1. Extreme Value Theory for High-Frequency Financial Data
Abhinay Sawant
April 1, 2009
Economics 201FS

2. Extreme Value Theory
Given i.i.d. log returns {r1, …, rn}, the maximum return converges weakly to the Generalized Extreme Value (GEV) distribution:
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.

3. 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

4. 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:
Still looked at both standardized and unstandardized data for analysis

5. 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.
Maximum likelihood estimation through the built-in MATLAB function “gevfit” is used to determine values of the parameters ξ, α, and β

6. Estimated GEV CDF Estimates: Citigroup (C) 8 minute frequency
Maximum Values (Right Tail)
Standardized Data
n = 48 (one day)
Estimates:
ξ =
α =
β =

7. Residuals From GEV CDF
Standardized: Max = ; Min = ; Unstandardized: Max = , Min =

8. Estimated GEV PDF

9. Shape Parameter

10. Scaling Parameter

11. Location Parameter

12. Value at Risk
In order to determine the 0.01% VAR, set P(rt < rn*) = The value rn* is the Value at Risk and can be determined from one of two ways:

13. 8-Minute 0.01%-VAR
8minVAR

14. 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 or higher with sample standard deviation of
Empirically, we find that about % of standardized 8-minute returns are above

15. 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:
Since there are 48 8-minute returns in a day, I multiplied my 8-minute VAR by (48)^(ξ) to determine a 1-day VAR.

16. One Day 0.01%-VAR

17. One Day 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 or higher with sample standard deviation of
Empirically, we find that about 2.32% of standardized daily log returns are above Empirically, about 0.01% of the data are above

18. Other VAR Methods
Sliding Window: Use previous 50 day’s worth of data to determine VAR for 51st day and calculate the number of exceedances, in order to determine the effectiveness of VAR model.

19. Points Over Thresholds
For a sufficiently high threshold u, the distribution function of the excesses (X – u) may be approximated by a Generalized Pareto Distribution:
The pth quantile can be computed from the parameters:

20. Further Research
Different Frequencies
Different Stocks