16 th Annual Conference Multinational Finance Society in Crete (2009)

Transcription

1 16 th Annual Conference Multinational Finance Society in Crete (2009)

2 Statistical Distributions in Finance (invited presentation) James B. McDonald Brigham Young University June 28- July 1, 2009 The research assistance of Brad Larsen and Patrick Turley is gratefully acknowledged as are comments from Richard Michelfelder and Panayiotis Theodossiou.

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4 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications 4. Qualitative response models 5. Option pricing 6. VaR (value at risk) 7. Conclusion

5 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions a. Families 3. Regression applications 4. Qualitative response models 5. Option pricing 6. VaR (value at risk) 7. Conclusion

6 Some families of statistical distributions a. Families f(y;θ), θ = vector of parameters i. GB: GB1, GB2, GG (0<Y)

7 GB distribution tree

8 Probability Density Functions ( a ( )( ) ) ap a y c y / b a a GB ( y; a, b, c, p, q) =, 0 y b / 1 c p+ q ap b B p q ( a ) (, ) 1 + c( y / b) q 1 ( ) ( ) ( ) GB1 y; a, b, p, q = GB y; a, b, c = 0, p, q = a y ap ( a ( y b) ) (, ) 1 1 / ap b B p q q 1

9 Probability Density Functions ( ) ( ) GB2 y; a, b, p, q = GB y; a, b, c = 1, p, q = ap 1 ( a ) (, ) 1 + ( y / b) ap b B p q ay p+ q ( ;,, p) GG y a = a y ap 1 ap e ( y / ) ( p) a a controls peakedness b is a scale parameter ( ) a a c domain 0 y b / 1 c p, q shape parameters

10 Probability Density Functions GB2 PDF evaluated at different parameter values:

11 Some families of statistical distributions a. Families i. GB: GB1, GB2, GG ii. EGB: EGB1, EGB2, EGG (Y is real valued)

12 EGB distribution tree

13 Probability Density Functions ( ;,, c, p, q) EGB y m = e ( 1 1 c e ) ( ) ( ) ( y m ) p y m / / ( / ) ( y m) (, ) 1+ ce B p q p+ q q 1 for y-m 1 - < n 1 c ( ) EGB1 y; m,, p, q = e ( 1 e ) ( ) ( y m) p y m / / (, ) B p q q 1

14 Probability Density Functions ( ) EGB2 y; m,, p, q = e ( ) B p, q 1 ( ) p y m / ( ( y m) / + e ) p+ q ( ;,, p) EGG y m = e ( ) ( y m / e ) p y m e ( p) / m controls location is a scale parameter c defines the domain p, q are shape parameters

15 Probability Density Functions EGB2 PDF evaluated at different parameter values:

16 Some families of statistical distributions a. Families i. GB: GB1, GB2, GG ii. iii. EGB: EGB1, EGB2, EGG SGT (Skewed generalized t): SGED, GT, ST, t, normal (Y is real valued)

17 SGT distribution tree 5 parameter SGT q λ=0 p=2 4 parameter SGED GT ST p=1 λ=0 q p=2 q p=2 λ=0 q=1/2 3 parameter SLaplace GED SNormal t SCauchy λ=0 p=1 p p=2 λ=0 q q=1/2 λ=0 2 parameter Laplace Uniform Normal Cauchy

18 Probability Density Functions ( ;,,, p, q) SGT y m = q+ 1/ p p 1/ p y m 2 q B 1/ p, q 1+ ( ) p p p (( 1+ sign( y m) ) q ) SGED y m ( ;,,, p) = pe (( sign( y m) ) ) p y m / 1+ ( p) 2 1/ p m = mode ( location parameter) = scale 1 = skewness area to left of m =, -1 < < 1 2 p, q = shape parameters tail thickness, moments of order pq = df ( )

19 Probability Density Functions SGT PDF evaluated at different parameter values:

20 Some families of statistical distributions a. Families i. GB: GB1, GB2, GG ii. EGB: EGB1, EGB2, EGG iii. SGT (Skewed generalized t): SGED, GT, ST, t, normal iv. IHS

21 Probability Density Functions IHS ( Y = a + b sinh ( + N ( 0,1 )/ k )) IHS y where ( ;,, k, ) = ke 2 k ln / / ln ( y ) ( y ) ( ) 2 2 ( 2 + ( y + ) ) /.5 ( ) ( ) ( ) k 2 2 k 2 2 e e e + e + e k 2 e k = 1/, = /, =.5, and = w w w w w = mean 2 = variance = skewness parameter k = tail thickness ;, = lim ( ;,,, = 0) N y IHS y k k

22 Probability Density Functions IHS PDF evaluated at different parameter values:

23 Some families of statistical distributions a. Families i. GB: GB1, GB2, GG ii. EGB: EGB1, EGB2, EGG iii. SGT (Skewed generalized t): SGED, GT, ST, t, normal iv. IHS v. g-and-h distribution (Y is real valued)

24 g-and-h distribution Definition: where Z ~ N[0,1] gz e 1 Y ( Z ) a b e gh, g = + h>0 h<0 hz 2 /2

25 g-and-h distribution Y0,0 Z = a + bz ~ N a, = b ( ) 2 2 gz e 1 Y ( ) gh, = 0 Z = a + b g Is known as the g distribution where the parameter g allows for skewness. Y Z a bze g ( ) = 0, h = + gz 2 /2 Is known as the h distribution Symmetric Allows for thick tails

26 Probability Density Functions g-and-h PDF evaluated at different parameter values with h>0:

27 Probability Density Functions g-and-h PDF evaluated at different parameter values with h<0:

28 Some families of statistical distributions a. Families f(y;θ) i. GB: GB1, GB2, GG ii. EGB: EGB1, EGB2, EGG iii. SGT (Skewed generalized t): SGED, GT, ST, t, normal iv. IHS v. g-and-h distribution vi. Other distributions: extreme value, Pearson family,

29 Some families of statistical distributions a. Families f(y;θ) i. GB: GB1, GB2, GG ii. EGB: EGB1, EGB2, EGG iii. SGT (Skewed generalized t): SGED, GT, ST, t, normal iv. IHS v. g- and h-distribution vi. Other distributions: extreme value, Pearson family, 1. = x, 2. Multivariate ( ) vii. Extensions: ( )

30 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions a. Families b. Properties 3. Regression applications 4. Qualitative response models 5. Option pricing 6. VaR (value at risk) 7. Conclusion

31 Some families of statistical distributions b. Properties i. Moments 1. GB family E GB ( h Y ) ( h / a, q) B( p, q) b B p + p + h / a, h/ a; c p + q + h / a; h = 2F1 for h < aq with c=1

32 Some families of statistical distributions b. Properties i. Moments 1. GB family a. GB1 E GB1 ( h Y ) = h b B p + h / a, q ( ) B( p, q)

33 Some families of statistical distributions b. Properties i. Moments 1. GB family a. GB1 b. GB2 h b B p + h / a, q h / a E Y = - p h / a q GB2 ( h ) ( ) B( p, q)

34 Some families of statistical distributions b. Properties i. Moments 1. GB family a. GB1 b. GB2 c. GG h p h / a E Y = for h / a p GG ( h ) ( ) ( p)

35 Some families of statistical distributions b. Properties i. Moments 1. GB family 2. EGB family (, ) (, ) t ( ) ( ty e B p + t q p + t, t; c M ) EGB t = E e = 2F1 B p q p+q+t for t q /σ with c = 1

36 EGB moments Mean Variance Skewness Excess kurtosis EGG EGB1 EGB2 + ( p) + ( p) ( p + q) + ( p) ( q) 2 2 ( ) '( p) '( p + q) '( p) + '( q) 2 ' p 3 3 ( ) ''( p) ''( p + q) ''( p) ''( q) 3 '' p 4 4 ( ) ''' ( p) ''' ( p + q) ''' ( p) + ''' ( q) 4 ''' p ( s) = d n ds ( s)

37 EGB2 moment space

38 Some families of statistical distributions b. Properties i. Moments 1. GB family 2. EGB family 3. SGT family

39 SGT family SGT h+ 1 h, p p 2 1 B, q p h/ p h q B q h h h h ( ) ( ) = ( 1+ ) + ( 1) ( 1 ) E y m for h < pq=d.f. SGED h + 1 h p 2 1 p + ( ) ( ) = ( 1+ ) + ( 1) ( 1 ) E y m h h h h

40 SGT moment space

41 SGT family moment space

42 Some families of statistical distributions a. Families b. Properties i. Moments 1. GB family 2. EGB family 3. SGT family 4. IHS

43 IHS moment space

44 Some families of statistical distributions a. Families b. Properties i. Moments 1. GB family 2. EGB family 3. SGT family 4. IHS 5. g-and-h family

45 g- and h-family i j i ( 1) n e n n n i i j 0 j = E ( X gh, ) = a b i i= i g 1 ih ( i j) g 21 ( ih) 2 Moments exist up to order 1/h (0<h)

46 g-and-h moment space (h>0) (visually equivalent to the IHS)

47 Moment space for g-and-h (h>0) and g-and-h (h real)

48 Moment space of SGT, EGB2, IHS, and g-and-h

49 Some families of statistical distributions b. Properties i. Moments ii. Cumulative distribution functions (see appendix) Involve the incomplete gamma and beta functions

50 Some families of statistical distributions b. Properties i. Moments ii. iii. Cumulative distribution functions (see appendix) Involve the incomplete gamma and beta functions Gini coefficients (G)

51 Gini Coefficients (G) Definition: 1 G = x y f ( x : ) f ( y : ) dxdy F ( y) G 0 1 ( ) 0 2 ( ) ( 1 ( )) dy = (Dorfman, 1979, RESTAT) F y dy = G( )

52 Gini Coefficients Interpretation: G = 2A

53 Gini Coefficients Application: Stochastic Dominance

54 Some families of statistical distributions b. Properties i. Moments ii. iii. iv. Cumulative distribution functions (see appendix) Gini coefficients (G) Incomplete moments

55 Incomplete moments Definition: ( yh ; ) = y h s f s ds ( ) ( h ) EY Applications: Option pricing formulas Lorenz Curves

56 Incomplete moments Convenient theoretical results: Distribution LN GG GB2 ( yh ; ) LN y ( 2 2 ; + h, ) ( ;,, + / ) GG y a p h a ( + ) GB2 y; a, b, p h / a, q h / a

57 Some families of statistical distributions b. Properties i. Moments ii. iii. iv. Cumulative distribution functions (see appendix) Gini coefficients (G) Incomplete moments v. Mixture models

58 Mixture Models ( ) Let f y;, denote a structural or conditional density of the random variable Y where and denote vectors of distributional parameters. Let the density of be given by the mixing distribution g ( ; ). The observed or mixed distribution can be written as h( y;, ) = f ( y;, ) g ( ; ) d

59 Mixture Models Observed model ( ;,,, p, q) SGT y m GT y ( ;, p, q) ( ) EGB2 y;,, p, q ( y a b p q) GB2 ;,,, LT y ( ;,, q) ( ;, q) t y Structural model SGED y m ( ;,, s, p) GED ( y; s, p) ( ;,ln ( ), ) EGG y s p GG ( y; a, s, p) LN y ( ;, s) ( ;, s) N y Mixing distribution 1/ p ( ;,, q) IGG s p q 1/ p ( ;,, q) IGG s p q 1 IGG s;, e, q IGG ( s; a, b, q) 1/ 2 ( ; = 1, ) IGG s a q 1/ 2 ( ; ) IGA s q

60 Some families of statistical distributions b. Properties i. Moments ii. iii. iv. Cumulative distribution functions (see appendix) Gini coefficients (G) Incomplete moments v. Mixture models vi. Hazard functions (Duration dependence)

61 Hazard functions Definition: ( ) Let f s denote the pdf of a spell (S) or duration of an event. 1 F( s) is the probability that that S>s. The corresponding hazard function is defined by ( ) ( ) f s hs () = 1 F s which can be thought of as representing the rate or likelihood that a spell will be completed after surviving s periods.

62 Hazard functions Applications: Does the probability of ending a strike, unemployment spell, expansion, or stock run depend on the length of the strike, unemployment spell, or of the run? With unemployment, A job seeker might lower their reservation wage and become more likely to find a job Increasing hazard function However, if being out of work is a signal of damaged goods, the longer they are out of work might decrease employment opportunities Decreasing hazard function. An alternative example might deal with attempts to model the time between stock trades. Engle and Russell (1998) Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica 66: Hazard function of time between trades is decreasing as t increases or the longer the time between trades the less likely the next trade will occur.

63 Hazard functions Applications: Bubbles McQueen and Thorley (1994) Bubbles, stock returns, and duration dependence. Journal of Financial and Quantitative Analysis, 29: Efficient markets hypothesis, stock runs should not exhibit duration dependence (constant hazard function) McQueen and Thorley argue that asset prices may contain bubbles which grow each period until they burst causing the stock market to crash. Hence, bubbles cause runs of positive stock returns to exhibit duration dependence the longer the run the less likely it will end (decreasing hazard function), but runs of negative stock returns exhibit no duration dependence Grimshaw, McDonald, McQueen, and Thorley. 2005, Communications in Statistics Simulation and Computation, 34: What model should we use to characterize duration dependence? Exponential constant Gamma the hazard function can increase, decrease, or be constant Weibull the hazard function can increase, decrease, or be constant Generalized Gamma: the hazard function can be increasing, decreasing, constant, -shaped, or -shaped

64 Hazard functions Possible shapes for the GG hazard functions

65 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions a. Families b. Properties c. Model selection 3. Regression applications 4. Qualitative response models 5. Option pricing 6. VaR (value at risk) 7. Conclusion

66 Some families of statistical distributions c. Model selection i. Goodness of fit statistics Log-likelihood values o o n ( i ) ( ) n f ( y : ) = i= 1 ( ) (!) ( ) i= 1 Partition the data into g groups, Empirical frequency: Theoretical frequency: g for individual data ( ) ( ) = n n + n n p n n i i i ) p n / n, n n for grouped data I,, 1,2,..., i = Yi 1 Yi i = g = = i i i i= 1 ( ) ( ; ) pi = f y dy I i g

67 Model Selection i. Goodness of fit statistics Log-likelihood values Possible Measures g i= 1 i i ( ) SAE = p p g i= 1 ( ( )) 2 i i SSE = p p g 2 2 ni 2 i i i= 1 n ( ) / ( ) ~ ( # 1) = n p p g parameters

68 Model Selection i. Goodness of fit statistics Log-likelihood values Possible Measures Akaike Information Criterion (AIC) AIC 2( ( ) k) = A tool for model selection Attaches a penalty to over-fitting a model

69 Model Selection i. Goodness of fit statistics ii. Testing nested models Examples: H : g = 0 O ( ) H : SGT = GT H : = 0 O O H : SGT = Normal H : p = 2, = 0, and q O O

70 Testing nested models Likelihood ratio tests Wald test 2 ( ) ( r) LR = 2 * ~ a of independent restrictions ( ) a 2 LR ( ) 1 = 2 SGT GT * ~ 1 a 2 LR2 = 2 SGT Normal * ~ 3 ( ) ( ) ( ( ))' var ( ) where r denotes the number ( ( )) 1 2 ( ) ~ a ( ) W = g g g r W 1 MLE MLE MLE ( ) 1 ( ) a 2 ( ) ( ) Var ( ) = ˆ 0 ˆ ˆ 0 ~ 1

71 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions a. Families b. Properties c. Model selection d. An example: the distribution of stock returns 3. Regression applications 4. Qualitative response models 5. Option pricing 6. VaR (value at risk) 7. Conclusion

72 An example: the distribution of stock returns y Pt + 1 Pt Pt + 1 = n( P / P ) ~ = 1 P P t t t 1 t Daily, weekly, and monthly excess returns (1/2/ /29/2006) from CRSP database (NYSE, AMEX, and NASDAQ) 4547 companies t H 0 : skewness = 0 H 0 : excess kurtosis = 0 H 0 : returns ~ N(μ, σ 2 ) ( ) ( ) CI.95 = 2 6/ n, 2 6/ n CI.95 = 2 24/ n, 2 24/ n ( JB ) CI.95 = JB = ( excess kurtosis ) 2 2 skew 2 n + ~.05 ( 2) =

73 An example: the distribution of stock returns (continued) % of stocks for which excess returns statistics are in 95% C.I. H O : Skewness=0 H O :Excess kurtosis=0 H O : Normal Daily 16.38% 0.04% 0.09% Weekly 30.61% 4.88% 4.75% Monthly 66.79% 56.65% 53.77%

74 Kurtosis An example: the distribution of stock returns (continued) Daily excess returns plotted with admissible moment space of flexible distributions CRSP daily stocks--excess returns CRSP stock EGB2 SGT IHS bound Skewness

75 Kurtosis An example: the distribution of stock returns (continued) Weekly excess returns plotted with admissible moment space of flexible distributions CRSP weekly stocks--excess returns CRSP stock EGB2 SGT IHS bound Skewness

76 Kurtosis An example: the distribution of stock returns (continued) Monthly excess returns plotted with admissible moment space of flexible distributions CRSP monthly stocks--excess returns CRSP stock EGB2 SGT IHS bound Skewness

77 An example: the distribution of stock returns (continued) Fraction of stocks in the admissible skewness-kurtosis space daily weekly monthly EGB % 43.81% 50.80% IHS 83.92% 84.39% 61.97% SGT 87.62% 89.00% 95.10% g-and-h % 99.98% 98.99%

78 An example: the distribution of stock returns (continued) Fitting a PDF to normal excess returns Company Name Skew Kurtosis Jb Stat US Steel Estimated PDF logl SSE SAE Chi^2 Normal EGB IHS SGT Estimated PDFs for US Steel daily excess returns Returns Normal EGB2 IHS SGT Excess returns

79 An example: the distribution of stock returns (continued) Fitting a PDF to leptokurtic excess returns Estimated PDFs for ishares daily excess returns 50 Company Name Skew Kurtosis Jb Stat ishares Estimated PDF logl SSE SAE Chi^2 40 Normal EGB IHS SGT Returns Normal EGB2 IHS SGT Excess returns

80 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications 4. Qualitative response models 5. Option pricing 6. VaR (value at risk) 7. Conclusion

81 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications a. Background 4. Qualitative response models 5. Option pricing 6. VaR (value at risk) 7. Conclusion

82 Regression applications-- background Model: Yt = Xt+ t X t 1xK vector of observations on the explanatory variables Kx1 vector of unknown coefficients t independently and identically distributed random disturbances with pdf f ( ; )

83 Regression applications-- background If the errors are normally distributed OLS will be unbiased and minimum variance However, if the errors are not normally distributed OLS will still be BLUE There may be more efficient nonlinear estimators

84 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications a. Background b. Alternative estimators 4. Qualitative response models 5. Option pricing 6. VaR (value at risk) 7. Conclusion

85 Alternative Estimators i. Estimation OLS LAD L p = arg min = ( Y X ) 2 OLS t t t= 1 = arg min = Y X LAD t t t= 1 = arg min = Y X Lp t t t= 1 n n n p

86 Alternative Estimators (continued) i. Estimation (continued) M-estimators: Includes OLS, LAD, and Lp as special cases Includes MLE (QMLE or partially adaptive estimators) as a special case where SGT SGED EGB2 IHS ( ; ) = nf( ; ) n ( Y X ) = arg min = M t t t= 1 ( Y X ) = arg min = ; MLE, t t t= 1 n

87 Alternative Estimators (continued) i. Estimation ii. Influence functions: ( ) = '( ) OLS LAD Redescending influence function

88

89 Alternative Estimators (continued) i. Estimation ii. iii. Influence functions Asymptotic distribution of extremum estimators ( ) min H ˆ ~ a N ; sandwich = A BA 1 1 where ( ) 2 dh dh dh A = E and B = E dd ' d d '

90 Alternative Estimators (continued) i. Estimation ii. Influence functions iii. Asymptotic distribution of extremum estimators iv. Other estimators Semiparametric (Kernel estimator, Adaptive MLE) n SP = arg min n fk ( t = Y X t ) t= 1 where f K ( ) e = e = Y X n 1 i K nh i= 1 h i i i OLS K ( ) denotes a kernel, and h is the window width

91 Regression applications (continued) iv. Other estimators (continued) Generalized Method of Moments (GMM) GMM = arg min g ' Qg where n ( ) ( ) ( ) ( ) g = Z h = Y X i= 1 i i i i Z denotes a vector of instruments (can be X) Q is a positive definite matrix Q = Var 1 ( g ) ( )

92 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications a. Background b. Alternative estimators c. A Monte Carlo comparison of alternative estimators 4. Qualitative response models 5. Option pricing 6. VaR (value at risk) 7. Conclusion

93 A Monte Carlo comparison of alternative estimators c. A Monte Carlo comparison of alternative estimators Model: y = 1+ X + t t t Error distributions: (zero mean and unitary variance) Normal: Mixture: N 0;1 + N.9* N 0,1/ 9.1* 0,9 Skewed: Skewness =0 Kurtosis =24.3 (.5 LN ( 0,1 ) e )/ e( e 1) Skewness=6.18 Kurtosis=113.9

94 A Monte Carlo comparison of alternative estimators Kurtosis Skewed Mixture Normal Skewness

95 A Monte Carlo comparison of alternative estimators Sample size = 50, T=1000 replications RMSE for slope estimators Estimators Normal Mixture-thick tails Skewed OLS LAD SGED ST GT SGT EGB IHS SP = AML GMM

96 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications a. Background b. Alternative estimators c. A Monte Carlo comparison of alternative estimators d. An application: CAPM i. Error distribution effects ii. ARCH effects 4. Qualitative response models 5. Option pricing 6. VaR (value at risk) 7. Conclusion

97 An application: CAPM i. CAPM and the error distribution Daily, weekly, and monthly excess returns (1/2/ /29/2006) from CRSP database (NYSE, AMEX, and NASDAQ) 4547 companies Percent of stocks for which OLS residual statistics are in 95% C.I. H O : Skewness=0 H O :Excess kurtosis=0 H O : Normal (JB) Daily 14.14% 0.02% 0% Weekly 28.13% 3.91% 3.43% Monthly 67.56% 57.14% 54.76%

98 An application: CAPM with and without ARCH effects (ST) i. CAPM and the error distribution Daily, weekly, and monthly excess returns (1/2/ /29/2006) from CRSP database (NYSE, AMEX, and NASDAQ) 4547 companies Percent of stocks for which ST residual statistics are in 95% C.I. H O : Skewness=0 H O :Excess kurtosis=0 H O : Normal (JB) Daily 14.05% 0.02% 0% Weekly 28.82% 3.83% 3.39% Monthly 64.04% 54.72% 51.48%

99 An application: CAPM with and without ARCH effects (IHS) i. CAPM and the error distribution Daily, weekly, and monthly excess returns (1/2/ /29/2006) from CRSP database (NYSE, AMEX, and NASDAQ) 4547 companies Percent of stocks for which IHS residual statistics are in 95% C.I. H O : Skewness=0 H O :Excess kurtosis=0 H O : Normal (JB) Daily 13.99% 0.02% 0% Weekly 27.89% 3.83% 3.36% Monthly 65.54% 55.71% 52.32%

100 An application: CAPM with alternative error distributions Statistics of OLS residuals Company Name Skewness Kurtosis JB stat UNITED NATURAL FOODS INC CENTS ONLY STORES Estimated Betas Company Name OLS T GT SGED EGB2 IHS ST SGT UNITED NATURAL FOODS INC CENTS ONLY STORES

101 An application: CAPM with and without ARCH effects i. CAPM and the error distribution ii. CAPM: how about ARCH effects? Review: If errors are normal and no ARCH effects, OLS is MLE If errors are not normal and no ARCH effects OLS is BLUE, but not MLE nor efficient If errors are normal and have ARCH effects OLS is BLUE, but not efficient If errors are not normal and have ARCH effects OLS is BLUE,but not efficient

102 An application: CAPM with and without ARCH effects ii. CAPM: ARCH effects (continued) Model: Yt = Xt+ t = u + 2 t t 0 1 t 1.5 Percent of stocks exhibiting ARCH(1) effects (OLS) (% rejecting HO : 1 = 0) 0.10 level 0.05 level Daily 63.2% 60.0% Weekly 29.2% 24.1% Monthly 18.7% 13.7%

103 An application: CAPM with and without ARCH effects Percent of stocks exhibiting ARCH(1) effects (ST) (% rejecting ) H : = level 0.05 level Daily 63.2% 59.9% Weekly 29.1% 23.9% Monthly 16.9% 12.3% Percent of stocks exhibiting ARCH(1) effects (IHS) (% rejecting H : = 0 ) 0.10 level 0.05 level Daily 63.3% 60.0% Weekly 29.3% 24.1% Monthly 18.9% 13.9% O O 1 1

104 An application: CAPM with and without ARCH effects ii. CAPM: ARCH effects (continued) ARCH Simulations ( ) y = X = excess market return + t t t, t= 1,, 60 X monthly excess market returns, 1/2002 to 12/31/2006 Error distributions 2 t ~ N 0, ( ) ( ) : = + where ~ 0,1 ARCH u u N t 1 N t t 0 1 t ( ) ( 2 = ) ARCH t 1 : t ut where u ~ (5) t 1 t t.5

105 An application: CAPM with and without ARCH effects ARCH Simulations (continued) Root Mean Square Error (RMSE) for 10,000 replications Erro rs N ( 0,σ^2 ) N ( 0,1), A rch( 1) t ( 5), A rch( 1) Est imat io n N o n- A R C H ARCH N o n- A R C H ARCH N o n- A R C H ARCH OLS/Normal LAD T GED GT SGED EGB IHS ST SGT

106 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications 4. Qualitative response models 5. Option pricing 6. VaR (value at risk) 7. Conclusion

107 Qualitative Response Models

108 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications 4. Qualitative response models a. Basic framework 5. Option pricing 6. VaR (value at risk) 7. Conclusion

109 Qualitative Response Basic Framework Model: y = X * i i i y = 1 i if y * i 0 Log-likelihood function: and 0 otherwise ( ) ( * y ) i = X i = yi = X i i X i Pr 1 Pr n X i ( X ) f ( s ) ds F ( X ) = Pr = ; = ; ( ) ( ) i= 1 i i i ( ) ( ) ( ( )), = y n F X ; + 1 y n 1 F X ; i i i i

110 Probit and logit estimators will be inconsistent if The error distribution is incorrectly specified heteroskedasticity exists, e.g. unmeasured heterogeneity is present relevant variables have been omitted The index appears in a nonlinear form Similar results are associated with Censored & Truncated regression models Qualitative Response Basic Framework (continued) ˆ MLE of will be consistent and asymptotically distributed as 2 ˆ ~ a d N ; = E d d ' if the model is correctly specified. 1

111 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications 4. Qualitative response models a. Basic framework b. An application: fraud detection 5. Option pricing 6. VaR (value at risk) 7. Conclusion

112 Qualitative response An application: fraud detection Prediction of corporate fraud (Y=1 fraud) Compare financial ratios of companies with averages of five largest companies ( virtual firm) 228 companies (114 fraud and 114 non-fraud) Variables: accruals to assets, asset quality, asset turnover, days sales in receivables, deferred charges to assets, depreciation, gross margin, increase in intangibles, inventory growth, leverage, operating performance margin, percent uncollectables, receivables growth, sales growth, working capital turnover. SGT, EGB2, & IHS formulations improve predictions

113 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications 4. Qualitative response models a. Basic framework b. An application c. Some related issues 5. Option pricing 6. VaR (value at risk) 7. Conclusion

114 Qualitative response Some related issues Cost of misclassification Choice-based sampling Heterogeneity Semi-parametric estimation procedures

115 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications 4. Qualitative response models 5. Option pricing: European call option 6. VaR (value at risk) 7. Conclusion

116 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications 4. Qualitative response models 5. Option pricing: European call option a. The Black-Scholes option pricing formula 6. VaR (value at risk) 7. Conclusion

117 Option pricing Black-Scholes a. The Black Scholes option pricing formula The equilibrium price of a European call option is equal to the present value of its expected return at expiration: where rt ( ) ( ) incomplete moments rt ( 0 ) ( ) ( ) C S, T, X = e E C S,0 = e S X f S S, T ds f T T X X rt X = ST ;1 e X ;0 S. T St ( yh) y ( ) h s f s ds ; = 1 = E y h ( ) h s f s ds y ( ) h ( ) E y involve normalized ( y; h) = 1 ( y; h) ( )

118 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications 4. Qualitative response models 5. Option pricing: European call option a. The Black-Scholes option pricing formula b. Some background and alternative formulations 6. VaR (value at risk) 7. Conclusion

119 Option pricing Some background and alternative formulations The Black Scholes (1973) option pricing formula corresponds to being the lognormal 2 2 ( ; ) = ( ; +, ) LN y h LN y h, the cdf for the lognormal ( ) f s The Black Scholes formula (Bookstaber and McDonald, 1991) corresponding to the Generalized Gamma is obtained from h y; h = GG y; a,, p + a ( ), the cdf for the GG GG The Black Scholes formula ( Bookstaber and McDonald, 1991) corresponding to the GB2 is obtained from h h GB2 ( y; h) = GB2 y; a, b, p +, q a a, the cdf for the GB2 ( ) Rebonato (1999) applied C to the Deutschemark GB2 ST, T, X

120 Option pricing Some background and alternative formulations ( ) Sherrick, Garcia, and Tirupattur (1996) used CBurr 3 ST, T, X to price soybean futures. Theodosiou (2000) developed the CSGED ( S,, T T X ) Savickas (2001) explored the use of CWeibull ( S,, T T X ) Dutta and Babbel (2005) explore the g- and h- family (4-parameter) of option pricing formulas, C ( ), based on Tukey s nonlinear transformation of a standard g & h ST, T, X normal. Applied the g-and-h to pricing 1-month and 3-month London Inter Bank Offer Rates (LIBOR) g- and- h distribution and GB2 perform much better (errors fairly highly correlated) than the Lognormal, Burr 3, and Weibull distributions

121 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications 4. Qualitative response models 5. Option pricing: European call option a. The Black-Scholes option pricing formula b. Some background and alternative formulations c. A comparison of pricing behavior 6. VaR (value at risk) 7. Conclusion

122 A comparison of pricing behavior c. A comparison of pricing behavior (Dutta and Babbel, Journal of Business, 2005) Calculates the difference between the market price and predicted price for the g-and-h, GB2, lognormal, Burr3, and Weibull distributions

123 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications 4. Qualitative response models 5. Option pricing: European call option 6. VaR (value at risk) 7. Conclusion

124 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications 4. Qualitative response models 5. Option pricing: European call option 6. VaR (value at risk) a. Background and definitions 7. Conclusion

125 VaR Background and definitions i. Value at risk (VaR) is the maximum expected loss on a portfolio of assets over a certain time period for a given probability level. R ( ) ( ; ) f R dr = R is the return on the asset θ denotes the distributional parameters α is the predetermined confidence level or coverage probability ( ) R is the corresponding maximum expected loss or conditional threshold ( ) 1 R = F R ( : )

126 VaR Background and definitions ii. Standardized returns R = + z Z ( ) ( ) ( R ) = ( ) f z; dz =, Z Z ( ) 1 = F ( : ) Z ( ) R = + F Z ( : ) 1

127 VaR Background and definitions iii. Unconditional VaR formulation Estimate f(r;θ)

128 VaR Background and definitions iv. Conditional VaR formulation (AR(1) ABS- GARCH(1,1)) R = + R + Z = + z t 0 1 t 1 t t t t t = + z + R t 0 1 t 1 t 1 2 t 1 ( ) = + F : t t t Z 1 ( )

129 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications 4. Qualitative response models 5. Option pricing: European call option 6. VaR (value at risk) a. Background and definitions b. Models and applications 7. Conclusion

130 VaR Models and applications i. Unconditional VaR formulation Exponential: (Hogg, R. V. and S. A. Klugman (1983)) Gamma: (Cummins, et al. 1990) Log-gamma: (Ramlau-Hansen (1988)), (Hogg, R. V. and S. A. Klugman (1983)) Lognormal: (Ramlau-Hansen (1988)) Stable: (Paulson and Faris (1985) Pareto: (Hogg, R. V. and S. A. Klugman (1983)) Log-t: (Hogg, R. V. and S. A. Klugman (1983)) Weibull: (Cummins et al. (1990))

131 VaR Models and applications i. Unconditional VaR formulation (continued) Burr: (Hogg, R. V. and S. A. Klugman (1983)) Generalized Pareto: (Hogg, R. V. and S. A. Klugman (1983)) GB2: (Cummins (1990, 1999, 2007) Pearson family: Aiuppa (1988) Extreme value distribution: Bali (2003), Bali and Theodossiou (2008) IHS: Bali and Theodossiou (2008)

132 VaR Models and applications ii. Conditional VaR formulations (Bali and Theodossiou, JRI, 2008) Data: S&P500 composite index, 1/4/50 12/29/2000 (n=12,832) Daily percentage log-returns: (Sample mean =.0341, maximum=8.71, minimum= standard deviation =.874 skewness =1.622 kurtosis=45.52

133 VaR Models and applications ii. Conditional distributions (Bali and Theodossiou, JRI, 2008) (continued) Models Generalized extreme value EGB2 SGT IHS Findings Out of sample VaR estimates are rejected for most unconditional specifications Thresholds exhibit time varying behavior Out of sample VaR estimates for the conditional specifications corresponding to the SGT, IHS, and EGB2 perform better than the extreme value distributions

134 Selected references for option pricing and VaR Aiuppa, T. A Evaluation of Pearson curves as an approximation of the maximum probable annual aggregate loss. Journal of Risk and Insurance 55, Bali, T. G., An Extreme Value Approach to Estimating Volatility and Value at Risk, Journal of Business, 76: Bali, T. G. and P. Theodossiou, A Conditional-SGT-VaR Approach with Alternative GARCH Models, Annals of Operations Research, 151: Bali, T. G. and P. Theodossiou, Risk Measurement Performance of Alternaitve Distribution Functions, Journal of Risk and Insurance, 75: Black, F (1976). The Pricing of Commodity Contracts. Journal of Financial Economics 3: Cummins, J. D., G. Dionne, J. B. McDonald, and B. M. Pritchett Applications of the GB2 family of distributions in modeling insurance loss processes. Insurance: Mathematics and Economics 9, Cummins, J. D., C. Merrill, and J. B. McDonald, Risky Loss Distributions and Modeling the Loss Reserve Pay-out Tail, Review of Applied Economics 3. Cummins, J. D., R. D. Phillips, and S. D. Smith Pricing Excess of Loss Reinsurance Contracts against catastrophic loss. In Kenneth Froot, ed., The Financing of Catastrophe Risk (Chicago: University of Chicago Press) Dutta, K. K. and D. F. Babbel Extracting Probabilistic Information from the Prices of Interest Rate Options: Tests of Distributional Assumptions. Journal of Business 78: Hogg, R. V. and S. A. Klugman, On the Estimation of Long Tailed Skewed Distributions with Actuarial Applications. Journal of Econometrics 23, McDonald, J. B. and R. M. Bookstaber (1991). Option Pricing for Generalized Distributions. Communications in Statistics: Theory and Methods, 20(12), Rebonato, R. (1999). Volatility and correlations in the pricing of equity. FX and interest-rate options. New York: John Wiley. Paulson, A. S. and N. J. Faris (1985). A Practical Approach to Measuring the Distribuiton of Total Annual Claims. In J. D. Cummins, ed., Strategic Planning and Modeling in Property-Liability Insurance. Norwell, MA: Kluwer Academic Publishers. Ramlau-Hansen, H. (1988). A Solvency Study in Non-life Insurance. Part 1. Analysis of Fire, Windstorm, and Glass Claims. Scandinavian Actuarial Journal, pp Rebonato, R Volatility and correlations in the pricing of equity, FX and interest-rate options. New York: John Wiley. Reid, D. H. (1978). Claim Reserves in General Insurance, Journal of the Institute of Actuaries 105: Savickas, R. (2001). A Simple option-pricing formula. Working paper, Department of Finance, George Washington University, Washington, DC. Sherrick, B. J., P. Garcia, and V. Tirupattur (1996). Recovering probabilistic information for options markets: Tests of distributional assumptions. Journal of Futures Markets 16: Theodossiou, Panayiotis, Skewed Generalized Error Distribution of Financial Assets and Option Pricing,

135 Statistical Distributions in Finance 1. Introduction 2. Some families of statistical distributions 3. Regression applications 4. Qualitative response models 5. Option pricing: European call option 6. VaR (value at risk) 7. Conclusion

136 Conclusion

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138

139 END OF PRESENTATION

140 Appendices Cumulative distribution functions 1. GB, GB1, GB2, GG 2. EGB2 3. SGT 4. SGED 5. IHS 6. g-and-h distribution Option pricing basics VaR Models and applications discussion

141 Appendices Cumulative distribution functions 1. GB, GB1, GB2, and GG ( y a b p q) GB1 ;,,, where z = B z ( p, q) = 0 = = and denotes the incomplete beta function p z 2F1 p,1 q; p + 1; z pb p q B z ( y / b) ( 1 ) ( p, q) a z p 1 q 1 s s ds (, ) B p q (, )

142 Appendices Cumulative distribution functions 1. GB, GB1, GB2, and GG (continued) ( y a b p q) GB2 ;,,, = = p z 2F1 p,1 q; p + 1; z pb p q B z ( p, q) (, ) where z a ( y/ b) = 1 + y/ b ( ) a

143 Appendices Cumulative distribution functions 1. GB, GB1, GB2, and GG (continued) where a ( y / ) ( y/ ) ( p + 1) ( ) e GG ( y; a, b, p) = 1F 11; p + 1; y / z = ( y/ ) a = z p ap ( ) a and ( p) = z z 0 s p 1 s e ds ( p) denotes the incomplete gamma function Abramowitz and Stegun (1970, p. 932), McDonald (1984), and Rainville (1960,p. 60 and 125)

144 Appendices Cumulative distribution functions 2. EGB2 ( ) = ( ) EGB 2 y; m,, p, q B p, q z where z e = 1 + e ( y m) / ( y m) / 3. SGT where ( 1+ sign( y m) ) 1 SGT ( y; m,,, p, q) = + sign y m Bz 1/ p, q 2 2 z = y m p ( 1 ( )) p p y m + q + sign y m p ( ) ( )

145 Appendix Cumulative distribution functions 4. SGED ( m) 1 1+ sign y SGED ( y; m,,, p) = + sign y m z 1/ p 2 2 ( ) ( ) where z = y m ( 1+ ( )) p sign y m p p

146 Appendices Cumulative distribution functions 5. IHS IHS y;,, k, = Pr Y y = Pr Z z where and ( ) ( ) ( ) ( 2 ; = 0, = 1) = Pr ( ) N z Z z 1 z 1 3 z = + 1F1 ; ; ( ) ( ) 1 sign z 1 = + ( z 2 /2 2 2 ) 2 2 y a y a z = k n k with b b b / k k k 2 w / e + e + 2 e 1 2.5k a = b = b.5 e e e.5.5 = = ( ) ( ) ( ( ) ) w

147 Appendices Cumulative distribution functions 6. g- and h-distribution Numeric procedures, based on the use of order statistics as outlined in Exploring Data Tables, Trends, and Shapes by Hoaglin,, Mosteller, and Tukey (1985), Wiley. For h > 0, the transformation gz e 1 Ygh, ( Z ) = a + b e g hz 2 /2 is one-to-one, (Martinez, J. and B. Iglewicz Some Properties of Tukey g and h family of distributions, Communications in Statistics Theory and Methods 13, ). Even without an explicit functional form for the inverse, numerical MLE estimates can be obtained.

148 Appendices Cumulative distribution functions Option pricing basics 1. European call option 2. Put option 3. Definitions of terms 4. Assumptions 5. Volatility 6. The Greeks VaR Models and applications discussion

149 Appendices Option pricing basics 1. European call option rt ( ) ( ) rt ( 0 ) ( ) ( ) C S, T, X, r = e E C S,0, X, r = e S X f S S, T ds f T T X X rt X = ST ;1 e X ;0 BS: S d e X d ST St rt ( T ( 1) ( 2 )) 2. Put option BS Put formula : e rt X d S -d ( ) ( ) 2 T 1

150 Appendices Option pricing basics 3. Definitions of terms: T = time to expiration ST = Current market price r = interest rate (risk free rate) X = strike price (or exercise price) call options: price at which the instrument can be purchased up to expiration ST X profit per share gained upon exercising or selling the option ST S X X >0 in the money <0 out of the money ( ) ( ) ( ) T put options: price at which the instrument can be sold up to expiration

151 Appendices Option pricing basics 4. Assumptions: Can short sell the underlying instrument No arbitrage opportunities Continuous trading in the instrument No taxes or transaction costs Securities are perfectly divisible Can borrow or lend at a constant risk free rate The instrument does not pay a dividend 5. Volatility (in the BS option pricing formula based on the LN)

152 Appendices Option pricing basics 6. The Greeks: (delta) measures the change in value of the instrument to a change in the current market price ( C f ( ST, T, X, r) ) X = = ;1 ST ST (kappa or vega) measures the responsiveness of the value of the instrument in response to a change in volatility ( ( )) C f ST, T, X, r = ( volatility) (theta) responsiveness of the value of the instrument to T (time to expiration) = ( C f ( ST, T, X, r) ) T (rho) responsiveness to changes in the risk free rate = ( C f ( ST, T, X, r) ) r

153 Appendices Cumulative distribution functions Option pricing basics VaR Models and applications discussion

154 Appendices VaR: Models and applications discussion Paulson and Faris (1985) used the stable family and Aiuppa (1988) used the Pearson family to model insurance losses Ramlau-Hansen (1988) modeled fire, windstorm, and glass claims using the log-gamma and lognormal Cummins, et al. (1990) modeled fire losses using the GB2 Cummins, Lewis, and Phillips (1999) used the LN, Burr 12, and GB2 to model hurricane and earthquake losses. Hogg, R. V. and S. A. Klugman, On the Estimation of Long Tailed Skewed Distributions with Actuarial Applications. Journal of Econometrics 23, Models loss distributions (a. Hurricaines ( ), b. malpractice claims paid for insured hospitals in 1975) Considers exponential, pareto (mixture of an exponential and inverse gamma), generalized pareto (mixture of gamma and inverse gamma), Burr distribution (mixture of a Weibull and inverse gamma), log-t (mixture of a lognormal and inverse gamma) and a log-gamma. Consider alternative estimation procedures: maximum likelihood and minimum distance estimators Many loss distributions are characterized by skewness and long tails such as associated with the flexible distributions coming from mixtures.

155 Appendices VaR: Models and applications discussion Cummins, J. D., G. Dionne, J. B. McDonald, and B. M. Pritchett, Applications of the GB2 family of distributions in modeling insurance loss processes. Insurance: Mathematics and Economics 9, Models fire losses Considers the GB2 and special cases GG, BR3, BR12, LN, W, and GA to model the fire loss data. MLE estimates of distributional parameters and Maximum Probably Yearly Aggregate Loss (MPY) were obtained at the.01 level. Important to use distributions which permit thick tails Bali, T. G., An Extreme Value Approach to Estimating Volatility and Value at Risk, Journal of Business, 76:83-108

156 Appendices VaR: Models and applications discussion Cummins, J. D., C. Merrill, and J. B. McDonald, Risky Loss Distributions and Modeling the Loss Reserve Pay-out Tail, Review of Applied Economics 3. Estimate aggregate loss distribution associated with claims incurred in a given year, but settled in different years Data: U.S. products liability insurance paid claims (Insurance Services Office (ISO)) Mixture model: Consider different GB2 distributions for each cell (year) Multinomial distribution for fraction of claims settled at different lags Single aggregate GB2 distribution for each year GB2 provides a significantly better fit to severity data than the LN, gamma, Weibull, Burr12, or generalized gamma The Aggregate GB2 distribution has a thicker tail than does the mixture distribution

157 Appendices VaR: Models and applications discussion Bali, T. G. and P. Theodossiou, Risk Measurement Performance of Alternative Distribution Functions, Journal of Risk and Insurance, 75: Models: Unconditional formulations Generalized Pareto Generalized extreme value Box-Cox extreme value SGED SGT EGB2 IHS Models: Conditional formulations (model time-varying VaR thresholds) Rt = 0 + 1Rt 1 + zt t = t + zt t = + z + t 0 1 t 1 t 1 2 t 1 t L

158 Appendices VaR: Models and applications discussion Bali, T. G. and P. Theodossiou, Risk Measurement Performance of Alternative Distribution Functions, Journal of Risk and Insurance, 75: (continued) Data S&P500 composite index (1/4/1950 to 12/29/2000) Daily percentage log-returns: (n=12,832 maximum=8.71 minimum= skewness =1.622 kurtosis=45.52 Findings Out of sample VaR estimates are rejected for most unconditional specifications Thresholds exhibit time varying behavior Out of sample VaR estimates for the conditional specifications corresponding to the SGT, IHS, and EGB2 perform better than the extreme value distributions

159 END OF APPENDICES