Applications Of Evt In Financial Markets

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Applications Of Evt In Financial Markets

  1. 1. Abstract Extreme Value Theory (EVT) has emerged as an important statisticaldiscipline for the applied sciences. It is useful because it provides techniques forestimating models that predict events occurring at extremely low probabilities.This paper describes EVT and tools such as quantile plots and mean excessplots used to determine the appropriateness of EVT for modeling given data.EVT techniques are then applied to model daily return of stock of three largecompanies: IBM, Ford and Nortel. The results show that Generalized ParetoDistribution (GPD) can appropriately model extreme daily returns, particularlyextreme daily losses. Finally the parameters of the appropriate GPD areestimated, and Value-at-Risk (VaR) and Expected Shortfall, the two key riskmeasures used by industry practitioners, are calculated based on the estimatedGPD.Keywords: Extreme Value Theory, Generalized Pareto Distribution, Value-at-Risk, Expected Shortfall 1 of 34
  2. 2. Introduction, Scope & Purpose EVT has been successfully applied in engineering, biology, meteorology,insurance and a myriad of other applied sciences1. This paper will focus onApplications of EVT in Financial Markets. It has been noted that EVT’s application to extreme risk in financialmarkets maybe motivated by the problem of daily determination of VaR forlosses incurred due to adverse market movements. Risk Managers areinterested in describing the tail of a loss distribution and measuring the expectedsize of a loss that exceeds VaR2. Until recently, most parametric methods usedthe Normal distribution to estimate VaR. However, under the assumption ofnormality, the risk of high quantiles is underestimated, especially for the fat-tailedseries common in financial data. This paper outlines the theoretical underpinnings of EVT and worksthrough examples illustrating how EVT can be applied to financial data. The firstpart of this paper introduces classical EVT, models for maxima/minima andthreshold models. However, modeling the maxima or minima of financial data isof little value to risk managers. Instead, a threshold model based on theGeneralized Pareto Distribution (GPD) is argued to be most suited for riskmanagers because it can be used to model the tail of a loss distribution3. Thus,the second part of this paper models daily returns price for IBM, Ford and Nortelstocks using the Generalized Pareto Distribution.1 Coles, Stuart. An Introduction to Statistical Modeling of Extreme Values.2 McNeil, A. “Extreme Value Theory for Risk Managers”. Pg 1-23 Ibid. 2 of 34
  3. 3. Theoretical Underpinnings4Classical EVT and Models The fundamental model of Extreme Value Theory is based on thebehavior of Mn where: Mn = max{X1, …, Xn}and Xi are independent identically distributed (iid) random variables. In theory,the distribution of Mn can be easily derived if the distribution of Xi are knownbecause if F(z) is the distribution of the Xi, then FMn(z) = P(Mn ≤ z) = P(X1 ≤ z, …, Xn ≤ z) = P(X1 ≤ z)…P(Xn ≤ z) = [F(z) ]n In practice this is not possible because the distributions of the Xi are notusually known. One possible solution to this problem is to estimate F(z) based onobserved values and then to derive FMn(z). However this approach is problematicbecause any estimation involves errors and small errors in estimating F(z) wouldlead to large errors in the estimation of FMn(z). Another solution is to accept F(z) as unknown and then to try and find afamily of functions that model FMn based only on extreme (maximal) data. Thearguments to justify this method are analogous to the justifications underlying theCentral Limit Theorem. Pursuing this method further, we consider: Mn* = (Mn – bn)/an where {an > 0} and {bn} are constantsAppropriate choices of bn and an stabilize location and scale of Mn* as n ∞. Allpossible limit distributions for Mn* are given by the Extremal Types Theorem:4 Adapted from book by Stuart Coles. See references. 3 of 34
  4. 4. In the above distribution, b is the location parameter, a is the scaleparameter and α is the shape parameter. The type I, type II and type III modelsabove are known as the Gumbel, Frechet, and Weibull distributions respectively.This theorem says that regardless of the distribution of the Xi, if we normalize Mnto Mn*, then the distribution of Mn* is of only 3 possible types. These three types of extreme models have distinct forms of behaviorcorresponding to different forms of the tail distribution of the Xi. Application ofEVT requires choosing one of the three models to estimate parameters. But thisraises two important problems. Firstly, how do you know which model type to use?A technique is needed to choose the appropriate model type for given data.Secondly, once the decision of model type has been made, all subsequentinferences will rest on the assumption that the decision of model type was correctand thus will not allow for any uncertainty in that decision. These problems are solved by combing the three model forms into theGeneralized Extreme Value Distribution: 4 of 34
  5. 5. In this combined model, µ is the location parameter, σ is the scaleparameter and ξ is the shape parameter. When ξ > 0, the GEV distributioncorresponds to the Frechet distribution. When ξ < 0, the GEV distributioncorresponds to the Weibull distribution. The case of ξ = 0 can be interpreted astaking the limit as ξ 0 and this corresponds to the Gumbel distribution. By doing statistical inference on ξ, the two problems associated withchoosing model types are solved. The data itself now determines the mostappropriate type of tail behavior. Furthermore, uncertainty in estimating ξcorresponds to uncertainty in choosing the correct model type. At this point we stop to consider that our original problem was to model Mn,not Mn*. In practice, the constants an and bn may not be known. However, wehave shown that P(Mn* ≤ z) = P((Mn - bn)/ an ≤ z) ≈ G(z). Then for large n, we canwrite P(Mn≤ z) ≈ G((z - bn)/ an) = G*(z)where G*(z) is also a member of the GEV family of distributions. Since theparameters of the distribution have to be estimated anyways, it is irrelevant thatthe parameters for G(z) are different than the parameters for G*(z) for a givendata set. 5 of 34
  6. 6. By the above argument, an approach to modeling extremes of Xi isdeveloped. First all the data is blocked into a sequence of observations of size n(n is large). From each block, we can derive block maxima: Mn 1, …, Mn m whereMn i is the block maxima from the block i of size n. The GEV can now be fittedusing Mn 1, …, Mn m as data. Often blocks are chosen according to time periods of1 year.Block Minima At this point let us note that the GEV distribution also provides asymptoticmodels for minima. Given data z1,…,zn , we can simply maximize -z1,…,-zn.Inference on GEV distribution: The aforementioned method for implementing the GEV distributionrequired us to divide the data into equal size blocks and fit to the set of blockmaxima. However choosing block size is always a trade off between bias andvariance. Overly small block sizes lead to bias because approximation by theGEV model is poor. Overly large block sizes generate fewer block maxima andthus lead to large estimation variance. For time series data sets, block sizes ofone year are commonly chosen because this usually makes plausible theassumption that block maxima have common distribution. Now we consider Z1 ,…,Zn where Zi are iid block maxima from a GEVdistribution whose parameters need estimation. The Zi are independent even ifthe Xi are not independent (as in the case of most time series). Likelihood-basedestimation provides one effective method of estimating these parameters. Thelog-likelihood for the GEV when ξ ≠ 0 is 6 of 34
  7. 7. The log-likelihood for the GEV when ξ = 0 (corresponding to the Gumbeldistribution) is: For any given data set, these log likelihood functions can be maximizedusing numerical methods. The estimators derived from this method can beassumed to be approximately multivariate normal and unbiased. Confidenceintervals and other inferences follow from this assumption of normality. Modelchecking can be done by plotting the empirical distribution function evaluated atzi for ordered block maxima against the model evaluated with the estimates. Agood fit will produce a linear graph lying close to the line y = x.Threshold Models In our quest to model extreme events, we may be given an entire timeseries of daily observations. Better use of this data is made by avoiding blocking.If X1, X2, … are iid, then extreme events can be defined as those Xi that exceedsome high threshold u. In modeling extreme events, we are interested in theconditional probability of X-u given X > u. Theory tells us that if block maxima ofthe data have an approximate GEV distribution, then this conditional probabilitycan be approximated by the Generalized Pareto Distribution. 7 of 34
  8. 8. Moreover, the parameters of the GPD (modeling excess over threshold)are uniquely determined by the parameters of the associated GEV modelregardless of the block size. This is because the shape parameter ξ isindependent of block size. Also β = σ + ξ(u – µ) where σ and µ are from theassociated GEV distribution. Changing block size n in the associated modeladjusts σ and µ in a self-compensating way so that β remains constant.Modeling Threshold Excesses Naturally the first step in modeling extreme data within the framework ofthreshold models is to choose an appropriate threshold. Choice of threshold isanalogous to the problem of choosing the appropriate block size in the GEVmodel. If the threshold is set too low, then the data beyond the threshold willdeviate significantly from the GPD. On the other hand, if the threshold is set toohigh, there will not be enough data to estimate the model, and a high variancewill result. So we must choose as low a threshold as possible provided that theGPD is still a reasonable approximation for excesses beyond the threshold. Onemethod to determine the threshold is to create the mean excess function plot.Theory tells us that if Y has GPD, then E(Y) = β/(1- ξ). Since the conditionaldistribution of excesses beyond threshold is approximated by GPD, we knowfrom theory that for a threshold u0: E( X– u0 | X > u0 ) = β(u0)/(1- ξ)where β(u0) is the value of β corresponding to the threshold u0. However, if theexcess beyond u0 can be modeled with GPD, then so can excess beyond anyother threshold u>u0. It has been shown that for u >u0, 8 of 34
  9. 9. E( X– u | X > u ) = (β(u0) + ξu) / (1- ξ)This is linear in u. Therefore we expect that for thresholds beyond which theexcesses follow a GPD, the conditional mean will be a linear function. Thus wehave the following method to determine the threshold. : Let x1, …, xn be the datato be modeled. 1. Order the data: x(1),…, x(n) 2. For each u in { x(1),…, x(n) } calculate the sample mean of the difference between the x’s and u for all x’s > u. In other words, calculate and plot: 3. Identify the point beyond which this graph is approximately linear and choose that as the threshold for the model. Ensure that there are sufficient points beyond the chosen threshold to make meaningful inferences.Estimating Parameters Once the threshold is chosen, likelihood techniques can be used toestimate parameters for the GPD model while considering only the data that liesbeyond the chosen threshold. Let y1,…, yk be the excesses of a threshold u (so yi= xi – u for xi > u). The log likelihood function for the GPD in the case where ξ ≠ 0is: ℓ(β,ξ) = -kln(β) – (1 + 1/ξ)∑ ln(1 + ξyi/β)provided that (1 + ξyi/β) > 0 for i = 1,…,k.In the case of ξ = 0, the log likelihood function is: 9 of 34
  10. 10. ℓ(β) = -k*ln(β) – (1/β)∑ yiThese log likelihood functions can be numerically maximized and the maximumlikelihood estimate (mle) for β, ξ can be found.Model Checking The models validity can be checked with probably plots, quantile plots anddensity plots. For a threshold u and threshold excesses y(1),…, y(k), the probabilityplot (for i = 1,…, k) consists of the pairs {i/(k+1), 1 – (1 + ξy(i)/β)-1/ξ }using the mle for ξThe quantile plot consists of the pairs {H-1(i/(k+1)), y(i)} for i= 1,…, kwhere H-1(t) = u + β/ξ[t-ξ – 1] using the mle for β and ξ. If the GPD is a good fit, the probability plot and the quantile plot will beapproximately linear. Also the density function of the fitted GPD can becompared to a histogram of the threshold excesses.Estimating VaR and Expected Shortfall5 As mentioned in the introduction, a major use of EVT in risk managementis to characterize the tail of a loss distribution using VaR and Expected Shortfall.VaR is a high quantile of a distribution of losses and can represent an upperbound for losses that is exceeded only rarely. Expected Shortfall is expected sizeof a loss that exceeds VaR.5 McNeil, A. “Extreme Value Theory for Risk Managers”. Pg 2-3. 10 of 34
  11. 11. Using historical simulation and maximum likelihood estimates of theparameters of GPD, the following tail estimator has been derived:where Nu is the number of data points that exceed the threshold u and n is thetotal number of data points. The VaR estimate for a probability q is calculated byinverting the tail estimation of the above formula to give:The expected shortfall is related to VaR by the following formula:where the second term is the mean of the excess distribution over the threshold.In practice, these can be calculated using the ‘riskmeasures’ function of the EVIRpackage in R.Application of EVT We now turn to three examples that illustrate how to fit the GPD tofinancial data and produce a model for extremes beyond a threshold. The rawdata for these examples are the historical prices of IBM, Ford, and Nortel stockswhich were downloaded from:IBM stock: http://finance.yahoo.com/q/hp?s=IBMFord stock: http://finance.yahoo.com/q/hp?s=FNortel stock: http://finance.yahoo.com/q/hp?s=NT The data analyzed however are historical daily returns price. The returnsprice for each day was calculated by the formula: 11 of 34
  12. 12. Returns price = (today’s price – yesterday’s price)/yesterday’s price Clearly, the returns price is a measure of daily gain or loss in stock priceregardless of the actual stock price. Modeling extremes of returns price is usefulbecause it can help risk managers determine what the maximum gain or moreimportantly, maximum loss that can be incurred in one day.Historical Daily Returns Prices of IBM The time series for IBM ranged between January 2 1980 and March 262004. The first step in analyzing this data is to see if it can be satisfactorilymodeled by the Normal distribution. We do this by creating a normal quantile plotand look for linearity: Notice that the data near the endpoints deviates from linearity significantly.This implies that the true distribution of the data is fat tailed and so the extremesare not normally distributed. This provides us with the impetus to model theextremes using EVT. 12 of 34
  13. 13. Now we must determine the threshold beyond which we can define thedata as “extreme”. To do this, we must first construct a plot of the mean excessfunction. Then we try to determine a high enough point beyond which the plotlooks linear but at the same time provides sufficient points for inference.Furthermore, we will omit the three largest losses because they tend to distortthe plot6. The mean excess function for different possible thresholds is:From this plot we can estimate the threshold to be approximately 0.06 becausethe data seems to kink downward at this point. This represents the threshold fordaily gain in stock prices. Now we can fit the GPD to data beyond this thresholdby using the gpd function in the EVIR software. We get the following results usinglikelihood methods of estimation:Total Number of Data Points: 6118Chosen Threshold: 0.06Number of Points Exceeding Threshold: 446 As mentioned in McNeil, A. & Saladin, T. “The Peaks over Threshold Method for Estimating HighQuantiles of Loss Distributions” Departement Mathematik ETH Zentrum. 13 of 34
  14. 14. Approximate Percentile at which threshold is located: 99th percentileParameter Estimate for Shape (ξ): -0.1376Parameter Estimate for β: 0.0253Variance-Covariance Matrix: Shape BetaShape 0.0631 -0.0017Beta -0.0017 5.406e-05Now we must do diagnostic checks to see if the model is a good fit. The quantileplot of the residuals is:Since this quantile plot is approximately linear, we conclude that the GPD is agood fit for this data. Thus extreme values (beyond 0.06) can be modeled by: G(x) = 1 – (1 + (-0.1376)x/(0.0253))1/(0.1376) We have just modeled the extremes for daily gain of IBM stock prices.However, in many financial situations, we have greater concern with negativevalues for daily returns because that implies that a loss is incurred. In order to 14 of 34
  15. 15. analyze extreme negative values, we need to multiply the given data by -1 andrepeat the procedure outlined above. We will now model the negative daily returns for IBM stock prices. Themodified mean excess function is:From this plot we choose can choose 0.05 as a threshold. So -0.05 representsthe threshold for daily loss. Now we can fit the GPD to data beyond this thresholdby using the gpd function in the EVIR software. We get the following results usinglikelihood methods of estimation: 15 of 34
  16. 16. Total Number of Data Points: 6118Chosen Threshold: 0.05Number of Points Exceeding Threshold: 46Approximate Percentile at which threshold is located: 99th percentileParameter Estimate for Shape (ξ): 0.4689Parameter Estimate for β: 0.0128Variance-Covariance Matrix: Shape BetaShape 0.0566 -0.0005Beta -0.0005 1.169e-05Now we must do diagnostic checks to see if the model is a good fit. The quantileplot of the residualsis: 16 of 34
  17. 17. Since this quantile plot is approximately linear, we conclude that the GPDis a good fit for this data. Thus the negative extreme values (beyond -0.025) canbe modeled by: G(x) = 1 – (1 + (0.4689)x/(0.0128))-1/(0.4689) In risk management, we are interested in estimates of VaR and ExpectedShortfall for different p-values of this model. In statistical language, VaR is simplya quantile estimate7. These are easily found for the above model using the EVIRsoftware: p-value Estimate of VaR Estimate of Expected Shortfall0.99 0.04658902 0.06763750.999 0.09292915 0.15488970.9999 0.22934078 0.41173420.99999 0.63089623 1.1678084Historical Daily Returns Prices of Ford Historical Prices of Ford Stocks was downloaded from Yahoo Finance.The time series ranged from January 2 1987 to March 26 2004. The returns pricewas calculated and forms the data for the present analysis. To see if the data hasa fat tail, we plot the normal quantiles for this data:7 McNeil, A. “Extreme Value Theory for Risk Managers” pg 7. 17 of 34
  18. 18. We see that the tail curves slightly away from the straight line, indicatingdeviation from normality. For confirmation, we also plot the empirical distributionfunction of the data on the log-log scale. A straight line on the double log scaleimplies Pareto tail behavior8:8 EVIR help document. 18 of 34
  19. 19. We see that the tail is approximately linear. So we are now justified in fittingthe GPD to the tails. As before, we must first find the threshold by plotting theMean Excess function. Furthermore, we will omit the three largest lossesbecause they tend to distort the plot9:9 As mentioned in McNeil, A. & Saladin, T. “The Peaks over Threshold Method for Estimating HighQuantiles of Loss Distributions” Departement Mathematik ETH Zentrum. 19 of 34
  20. 20. We see that the graph is approximately linear beyond 0.06; so we will choosethis as our threshold. This represents the threshold for maximum possible gainper day. Now we will fit the GPD for the data beyond this threshold using the gpdfunction of the EVIR package. We get the following results using likelihoodmethods of estimation:Total Number of Data Points: 4348Chosen Threshold: 0.03Number of Points Exceeding Threshold: 47Approximate Percentile at which threshold is located: 98th percentileParameter Estimate for Shape (ξ): 0.4607Parameter Estimate for β: 0.0158Variance-Covariance Matrix: Shape BetaShape 0.03637 -0.0003453Beta -0.0003453 1.306e-05 20 of 34
  21. 21. Now we look at the quantile plot of the residuals to see if the GPD model isa good fit:The quantile plot seems to be approximately linear, indicating that we have founda good fit for this data. So we conclude that negative extreme values (beyond0.03) can be modeled by: G(x) = 1 – (1 + (0.4607)x/(0.0158))-1/(0.4607) The above model corresponds to daily gain of Ford stock prices. In orderto analyze extreme negative values, we need to multiply the given data by -1and repeat the procedure outlined above. We will now model the negative dailyreturns for Ford stock prices. The last 3 data points in the mean excess functionare omitted as before. The mean excess function is now: 21 of 34
  22. 22. From this plot we can estimate the threshold to be approximately 0.05. So-0.05 represents the threshold for daily loss. Now we can fit the GPD to databeyond this threshold by using the GPD function in the EVIR software. We getthe following results using likelihood methods of estimation:Total Number of Data Points: 4348Chosen Threshold: 0.05Number of Points Exceeding Threshold: 58Approximate Percentile at which threshold is located: 98th percentileParameter Estimate for Shape (ξ): 0.2646Parameter Estimate for β: 0.01145 22 of 34
  23. 23. Variance-Covariance Matrix: Shape BetaShape 0.02022 -1.619e-04Beta -1.619e-04 4.577e-06Now we must do diagnostic checks to see if the model is a good fit. The quantileplot of the residuals is:Since this quantile plot is approximately linear, we conclude that the GPD is agood fit for this data. Thus negative extreme values (beyond -0.05) can bemodeled by: G(x) = 1 – (1 + (0.2646)x/(0.01145))-1/(0.2646) In risk management, we are interested in estimates of VaR and ExpectedShortfall for different p-values of this model. These are easily found for the abovemodel using the EVIR software: 23 of 34
  24. 24. p-values Estimate of VaR Estimate of Expected Shortfall0.99 0.05343 0.07020.999 0.09260 0.12350.9999 0.1646 0.22140.99999 0.2971 0.4015Historical Daily Returns Prices of Nortel Historical Prices of Nortel stocks were downloaded from Yahoo Finance.The time series ranged from December 16 1991 to March 26 2004. The returnsprice was calculated and forms the data for the present analysis. To see if thedata has a fat tail, we plot the normal quantiles for this data: 24 of 34
  25. 25. Notice that the data near the endpoints deviates significantly from linearity.This implies that the true distribution of the data is a fat tailed distribution and sothe extremes are not normally distributed. This provides us with the impetus tomodel the extremes using EVT. To choose the threshold we now consider the plot of the mean excessfunction:From this plot we estimate the threshold to be approximately 0.105. Thisrepresents the threshold for the maximum possible gain everyday. Now we can 25 of 34
  26. 26. fit the GPD to data beyond this threshold by using the gpd function in the EVIRsoftware. We get the following results using likelihood methods of estimation:Total Number of Data Points: 3093Chosen Threshold: 0.105Number of Points Exceeding Threshold: 40Approximate Percentile at which threshold is located: 98th percentileParameter Estimate for Shape (ξ): -0.1379Parameter Estimate for β: 0.04853Variance-Covariance Matrix: Shape BetaShape 0.0329 -0.0017Beta -0.0017 0.0001349 Now consider the following quantile plot of the residual to verify whether theaforementioned model is a good fit for the data: 26 of 34
  27. 27. The quantile plot seems to be approximately linear, indicating that we have founda good fit for this data. Thus we conclude that negative extreme values (beyond0.105) can be modeled by: G(x) = 1 – (1 + (-0.1379)x/(0.04853))-1/(-0.1379) The above model corresponds to the extremes for daily gain of Nortelstock prices. In order to analyze extreme negative values, we need to multiplythe given data by -1 and repeat the procedure outlined above. We will now model 27 of 34
  28. 28. the negative daily returns for Nortel stock prices. The modified mean excessfunction is:From this plot we can estimate the threshold to be approximately 0.07. So -0.07represents the threshold for daily loss. Now we can fit the GPD to data beyondthis threshold by using the gpd function in the EVIR software. We get thefollowing results using likelihood methods of estimation:Total Number of Data Points: 3093Chosen Threshold: 0.07Number of Points Exceeding Threshold: 77 28 of 34
  29. 29. Approximate Percentile at which threshold is located: 97th percentileParameter Estimate for Shape (ξ): 0.2601Parameter Estimate for β: 0.0285Variance-Covariance Matrix: Shape BetaShape 0.02658 -6.342e-04Beta -6.342e-04 3.123e-05Now we must do diagnostic checks to see if the model is a good fit. The quantileplot of the residuals is: 29 of 34
  30. 30. Since this quantile plot is approximately linear, we conclude that the GPD is agood fit for this data. So we conclude that negative extreme values (beyond-0.03) can be modeled by: G(x) = 1 – (1 + (0.2601)x/(0.0285))-1/(0.2601) In risk management, we are interested in estimates of VaR and ExpectedShortfall for different p-values of this model. These are also easily found for theabove model using the EVIR software: p-values Estimate of VaR Estimate of Expected Shortfall0.99 0.09933 0.14820.999 0.2132 0.30210.9999 0.4206 0.58230.99999 0.7980 1.092 30 of 34
  31. 31. R Software and Code Used to Model Data First of all, the EVIR and EVD packages must be downloaded andinstalled into the R program in order to utilize functions pertaining to EVT. Thesecan be procured for free from the following website: http://www.maths.lancs.ac.uk/~stephena/software.htmlThe following is a generic version of the specific code used to do model the threecases above.//Imports the EVD and EVIR packages needed for modeling.>library(evd)>library(evir)//Reads the file “data.txt” containing data to be modeled.>data = scan(“data.txt”)//Plots the sample normal quantiles against the theoretical normal quantiles>qqnorm(data)//Adds a line to the normal QQ-plot.>qqline(data, col=2)//Plots empirical distribution of data on a log-log scale>emplot(data, alog = “xy”, labels=TRUE)//Creates the plot of the mean excess function for the data.>meplot(data, omit=3, labels=TRUE)//Fits the GPD for data beyond threshold t, and uses maximum likelihood method to//estimate parameters.>FittedData=gpd(data, threshold = t, method = c(“ml”)) 31 of 34
  32. 32. //Displays the details of the GPD fit to the data.>FittedData//Provides 4 different plots to assess the fit of the GPD model. The user can choose the//requisite plot from a menu.>plot.gpd(FittedData, labels=TRUE)//Once data is fitted, calculates estimates of quantiles and expected shortfall for the model//for a given vector of probability levels p.>riskmeasures(FittedData, p) 32 of 34
  33. 33. Conclusion EVT is here to stay as a technique for the risk managers toolkit. Wheneverthe tails of probability distributions are of interest, it is natural to consider applyingthe theoretically supported techniques of EVT. Methods based around theassumptions of normal distributions are likely to underestimate tail risk. Althoughnot perfect, EVT provides the best available models to predict extreme events. In the second part of the study, it was shown that EVT can be used tosuccessfully model the daily returns of the stock prices. It illustrates how EVT canbe used as a day-to-day exploratory risk management tool. 33 of 34
  34. 34. References 1. Bensalah, Younes. (November 2000) “Steps in Applying Extreme Value Theory to Finance: A Review” Bank of Canada Working Paper 2000-20. 2. Coles, Stuart. An Introduction to Statistical Modelling of Extreme Values. London: Springer, 2001. 3. Embrechts P., Klüppelberg C., & Mikosch T. Modelling Extremal Events for Insurance and Finance. Heidelberg: Springer-Verlag, 1999. 4. McNeil, A. & Saladin, T. (April 24, 1997), “The Peaks over Threshold Method for Estimating High Quantiles of Loss Distributions” Departement Mathematik ETH Zentrum. 5. McNeil, A. (May 17, 1999), “Extreme Value Theory for Risk Managers” Departement Mathematik ETH Zentrum. 6. Stephenson, A. (2003), “EVD Documentation”. Documentation for Extreme Value Distributions package for R Statistical Program. (http://www.maths.lancs.ac.uk/~stephena/software.html) 7. Stephenson, A. (2003), “EVD Documentation”. Documentation for Extreme Value Distributions package for R Statistical Program. (http://www.maths.lancs.ac.uk/~stephena/software.html) 8. Stephenson, A. (2002), “EVIR”. Documentation for Extreme Value In R package for R Statistical Program. (http://www.maths.lancs.ac.uk/~stephena/software.html) 34 of 34

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