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# Roll 1984 looked at the prices of orange juice

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- Roll (1984) looked at the prices of orange juice futures. By far the most important news for orange juice futures prices is news about the weather and news about the weather is equally likely to arrive at any time. He found that the Friday-to-Monday variance is only 1.54 times the first variance. - The only reasonable conclusion from all this is that volatility is to a large extent caused by trading itself . 2. Variance rate : risk managers often focus on the variance rate rather than the volatility. It is defined as the square of the volatility. 3. Implied volatilities are used extensively by traders. However, risk management is largely based on historical volatilities. 4. Suppose that most market investors think that exchange rates are log normally distributed. They will be comfortable using the same volatility to value all options on a particular exchange rate. But you know that the lognormal assumption is not a good one for exchange rates. What should you do? – You should buy deep-out-of-the-money call and put options on a variety of different currencies – and wait . These options will be relatively inexpensive and more of them will close in the money than the lognormal model predicts. The present value of your payoffs will on average be much greater than the cost of the options. In the mid-1980s, the few traders who were well informed followed the strategy and made lots of money. By the late 1980s everyone realized that out-of-the-money options should have a higher implied volatility than at the money options and the trading opportunities disappeared. 5. An alternative to normal distributions: the power law- has been found to be a good descriptions of the tails of many distributions in practice. - The power law : for many variables, it is approximately true that the value of v of - 5 -

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Study Notes: Risk Management and Financial Institutions By Zhipeng Yan the variable has the property that, when x is large , Pr ( ) ob v x Kx α > = , where K and alpha are constants. 6. Monitoring volatility - The exponentially weighted moving average model (EWMA). 2 2 1 (1 ) n n u σ λσ λ = + 2 1 n , RiskMetrics use λ =0.94. - The GARCH(1,1) MODEL 2 2 1 n L n V u 2 1 n σ γ α βσ = + + , Where 1 γ α β + + = , if γ =0, then GARCH model is EWMA - ML method to estimate GARCH (1,1) 7. How good is the model: - The assumption underlying a GARCH model is that volatility changes with the passage of time. If a GARCH model is working well, it should remove the autocorrelation of 2 i u . We can consider the autocorrelation of the variables 2 2 / i i u σ . If these show very little autocorrelation, the model for volatility has succeed in explaining autocorrelations in the . We can use 2 i u Ljung-Box statistic. If this statistic is greater than 25, zero autocorrelation can be rejected with 95% confidence.
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