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Quantitative Finance Collector is a blog on Quantitative finance analysis, financial engineering methods in mathematical finance focusing on derivative pricing, quantitative trading and quantitative risk management. Random thoughts on financial markets and personal staff are posted at the sub personal blog.

Nov 15
We all know one assumption of Black Scholes model is constant volatility during option period, which has been relaxed by several methods including Heston stochastic volatility, SABR stochastic volatility, etc. Here is another way proposed by Jin-Chuan Duan, Geneviève Gauthier, and Jean-Guy Simonato in their paper "An analytical approximation for the GARCH option pricing model" published at the Journal of computational finance in 1999, http://www.journalofcomputationalfinance.com/public/showPage.html?page=1112.
GARCH option pricing framework has been developed in recent years. However, an efficient method for computing option prices in this framework remains lacking. In this article, a fast analytical approximation is developed for computing European option prices in the GARCH framework. The approach, following that of Jarrow and Rudd (1982), uses the Edgeworth expansion of the risk-neutral density function. Analytical expressions for the first four moments of the cumulative asset return over any horizon under the GARCH model are derived in this paper. A numerical analysis shows that these moment formulas are accurate under fairly general conditions. The analytical GARCH option pricing formula based on the Edgeworth expansion is found to work well for short-maturity options. For long-maturity options, the approximate formula is generally satisfactory, except when the volatility dynamic of the GARCH model exhibits an extremely high level of persistence.

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Mar 19
The Oxford MFE Toolbox is the follow on to the UCSD GARCH toolbox. It has been widely used by students here at Oxford, and represents a substantial improvement in robustness over the original UCSD GARCH code, although in its current form it only contains univariate routines.

Contents include:
1 Stationary Time Series 5
1.1 ARMA Simulation
1.1.1 Simulation: armaxfilter_simulate . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 ARMA Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Estimation: armaxfilter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Residual Plotting: tsresidualplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.2.3 Characteristic Roots: armaroots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.2.4 Information Criteria: aicsbic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.3 ARMA Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.3.1 Forecasting: arma_forecaster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4 Sample autocorrelation and partial autocorrelation . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 Sample Autocorrelations: sacf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.2 Sample Partial Autocorrelations: spacf . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.5 Theoretical autocorrelation and partial autocorrelation . . . . . . . . . . . . . . . . . . . . . 27
1.5.1 ARMA Autocorrelations: acf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5.2 ARMA Partial Autocorrelations: pacf . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.6 Testing for serial correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6.1 Ljung-BoxQ Statistic: ljungbox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.6.2 LM Serial Correlation Test: lmtest1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2 Nonstationary Time Series 37
2.1 Unit Root Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.1 Augmented Dickey-Fuller testing: augdf . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.2 Augmented Dickey-Fuller testing with automated lag selection: augdfautolag . . . . 40
3 Vector Autoregressions 43
3.1 Stationary Vector Autoregression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.1 Vector Autoregression estimation: vectorar . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.2 Granger Causality Testing: grangercause . . . . . . . . . . . . . . . . . . . . . . . . 50
3.1.3 Impulse Response function calculation: impulseresponse . . . . . . . . . . . . . . 53
4 Volatility Modeling 57
4.1 GARCH Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1.1 ARCH/GARCH/GJR-GARCH/TARCH/AVGARCH/ZARCH Estimation: tarch . . . . . . 57
4.1.2 Some behind the scenes choices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1.3 EGARCH Estimation: egarch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.1.4 APARCH Estimation: aparch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5 Density Estimation 71
5.1 Kernel Density Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Code and documention are available at: http://www.kevinsheppard.com/wiki/MFE_Toolbox
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Jul 25
The primary feature that differentiates GARCHKIT from other GARCH implementations in Matlab is its ability to incorporate covariates into the second moment. The current version of GARCHKIT, 1.0b3, allows univariate ARMA(P,Q)-GARCH(R,S) estimation and simulation using maximum likelihood. The conditional distribution may be normal, student's t or a mixture of two normals.

Version 1.1 now estimates and simulates FIGARCH and GARCH-in-Mean models.

Code can be downloaded at http://www-agecon.ag.ohio-state.edu/people/roberts.628/papers/research/garchkit/garchkit.html
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Jul 24
Parameters estimation of GARCH model.

http://w3.uniroma1.it/passalac/buffer/GARCH.xls

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