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Modeling Volatility with ARCH and GARCH Models

Learn about ARCH and GARCH models for modeling the volatility of asset returns. Understand the importance of volatility in options trading and how volatility modeling can improve parameter estimation and forecast accuracy. Explore the characteristics of volatility and the properties of ARCH models.

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Modeling Volatility with ARCH and GARCH Models

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  1. Tutorial 8 SEG 7550 6th Nov.

  2. Outline • ARCH and GARCH model (part one) • The project • About assignment 2

  3. ARCH and GARCH model • Modeling volatility of a asset return • Volatility is an important factor in options trading • Here volatility means conditional standard deviation of the underlying asset return • Why? • Volatility modeling provides a simple approach of calculating value at risk • Modeling volatility can improve the efficiency in parameter estimation and the accuracy in interval forecast • Volatility index of market is a financial instrument

  4. Character of volatility • Cannot be observed directly • Make it difficult to evaluate forecasting performance • Specific model assumption may not hold in practice • Blac-scholes formula assume geometric Brownian motion • Volatility clusters • Volatility evolves over time in a continuous manner • Volatilities jumps are rare • Volatility does diverge to infinity • Volatility seems to react differently to a big price increase or a big price drop (leverage effect)

  5. ARCH • Basic idea • The shock of an asset return is serially uncorrelated • The dependence of can be described by a simple quadratic function of its lagged values, i.e. an ARCH(m) model: Where is iid sequence with mean 0 and variance 1

  6. ARCH • The parameters • , for • must satisfy some regularity conditions to ensure the unconditional variance of is finite • In practice, is often assumed to follow normal distribution or student’s t distribution.

  7. Properties of ARCH • Take ARCH(1) as example: • Where and • The conditional mean of is 0 because:

  8. Properties of ARCH • Conditional variance of : Because is stationary process which means: We have

  9. Properties of ARCH • Variance must be positive, therefore • In practice we want higher moments of exist and, hence, must satisfy some additional constraints. • An example: if we want to examine the tail behavior, we want the fourth moment of is finite. Under the assumption follow normal distribution, we have: if is fourth-order stationary with then:

  10. Properties of ARCH • Example con’d:

  11. Properties of ARCH • Example con’d: • Since the fourth moment of is positive, we have • The unconditional kurtosis of is: tail distribution of is heavier than normal

  12. Weakness of ARCH • Positive shocks and negative shocks have the same effect • Rather restrictive (constraints on parameters) • Does not provide any insight for understanding the source of variations of a financial time series • ARCH tends to overpredict the volatility because they response slowly to large isolated shock to the return series

  13. Project • There would be no exams • Please submit your project title and description to hding@se.cuhk.edu.hk • A sample is provided on course website

  14. How to work on your Course Project? • (1) Data Collection • This phase is quite time consuming as the location of data source is not easy. Also, your model is limited by your data. • Please refer to your tutorial website for the collection of daily data. • Or try these two website for data: • http://www.hkgem.com • http://www.hkex.com.hk • For the intraday data, you can collect from Market Browser. Please note that they only provides data up to 14:50 and 5 days data. • http://www.marketbrowser.com/mbzzzq2.asp

  15. How to work on your Course Project? • (2) Background study • You need to find out the related works for your project. Is the result of the related works better than yours. What is the merits and weaklessness of them? Also, please mark the reference papers and books and include them in the reference. ... • (3) Construction of your model • What is your model/strategy? Please describe your model using diagrams or equations. Please state any assumptions. ...What is your model/strategy? Please describe your model using diagrams or equations. Please state any assumptions. ...

  16. (4) Experimental Result • Please provide the significant figure. Does your result agree with your prediction. ... • (5) Conclusion and Future Work • What conclusion can you draw? Does your work perform better than that of the related works? If not, please state any improvement. If yes, how can you enhance the model to provide better result? What is the potential use of your model? ...

  17. (7) Reference • Please provide the list of references. The format of the reference is like that • Book • Format : Author Name, "Book title", Edition, publisher, publishing year. • e.g. Francis X. Diebold,"Elements of Forecasting", Fouth Edition, Thomson South-Western, 2007. • Conference Paper • Format : Author name, paper title, Name of Conference, pape number, Conference venue, date, year. • e.g. H.S. Ng and K.P. Lam, "Incremental Intraday Prediction of Extreme Values and Range-based Volatility", In Proc. of The Third IASTED International Conference on Financial Engineering and Applications, pp90-98, Boston, October 9-11, 2006. • Journal Paper • Format : Author Name, Paper title, Journal, Vol(issue), page number, year. • e.g. E. F. Fama, The Behavior of Stock-Market Prices, The Journal of Business, 38(1), 34–105, 1965.

  18. About assignment 2Question one: (a) • (i) Please capture the HSBC and Cheung Kong financial charts from January to mid-Februray from internet. • (ii) Trace through the major ups and downs of HSBC and Cheung Kong price movement for the above period and relate them to any relevant changes in fundamental, major events and news. Please identify the two or three major events, news, or relevant changes in fundamental. e.g. For China Telecom (0941) in 2000,

  19. Question one:

  20. Question one: • Mark will be counted based on • (1) the date, • (2) the price change (approximation), and • (3) the relationship between the news and the major ups and downs of PCC price movement

  21. Question one: • (b) • Please use gsl.dat as the input of your Matlab program. • (i) Modify the program "candle_test.m" to detect "The Three Knights" and "The Three Crows". List the date detected these signal. (Note: Since the close price is available after market open, the detection date should be one day after "The Three Knights" or "The Three Crows". Please define your candle's body.) • (ii) Please state your trading strategy such as assumption, consideration and trading rules. • (iii) Please submit the amended program.

  22. Question one: • (b) • (iv) Please list your buy date and sell date.

  23. Question one: • (b) • (iv) Please provide the (position or negative) return. • (v) Please plot the buy/sell point and the stock price at the same figure. • (vi) Please explain your result, if applicable, and state its potential use.

  24. Question two (a) • i. Generate a set of random number (dataset A). • ii. For model M1, construct an AR(2) model based on dataset A, obtain the output of the AR(2) model, name the output dataset as dataset Ao. • iii Generate another set of random number (dataset B).

  25. iv. For model M2, construct and AR(4) model based on dataset B, obtain the output of the AR(4) model, name the output dataset as dataset Bo. • v. Repeat i to iv to generate another dataset Ao and Bo and call them as Ao1 and Bo1 respectively. • vi. Construct a dataset with the sequence [M1|M2|M1|M2].

  26. Recursive Least Squares • Matlab Function: • rarx(z,nn,adm,adg) - Refer to Page 4-80 of System Identification Toolbox Manual • Matlab program: • Cheungkong_rtest.m • Load data • Obtain the return series using the loaded data • Estimate the four order ot AR for dly • Plot the parameter estimates • Plot the predicted differential-log output and actual differential-log output with time • Recover the actual output • Plot the predicted output, the actual output and the prediction error

  27. Question three • (i) Please submit the amended program for (2-20) MA rule, and (20-50) MA rule. • (ii) Please use the following equation to calculate the percentage of return. • ( y_typ(buy_date) - y_typ(sell_date) ) / y_typ(buy_date) where y_typ is gsl_typ or cheungkong_typ

  28. Question three • Please find the mean of the percentage of return and compare the mean of the percentage of return for the two stocks. • If the first signal is sell, you can ignore the first signal. If the last signal is buy, you also need to ignore the last signal. • You need to submit the list for buy date, sell date and the return for the two stocks.

  29. Question three • For GSL

  30. Question three • For Cheung Kong

  31. Question three • (iii) Please submit the graph which indicates the buy prices, sell prices and the gsl_typ/cheung_typ. You can use the following Matlab function to plot the graph. • plot(1:333,gsl_typ,gsl_sell,gsl_typ(gsl_sell),'o',gsl_buy,gsl_typ(gsl_buy),'x') where gsl_typ is 0.25*(gsl_open+ gsl_close+ gsl_high + gsl_low), gsl_sell is the row of gsl_typ where you sell the stock at that date, and gsl_buy is the row of gsl_typ where you buy the stock at that date.

  32. Question three • (iv) Please state which stock is more profitable.

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