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20 Pages

### 2009-04-01-d

Course: ECON 201, Spring 2009
School: Duke
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Word Count: 533

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Value Extreme Theory for High-Frequency Financial Data Abhinay Sawant April 1, 2009 Economics 201FS Extreme Value Theory Given i.i.d. log returns {r1, ..., rn}, the maximum return converges weakly to the Generalized Extreme Value (GEV) distribution: F (x ) = Two methods of estimation: (1) Block Maxima, (2) Points Over Thresholds Applications include financial risk management measures involving tail...

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Value Extreme Theory for High-Frequency Financial Data Abhinay Sawant April 1, 2009 Economics 201FS Extreme Value Theory Given i.i.d. log returns {r1, ..., rn}, the maximum return converges weakly to the Generalized Extreme Value (GEV) distribution: F (x ) = Two methods of estimation: (1) Block Maxima, (2) Points Over Thresholds Applications include financial risk management measures involving tail estimations such as Value at Risk and Expected Shortfall. EVT typically provides better tail estimations compared to other methods, especially for very high quantiles. Why High Frequency Data? 1. Better Estimation More data increases availability of extreme values Captures risk from intraday movements More data from more recent years 2. Intraday Measures Concepts such as intraday VaR have been proposed recently in current literature Standardization of Data In order to generate a data set of i.i.d. data {x1, ..., xn}, adjust all of the intraday returns by previous day's daily realized volatility: rt ,i x t ,i = RV 10,000 t -1 Still looked at both standardized and unstandardized data for analysis Block Maxima Method Divide time series data set into n subgroups and determine the maximum/minimum of each subgroup: {x1,...,xt| xt+1,...,x2t|...| x(g-1) t+1,...,xtg} If rn,i is the maximum/minimum of the the ith subsample, then xn,I should follow a GEV distribution for sufficiently large n. x n ,i rn ,i - n = n Maximum likelihood estimation through the built-in MATLAB function "gevfit" is used to determine values of the parameters , , and Estimated GEV CDF Citigroup (C) 8 minute frequency Maximum Values (Right Tail) Standardized Data n = 48 (one day) Estimates: = 0.1198 = 0.0012 = From 0.0013 Residuals GEV CDF Estimated GEV PDF Shape Parameter Scaling Parameter Location Parameter Value at Risk In order to determine the 0.01% VAR, set P(rt < rn*) = 0.999. The value rn* is the Value at Risk and can be determined from one of two ways: P( rn ,i r ) = P ( rt r ) * n * n * n [ ] n n r = n + (1 - ( -n ln(1 - p )) n ) n 8-Minute 0.01%-VAR 8-Minute 0.01%-VAR According to our EVT estimation, there is a 0.01% chance that we'll see an 8-minute standardized log return of 0.00726 or higher with sample standard deviation of 0.00004. Empirically, we find that about 0.0105% of standardized 8-minute returns are above 0.00726. Value at Risk for One Day Given a VAR with a one-day horizon, the VAR with a time-horizon of L days can be calculated as follows: VAR L = ( L ) VAR1 ^ Since there are 48 8-minute returns in a day, I multiplied my 8-minute VAR by (48)^() to determine a 1-day VAR. One Day 0.01%-VAR One Day 0.01%-VAR According to our EVT estimation, there is a 0.01% chance that we'll see an 8-minute standa...

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