covariances and correlations, and Section 7 discusses volatility forecast evaluation methods in both univariate and multivariate cases.

SAS/ETS 9.22 User's Guide 132

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Andersen, T.G., Chung, H-J., and Sorensen, B.E. (1999), “Efficient Method of Moments Estimation of a Stochastic Volatility Model: A Monte Carlo Study,” Journal of Econometrics . (1996), “GMM Estimation of a Stochastic Volatility Model: A Monte Carlo Study,” Journal of Business and Economic Statistics . (1991), “Heteroscedasticity and Autocorrelation Consistent Covariance Matrix Estimation,” Econometrica . (1992), “Improved Heteroscedasticity and Autocorrelation Consistent Covariance Matrix Estimator

Handbook of Economic Forecasting part 82

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Sections 3–5 present a variety of alternative procedures for univariate volatility forecasting based on the GARCH, stochastic volatility and realized volatility paradigms, respectively. Section 6 extends the discussion to the multivariate problem of forecasting conditional covariances and correlations, and Section 7 discusses practical volatility forecast evaluation techniques..

Handbook of Economic Forecasting part 83

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One manifestation of this is the famous volatility-smiles which indicate systematic underpricing by the BSM model for in- or out-of-the-money options. The direction of these deviations, however, are readily explained by the presence of stochastic volatility, which creates fatter tails than the normal distribution, in turn increasing the value of in- and out-of- the-money options relative to the constant-volatility BSM model..

Handbook of Economic Forecasting part 85

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For example, Hull and White (1987) develop a theory for option pricing under stochastic volatility using a model much in the spirit of Taylor’s discrete-time log SV in Equation (4.6). The strength of the mean reversion in (log) volatility is given by β and the volatility is governed by v. Positive but low values of β induces a pronounced volatility persistence, while large values of v increase the idiosyncratic variation in the volatility series.

The perception of time, risk and return during periods of speculation

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In the short run, expected return is proportional to the temperature of the stock, where temperature is the product of the standard volatility and the square root of trading frequency. Other authors have used the stochastic nature of the time between trades to attempt to account for stochastic volatility and the implied volatility skew 7.

SAS/ETS 9.22 User's Guide 101

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Example 18.17: Simulated Method of Moments—Stochastic Volatility Model 1285 Example 18.18: Duration Data Model with Unobserved Heterogeneity. Example 18.19: EMM Estimation of a Stochastic Volatility Model. Example 18.20: Illustration of ODS Graphics. The MODEL procedure analyzes models in which the relationships among the variables comprise a system of one or more nonlinear equations.

Handbook of Economic Forecasting part 88

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More recent empirical studies and numerically efficient algorithms for the estimation of latent multivariate volatility structures include Aguilar and West (2000), Fiorentini, Sentana and Shephard (2004), and Liesenfeld and Richard (2003). A recent insightful discussion of the basic features of multivariate stochastic volatility factor models, along with a discussion of their ori-.

Handbook of Economic Forecasting part 86

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An early advocate for the Mixture-of-Distributions-Hypothesis (MDH), beyond Clark (1973), is Praetz (1972) who shows that an i.i.d. mixture of a Gaussian term and an inverted Gamma distribu- tion for the variance will produce Student t distributed returns. mixture is indistinguishable from a standard fat-tailed error distribution and the associated volatility process is not part of the genuinely stochastic volatility class..

Handbook of Economic Forecasting part 10

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Stochastic volatility. Moreover, in the case of financial assets, conditional vari- ances are fundamental to portfolio allocation. Stochastic volatility models provide an alternative approach, first motivated by autocorrelated information flows [see Tauchen and Pitts (1983)] and as discrete approximations to diffusion processes utilized in the continuous time asset pric- ing literature [see Hull and White (1987.

On a stochastic linear dynamical system driven by a volterra process

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Veraart, “On stochastic integration for volatility modulated L´ evy-driven Volterra processes”, Stochastic Processes and their Applica- tions , vol. https://doi.org/10.1016/j.spa Ole E. https://doi.org/10.4213/rm701. [5] Fabrice Baudoin, David Nualart, “Equivalence of Volterra processes”, Stochastic Processes and their Applications, vol. https://doi.org/10.1016/S . https://doi.org . [7] M.Fujisaki, G.Kallianpur and H.Kunita, “Stochastic Differential Equations for The Non Linear Filtering Problem”