cht10 相关系数与Copula函数.
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相关系数与 Copula 函数
Chapter 10
,金融风险管理
张学功
1
Correlation and Covariance
The coefficient of correlation between two variables V1 and V2 is defined as
E (VV 1 2 ) E (V1 ) E (V2 ) = SD(V1 ) SD(V2 )
The covariance is E(V1V2)−E(V1 )E(V2)
张学功
金融风险管理
2
Independence
V1 and V2 are independent if the knowledge of one does not affect the probability distribution for the other
f (V2 V1 x ) f (V2 )
where f(.) denotes the probability density function
金融风险管理 张学功
3
Independence is Not the Same as Zero Correlation
Suppose V1 = –1, 0, or +1 (equally likely) If V1 = -1 or V1 = +1 then V2 = 1 If V1 = 0 then V2 = 0 V2 is clearly dependent on V1 (and vice versa) but the coefficient of correlation is zero
金融风险管理
张学功
4
Types of Dependence (Figure 11.1, page 235)
E (Y ) X E (Y ) X
(a)
E (Y )
(b)
X
(c)
金融风险管理 张学功
5
Monitoring Correlation Between Two Variables X and Y
Define xi=(Xi−Xi-1)/Xi-1 and yi=(Yi−Yi-1)/Yi-1 Also varx,n: daily variance of X calculated on day n-1 vary,n: daily variance of Y calculated on day n-1 covn: covariance calculated on day n-1 The correlation is
covn varx,n vary ,n
金融风险管理 张学功
6
Covariance
The covariance on day n is E(xnyn)−E(xn)E(yn) It is usually approximated as E(xnyn)
金融风险管理
张学功
7
Monitoring Correlation continued
EWMA:
covn covn1 (1 ) xn1 yn1
GARCH(1,1)
covn xn1 yn1 covn1
金融风险管理
张学功
8
假设 λ=0.95 ，m变量X 及 Y 在第 n -1 天的相关系数估计为 0.6 ，同 时我们又假设变量X 及 Y 在第n -1天的技动率估计分别为 1% 及 2% 。 由协方差及 相关系数的关系式，在第 n -1 天的协方差估计为0.6 x 0. 01 x 0. 02 = 0.00012 假定变量X及 Y 在第 n -1 天的百分比变化分别为0.5% 及 2.5% ， 在第n 天的方差及协方差的估计分别为 σ2X，n = 0. 95 x 0. 012 + 0. 05 x 0. 005= 0. 000096 25 σ2Y，n = 0. 95 x 0 . 02+ 0. 05 x 0. 0252= 0.000411 25 COVn = 0.95 x 0.00012 + 0. 05 x 0. 005 x 0. 025= 0.000 12025 变量 X 的最新波动率估计为 0.00009625=0.981%
;变量Y 的最新波动率估计为 ，X 及 Y 的最新相关系数为
0.00041125=2.2028%
0.00012025 =0.6044 0.00981 0.02028
金融风险管理
张学功
9
Positive Finite Definite Condition
A variance-covariance matrix, W, is internally consistent if the positive semidefinite condition wTWw ≥ 0 holds for all vectors w
金融风险管理
张学功
10
Example
The variance covariance matrix
1 0 0.9 0 1 0.9 0.9 0.9 1
is not internally consistent
金融风险管理
张学功
11
V1 and V2 Bivariate Normal
Conditional on the value of V1, V2 is normal with mean
V1 m1 m 2 s 2 s1
and standard deviation s 2 1 2 where m1,, m2, s1, and s2 are the unconditional means and SDs of V1 and V2 and is the coefficient of correlation between V1 and V2
金融风险管理 张学功
12
N元联合正态分布的随机抽样生成 n 个相互独立，并 且服从正态分布的随机抽样 Zi 1 i N ，则n元正态
随机分布变量为：
i
i ik zk , 其中 ik =1， ik z jk ij i j
Chapter 10
,金融风险管理
张学功
1
Correlation and Covariance
The coefficient of correlation between two variables V1 and V2 is defined as
E (VV 1 2 ) E (V1 ) E (V2 ) = SD(V1 ) SD(V2 )
The covariance is E(V1V2)−E(V1 )E(V2)
张学功
金融风险管理
2
Independence
V1 and V2 are independent if the knowledge of one does not affect the probability distribution for the other
f (V2 V1 x ) f (V2 )
where f(.) denotes the probability density function
金融风险管理 张学功
3
Independence is Not the Same as Zero Correlation
Suppose V1 = –1, 0, or +1 (equally likely) If V1 = -1 or V1 = +1 then V2 = 1 If V1 = 0 then V2 = 0 V2 is clearly dependent on V1 (and vice versa) but the coefficient of correlation is zero
金融风险管理
张学功
4
Types of Dependence (Figure 11.1, page 235)
E (Y ) X E (Y ) X
(a)
E (Y )
(b)
X
(c)
金融风险管理 张学功
5
Monitoring Correlation Between Two Variables X and Y
Define xi=(Xi−Xi-1)/Xi-1 and yi=(Yi−Yi-1)/Yi-1 Also varx,n: daily variance of X calculated on day n-1 vary,n: daily variance of Y calculated on day n-1 covn: covariance calculated on day n-1 The correlation is
covn varx,n vary ,n
金融风险管理 张学功
6
Covariance
The covariance on day n is E(xnyn)−E(xn)E(yn) It is usually approximated as E(xnyn)
金融风险管理
张学功
7
Monitoring Correlation continued
EWMA:
covn covn1 (1 ) xn1 yn1
GARCH(1,1)
covn xn1 yn1 covn1
金融风险管理
张学功
8
假设 λ=0.95 ，m变量X 及 Y 在第 n -1 天的相关系数估计为 0.6 ，同 时我们又假设变量X 及 Y 在第n -1天的技动率估计分别为 1% 及 2% 。 由协方差及 相关系数的关系式，在第 n -1 天的协方差估计为0.6 x 0. 01 x 0. 02 = 0.00012 假定变量X及 Y 在第 n -1 天的百分比变化分别为0.5% 及 2.5% ， 在第n 天的方差及协方差的估计分别为 σ2X，n = 0. 95 x 0. 012 + 0. 05 x 0. 005= 0. 000096 25 σ2Y，n = 0. 95 x 0 . 02+ 0. 05 x 0. 0252= 0.000411 25 COVn = 0.95 x 0.00012 + 0. 05 x 0. 005 x 0. 025= 0.000 12025 变量 X 的最新波动率估计为 0.00009625=0.981%
;变量Y 的最新波动率估计为 ，X 及 Y 的最新相关系数为
0.00041125=2.2028%
0.00012025 =0.6044 0.00981 0.02028
金融风险管理
张学功
9
Positive Finite Definite Condition
A variance-covariance matrix, W, is internally consistent if the positive semidefinite condition wTWw ≥ 0 holds for all vectors w
金融风险管理
张学功
10
Example
The variance covariance matrix
1 0 0.9 0 1 0.9 0.9 0.9 1
is not internally consistent
金融风险管理
张学功
11
V1 and V2 Bivariate Normal
Conditional on the value of V1, V2 is normal with mean
V1 m1 m 2 s 2 s1
and standard deviation s 2 1 2 where m1,, m2, s1, and s2 are the unconditional means and SDs of V1 and V2 and is the coefficient of correlation between V1 and V2
金融风险管理 张学功
12
N元联合正态分布的随机抽样生成 n 个相互独立，并 且服从正态分布的随机抽样 Zi 1 i N ，则n元正态
随机分布变量为：
i
i ik zk , 其中 ik =1， ik z jk ij i j