EXTREME VALUE THEORYAND APPLICATIONS TOFINANCIAL MARKET

VIETNAM NATIONAL UNIVERSITY UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS Dao Minh Phuong EXTREME VALUE THEORY AND APPLICATIONS TO FINANCIAL MARKET Undergraduate Thesis Advanced Undergraduate Program in Mathematics Hanoi - 2012 VIETNAM NATIONAL UNIVERSITY UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS Dao Minh Phuong EXTREME VALUE THEORY AND APPLICATIONS TO FINANCIAL MARKET Undergraduate Thesis Advanced Undergraduate Program in Mathematics Thesis advisor: Dr. Luu Hoang Duc Hanoi - 2012 Contents Chapter 1. Extreme Value Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2. Block Maxima method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2.1. L imiting Behavior of Maxima and Extrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2.2. F ish er- Tippett Theorem (1928) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3. The Peaks- over- Thresholds (POT) method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.1. The Generalized Pareto Distribution (GPD). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2. The POT meth od. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.3. Pickands- Balkema- de Hann theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Chapter 2. Applications: Some theoretical computations . . . . . . . . . . . . . . . . . . . 13 2.1. Block Maxima Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.1. Maximum Likelihood Est imation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.2. Hill estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.1.3. Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.1.4. B lock Maxima Method Ap proach to VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.1.5. Multipe r iod VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.6. Return Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.2. The POT method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2.1. The selection of threshold u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.2. E x pected Shortfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2.3. POT method approach to VaR and Expected Sho r fall . . . . . . . . . . . . . . . . . . . . . . 22 Chapter 3. Applications: Empirical computations . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1. Stock market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2. Case study: the Crash in 1987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3. Risk measure computations using R: case study for Coca-Cola stock . . 30 3.3.1. B lock Maxima Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 i 3.3.2. POT method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ii ACKNOWLEDGM ENTS I am grateful to all those who spent their time and support for this the- sis. Foremost among this group is my advisor and instructor- Dr Luu Hoang Duc. Thank you for your dedication, patience, enthusiasm, motivation and immense knowledge. I am highly appreciate for your encouragement to the thesis. Second, my sincere thanks to all professors and lecturers of Faculty of Math- ematics, Mechanics and Informatics for their help throughout my university’s life at Hanoi University of Science. Last but not least, I would like to thank my family and friends from K53- Ad- vanced Mathematics program who always support and give me advice during my university’s life. iii Introduction Conceptually, mathematics and finance have a lot of things in common. Both speak everyday about the changes of economy and factors that lead to those changes. The subject ”financial mathematics” was created to apply mathematical theories and results to develop and to predict the quantitative changes which might lead to se- rious damages in financial world and economies, and to suggest best solutions for those phenomena. The purpose of this thesis is to highlight the significant conceptual overlap between mathematics and finance and represent the possibility for advancement of financial resea rch through the applications of a mathematical method calle d ”Ex- treme Value Theory”. Extreme Value Theory is a branch of statistical mathematics that studies extremal de viation from the median of probability distribution. It seeks to assess, from a given ordered sample of a given random variable, the probability of events that are more extreme than any observed prior. These values are definitely important since they usually describe times of the greatest gains or losses. It pro- vides the solid fundamental needed for the statistical modeling of such events and the computation of extremal risk finance. Devastating floods, tornadoes, seismic, market crashes, etc are all real world phenomena modeled by extreme value theory. This thesis consists of three chapters. The first chapter represents Extreme Value Theory and some theoretical results, the block maxima and the Peaks-over- Threshold method. The next chapter introduces some a p p lications in financial mar- ket such as Value at Risk, return levels, the choice of threshold . And the la st chapter is some empirical computations using R. 1 CHAPTER 1 Extreme Value Theory 1.1. Introd uction In this chapter, we introduce the e xtreme value theory in statistic literatures. Basically, extreme value theory, when modeling the maxima of a random variable, plays the same role as the central limit theorem does for modeling the sum of vari- ables. Both two theories represent the limiting distributions. Let’s consider a random variable showing daily returns. Basically, there are two methods for id entifying extremes in real data. The first method considers the maximum of variable in consecutive periods, such as months or year. These selected observations from the extreme events, called block maxima. In the left of Figure1.1, the observations X 2 , X 5 , X 7 , X 11 perform the block maxima for four periods of each three observations. In contrast, the second method focuses on the realizations sur- passing a given threshold. All the observations X 1 , X 2 , X 7 , X 8 , X 9 , X 11 of the right panel exceed to threshold u and compose the extreme events. The block maxima is the traditional approach used to analyze data with oc- casional as for instance hydrologic data. Nevertheless, the Peaks-over- thresholds (POT) method use data more efficiently so it has been used popularly in recent ap- plications. In the following sections, the block maxima and POT methods are presented. They are mostly based on Embrechts [2]. 1.2. Block Maxima method In this part, we will consider the so called Generalized Extreme Value distri- bution (in short GEV) and the most important result: the Fisher theorem. In order 2 Figure 1.1: Block maxima (left panel) and peaks- over- threshold u (right panel) to do that, let’s consider the behaviour of extrema and maxima: 1.2.1. Limiting Behavior of Maxima and Extrema Let X 1 , X 2 , . . . , be iid random variables with distribution function (df) F. In risk management applications, these may exhibit operational losses, financial losses or insurance losses. Let M n = max(X 1 , . . . , X n ) be worst-case loss in a sample of n losses. Let the range of X 1 , . . . , X n be (l,u). Obviously P (M n ≤ x) = P(X 1 ≤ x, . . . , X n ≤ x) (1.1) = n ∏ i=1 P (X i ≤ x) = F n (x). (1.2) It can be shown that, almost certainly, M n n→ ∞ −→ x F , where x F : = sup{x ∈ R : F(x) < 1} ≤ ∞ is the right endpoint of F. In practice, the distribution F(x) is unknown and therefore, the cummulative dis- 3 tribution function (in short cfd) of M n is also unknown. Nevertheless, as n goes to infinity, F n (x) → 0 if x < u and F n (x) → 1 if x > u or we can say that F n (x) be- comes degenerated. This degenerated cdf does not have any practical value. Hence, the extreme value theory is interested in finding sequences of real numbers a n > 0 and b n such that (M n − b n )/a n , the seq uence of normalized maxima and converges in distribution, i.e: P  (M n − b n ) a n ≤ x  = F n (a n x + b n ) n→ ∞ −→ H(x), (1.3) for some non degenerate distribution function H(x). If this condition holds, then F is in the maximum domain of attraction of H, and we write F ∈ MDA(H). H depends on location series b n and scale a n , thus it creates a unique type of distribution. For more details and comprehensive treatment of extreme value theory, we refer to Embrechts [2]. Let M n∗ = (M n − b n )/a n . Now under the independent assumption, the lim- iting distribution of normalized minima M n∗ is given as: H ξ (x) =  exp(−(1 + ξx) −1 /ξ ) ξ = 0, exp(−e −x ) ξ = 0, (1.4) with x such that 1 + ξx > 0 and ξ is the shape parameter. This parametrization is continuous in ξ. The one-parameter representation in equation (1.4) was suggested by Jenkinson (1955) and Von Mises (1954), known as Generalized Extreme Value Distribution (GEV). It consists three types of limiting distribution of Gnedenko(1943) : ξ > 0 H ξ corresponds to classical Frechet df ξ = 0 H ξ corresponds to classical Gumbel df ξ < 0 H ξ corresponds to classical Weibull df Below, we present now the most important and funda mental result - the Fisher- Tippe tt theorem. 4 Figure 1.2: GEV: distribution functions for various ξ 1.2.2. Fisher- Tippett Theorem (1928 ) Theorem 1.1. If appropriately normalized maxima converge in distribution to a non-degenerate limit, then the limit distribution must be an extreme value distribution, abbreviated: F ∈ MDA (H) then H is of typ e H ξ for some ξ. where H ξ is Generalized Extreme Value Distribution. The Fisher- Tippett theorem essentially says that the GEV is the only p ossible limiting distribution for normalized block maxima. One of the main part to apply Fisher- Tippett theorem is to determine in which case F ∈ MDA (H ξ ) holds. The following remarks would help us to do that. 1. Fr ´ echet case: (ξ > 0) Gnedenko (1943) pointed out that for ξ > 0 5 [...]... C HAPTER 2 Applications: Some theoretical computations Extreme value theory has many applications in financial market and risk management However, within the framework of this thesis, we can only present some applications of extreme value theory in computation of value at risk, return levels and expected shortfall As we mentioned above, there are two main methods are used to estimate these values This... and variance ξ 2 15 2.1.3 Value at Risk There are various kinds of risk in financial markets Three main kinds are credit risk, market risk and operational risk The concept of Value at Risk (VaR) is applicable to all types, but it mostly concerned with market risk VaR is single measure of the amount by which an company’s position in a risk category might decrease due to general market movements during... case under the extreme value theory The proper relationship between l-day and 1-day horizons is: VaR(l ) = l 1/σ VaR = l ξ VaR (2.18) Here, σ is the tail index and ξ is the shape parameter of the extreme value distribution This relationship is refered to as the σ-root of the time rule Note that σ = 1/ξ , not the scale parameter σn in equation (2.17) 2.1.6 Return Levels Often when examining extreme data... into equation (1.5) for x = (r − µn )/σn to get the quantile of a given probability of the generalized extreme value distribution Let p∗ be a small upper tail probability that expresses the potential loss ∗ and rn be the (1 − p∗ )th quantile of the block maxima under the limiting generalized extreme value distribution Then we have: ∗ ξ n (rn − µ n ) −1/ξ n ) ] σn ∗ (r n − µ n ) exp[− exp(− σn )] exp[−(1... maxima method is that if we observe data over a period of a few years and then take the maximum value for each period, we might loose the potential extreme events For instance, we are interested in modeling the rainfall of Vietnam Given that heavy rain often occurs over summer period, we expect that the most extreme rainfall to take place over a few months in summer However, if we divide a year into twelve... Then, to estimate VaR in long position, we can use equation (2.17) for n = D =252 23 C HAPTER 3 Applications: Empirical computations R, also called GNU S, is an useful functional language and environment to statistical explore data sets Therefore, in this chapter, we introduce some R commands to calculate extreme values and apply it into stock of the biggest company: Coca-cola Inc (KO) from January, 2nd,... extreme values and apply it into stock of the biggest company: Coca-cola Inc (KO) from January, 2nd, 1980 to September, 28th, 2012 First, let us introduce some basic information about stock market 3.1 Stock market A stock market is a public entity for the trading of company stock shares and derivatives at an agreed price The stock of the business entity represents the invested money in the business by its... property and the assets of a business which may fluctuate in quantity and value A derivative instrument is a contract between two parties that specifies conditions (especially the dates, resulting values of the underlying variables, and notional amounts) under which payments are to be made between the parties Participants in stock market include individual retail investors, institutional investors such... Standard & Poor’s 500, is a stock market index based on the common stock prices of 500 top publicly traded American companies, as determined by S&P It differs from other stock market indices because it tracks a different number of stocks and weights the stocks differently It is one of the most commonly followed indices and many consider it the best representation of the market and a bellwether for the... October 1987, there was a crash in the stock market, which has seen the S&P 500 index drop by 9.21% On that Friday alone the index is decreased 5.25% on the previous day, the largest one- day fall since 1962 At our disposal are all daily closing values of the index since 1960 We analyze annual maxima of daily percentage go down in the index These (1) (28) values M260 , , M260 are supposed to be iid . NATIONAL UNIVERSITY UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS Dao Minh Phuong EXTREME VALUE THEORY AND APPLICATIONS TO FINANCIAL MARKET Undergraduate Thesis Advanced. NATIONAL UNIVERSITY UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS Dao Minh Phuong EXTREME VALUE THEORY AND APPLICATIONS TO FINANCIAL MARKET Undergraduate Thesis Advanced