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Potential PCA Interpretation Problems for Volatility Smile Dynamics

Robert Tompkins, Dimitri Reiswich July, 16th 2010 Analysis, Stochastics, and Applications

A Conference in Honour of Walter Schachermayer Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 1 / 34

Introduction

The goal of PCA is to reduce the dimensionality of multiple correlated random variables to a parsimonious set of uncorrelated components. Suppose that the correlated random variables are summarized in a n × 1 vector x with covariance matrix Σ.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 2 / 34

Introduction

The goal of PCA is to reduce the dimensionality of multiple correlated random variables to a parsimonious set of uncorrelated components. Suppose that the correlated random variables are summarized in a n × 1 vector x with covariance matrix Σ. Initially, PCA determines a new random variable y1 which is a linear combination of the components of x weighted by the components of a vector γ1 ∈ Rn×1 y1 = γT

1 x = n

• i=1

γ1ixi

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 2 / 34

Introduction

The goal of PCA is to reduce the dimensionality of multiple correlated random variables to a parsimonious set of uncorrelated components. Suppose that the correlated random variables are summarized in a n × 1 vector x with covariance matrix Σ. Initially, PCA determines a new random variable y1 which is a linear combination of the components of x weighted by the components of a vector γ1 ∈ Rn×1 y1 = γT

1 x = n

• i=1

γ1ixi The vector γ1 is chosen such that y1 has maximum variance Var(y1) and γ1 has a length of 1. In the next step, a new variable y2 = γT

2 x is determined, which is

not correlated with y1 and has maximum variance. The kth derived variable yk = γT

k x is called kth Principal Component (PC).

We hope that most of the variation in x will be accounted for by m PCs, where m << n.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 2 / 34

Introduction

The maximization problem we have to solve at stage k is max

||γk||=1 Var(yk)

subject to Cov(yk, yk−1) = Cov(γT

k x, γT k−1x) = 0.

This problem can be solved by the choice of γk as the eigenvector corresponding to the k-th largest eigenvalue λk of Σ - the covariance matrix of x. We then have Var(yk) = λk The explained variance associated with the k-th PC can be expressed as: λk n

i=1 λi

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 3 / 34

Introduction

The maximization problem we have to solve at stage k is max

||γk||=1 Var(yk)

subject to Cov(yk, yk−1) = Cov(γT

k x, γT k−1x) = 0.

This problem can be solved by the choice of γk as the eigenvector corresponding to the k-th largest eigenvalue λk of Σ - the covariance matrix of x. We then have Var(yk) = λk The explained variance associated with the k-th PC can be expressed as: λk n

i=1 λi

The empirical financial literature shows that most of the variance is explained by 3 factors, see for example Alexander (2003), Alexander (2001), Cont and da Fonseca (2002), Daglish et al. (2007), Fengler et al. (2003), Schmidt et al. (2002), Skiadopoulos et al. (2000), Zhu and Avellaneda (1997), P´ erignon et al. (2007).

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 3 / 34

Introduction

1.0 1.5 2.0 2.5 3.0 −1.0 −0.5 0.0 0.5 1.0 Loading Index Loading 1.0 1.5 2.0 2.5 3.0 −1.0 −0.5 0.0 0.5 1.0 Loading Index Loading 1.0 1.5 2.0 2.5 3.0 3.5 4.0 −1.0 −0.5 0.0 0.5 1.0 Loading Index Loading 1 2 3 4 5 −1.0 −0.5 0.0 0.5 1.0 Loading Index Loading

Figure: Level (upper left), Skew (upper right), Twist (lower left), Curvature (lower right) Patterns in PCA

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 4 / 34

Potential Interpretation Problems

A well known theorem stated by Perron and Frobenius reveals a priori information about the structure/shape of the eigenvectors (see (Meyer, 2000, Chapter 8), Lord and Pelsser (2007)) Theorem (Frobenius-Perron) For a n × n strictly positive matrix Σ, the eigenvector associated with the largest eigenvalue is strictly positive.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 5 / 34

Potential Interpretation Problems

A well known theorem stated by Perron and Frobenius reveals a priori information about the structure/shape of the eigenvectors (see (Meyer, 2000, Chapter 8), Lord and Pelsser (2007)) Theorem (Frobenius-Perron) For a n × n strictly positive matrix Σ, the eigenvector associated with the largest eigenvalue is strictly positive. Lord and Pelsser (2007) provide another theorem based on the work of Gantmacher and Krein (1960) Theorem (Sign Change Pattern) Assume Σ is a valid n × n covariance matrix that is strictly totally positive of order k. Then we have λ1 > λ2 > ... > λk. The eigenvector corresponding to the eigenvalue λj has j − 1 changes of sign for j = 1, ..., k (zero terms in the eigenvector are discarded).

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 5 / 34

Potential Interpretation Problems

1.0 1.5 2.0 2.5 3.0 −1.0 −0.5 0.0 0.5 1.0 Loading Index Loading 1.0 1.5 2.0 2.5 3.0 −1.0 −0.5 0.0 0.5 1.0 Loading Index Loading 1.0 1.5 2.0 2.5 3.0 3.5 4.0 −1.0 −0.5 0.0 0.5 1.0 Loading Index Loading 1 2 3 4 5 −1.0 −0.5 0.0 0.5 1.0 Loading Index Loading

Figure: Level (upper left), Skew (upper right), Twist (lower left), Curvature (lower right) Patterns in PCA

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 6 / 34

Potential Interpretation Problems

Consider the following empirical correlation matrix of implied volatilities of the Euro

• vs. US Dollar across five moneyness levels (in delta terms) for 1 month maturity
• ptions, using Bloomberg data from 03.10.2003 to 21.01.2009.

      1.000 0.968 0.953 0.927 0.898 0.968 1.000 0.989 0.968 0.923 0.953 0.989 1.000 0.991 0.951 0.927 0.968 0.991 1.000 0.966 0.898 0.923 0.951 0.966 1.000      

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 7 / 34

Potential Interpretation Problems

Consider the following empirical correlation matrix of implied volatilities of the Euro

• vs. US Dollar across five moneyness levels (in delta terms) for 1 month maturity
• ptions, using Bloomberg data from 03.10.2003 to 21.01.2009.

      1.000 0.968 0.953 0.927 0.898 0.968 1.000 0.989 0.968 0.923 0.953 0.989 1.000 0.991 0.951 0.927 0.968 0.991 1.000 0.966 0.898 0.923 0.951 0.966 1.000       The matrix represents the class of bisymmetric matrices.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 7 / 34

Potential Interpretation Problems

Definition Let J ∈ Rn×n be a matrix which has ones on its anti-diagonal, and zeros everywhere else J =        ... ... 1 ... ... 1 . . . ... . . . 1 ... ... 1 ... ...        . (1) A bisymmetric matrix A ∈ Rn×n is a matrix which is symmetric with respect to both

• f its diagonals and thus fulfills the following condition

JAJ = A.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 8 / 34

Potential Interpretation Problems

The matrix J can be used to define symmetric and skew-symmetric vectors, which will be important objects in the following analyses. Definition A vector γs ∈ Rn is called symmetric, if Jγs = γs. A vector γss ∈ Rn is called skew-symmetric, if Jγss = −γss. Examples of these classes are γs =   1 2 1   , γs = 3 3

• ,

γss =   1 −1   , γss = 1 −1

• .

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 9 / 34

Potential Interpretation Problems

It can be shown that the eigenvectors of bisymmetric classes are either symmetric or skew-symmetric, see Cantoni and Butler (1976). We will summarize this in the following theorem, distinguishing between a quadratic matrix with an odd or even number of rows.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 10 / 34

Potential Interpretation Problems

It can be shown that the eigenvectors of bisymmetric classes are either symmetric or skew-symmetric, see Cantoni and Butler (1976). We will summarize this in the following theorem, distinguishing between a quadratic matrix with an odd or even number of rows. Theorem Suppose A ∈ Rn×n is bisymmetric and n is even. Matrix A has n/2 skew-symmetric and n/2 symmetric orthonormal eigenvectors. Let ⌈x⌉ denote the smallest integer ≥ x, and ⌊x⌋ denote the largest integer ≤ x. Define u =: n 2

• (2)

l := n 2

• (3)

to be the upper and lower integer of n

2 respectively. Suppose A ∈ Rn×n is bisymmetric

and n is odd. Matrix A has l skew-symmetric and u symmetric orthonormal eigenvectors.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 10 / 34

Potential Interpretation Problems

It can be shown that the eigenvectors of bisymmetric classes are either symmetric or skew-symmetric, see Cantoni and Butler (1976). We will summarize this in the following theorem, distinguishing between a quadratic matrix with an odd or even number of rows. Theorem Suppose A ∈ Rn×n is bisymmetric and n is even. Matrix A has n/2 skew-symmetric and n/2 symmetric orthonormal eigenvectors. Let ⌈x⌉ denote the smallest integer ≥ x, and ⌊x⌋ denote the largest integer ≤ x. Define u =: n 2

• (2)

l := n 2

• (3)

to be the upper and lower integer of n

2 respectively. Suppose A ∈ Rn×n is bisymmetric

and n is odd. Matrix A has l skew-symmetric and u symmetric orthonormal eigenvectors. In particular, the eigenvectors of any non-trivial 2 × 2 correlation matrix are γs = ( 1 √ 2 , 1 √ 2 ) γss = ( 1 √ 2 , − 1 √ 2 ).

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 10 / 34

Potential Interpretation Problems

1 2 3 4 5 −1.0 −0.5 0.0 0.5 1.0 Index Loadings 1 2 3 4 5 −1.0 −0.5 0.0 0.5 1.0 Index Loadings 1 2 3 4 5 −1.0 −0.5 0.0 0.5 1.0 Index Loadings

Figure: Level (left), skew (center), curvature (right) patterns in a PCA on EURUSD 1 month implied vols.

As stated in the previous theorem, one can observe 2 symmetric and 1 skew-symmetric eigenvector in the empirical data

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 11 / 34

Potential Interpretation Problems

The theorem reveals that there is an a priori known, systematic structure in some matrix classes. The symmetric and skew-symmetric eigenvectors have the shape of the typical level, skew, curvature vectors observed in the empirical literature.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 12 / 34

Potential Interpretation Problems

The theorem reveals that there is an a priori known, systematic structure in some matrix classes. The symmetric and skew-symmetric eigenvectors have the shape of the typical level, skew, curvature vectors observed in the empirical literature. We will stick to the bisymmetric case and analyze potential problems in the 3 × 3 case for the following general bisymmetric matrix A =   a1 b c b a2 b c b a1   (4) From the previous result we know that this matrix will have 2 symmetric and 1 skew-symmetric eigenvector. More concrete results can be derived for this case.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 12 / 34

Potential Interpretation Problems

Lemma Matrix A has the following skew-symmetric, orthonormal eigenvector vss and its corresponding eigenvalue λss vss =  

1 √ 2

− 1

√ 2

  λss = a1 − c. (5)

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 13 / 34

Potential Interpretation Problems

Lemma Matrix A has the following skew-symmetric, orthonormal eigenvector vss and its corresponding eigenvalue λss vss =  

1 √ 2

− 1

√ 2

  λss = a1 − c. (5) Of particular interest is, that the explicit form of the eigenvector does not depend on any of the variables a, b, c! It is known in advance, independent of the underlying system. Also, note that the form of the vector is the typical form of a skew vector produced by a PCA for financial market data.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 13 / 34

Potential Interpretation Problems

Lemma Define d :=

• a2

1 − 2a1a2 + a2 2 + 8b2 + 2a1c − 2a2c + c2

and ws1 =  

a1−a2+c+d 4b

1

a1−a2+c+d 4b

  ws2 =  

a1−a2+c−d 4b

1

a1−a2+c−d 4b

  . (6) Matrix A has the following symmetric, orthonormal eigenvectors vs1, vs2 and corresponding eigenvalues λs1, λs2 vs1 = 1 ws1ws1, λs1 = 1 2(a1 + a2 + c + d) (7) vs2 = 1 ws2ws2, λs2 = 1 2(a1 + a2 + c − d) (8) where x denotes the Euclidian norm of vector x.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 14 / 34

Potential Interpretation Problems

Assume that we observe three random variables defined as follows σ+ = a + b+ σ0 = a + b0 σ− = a + b− (9) where b+, b0, b− and a are independent random variables. Furthermore, assume that the variables b+ and b− have the same variance: V (b+) = V (b−).

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 15 / 34

Potential Interpretation Problems

Assume that we observe three random variables defined as follows σ+ = a + b+ σ0 = a + b0 σ− = a + b− (9) where b+, b0, b− and a are independent random variables. Furthermore, assume that the variables b+ and b− have the same variance: V (b+) = V (b−). The covariance matrix will then have the following form Σ =   V (σ+) V (a) V (a) V (a) V (σ0) V (a) V (a) V (a) V (σ+)   . This matrix is bisymmetric, as is the corresponding correlation matrix. Consequently, the system will have an eigenvector which can be interpreted as ”skew”. Looking at the original system does not reveal any skew effect. The only effect that makes sense from an intuitive point of view is a level/shift factor.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 15 / 34

Potential Interpretation Problems

To verify these theoretical results, consider the following simulation. We define a ∼ N(0, 0.2) b+ ∼ N(0, 0.25) b0 ∼ U[0, 1] b− ∼ exp(2) We generated n = 50.000 realizations of the variables σ−, σ0, σ+ by using independent variables a, b−, b0, b+ with the specifications above. The estimated covariance matrix contained the following values: Σ =   0.450 0.202 0.201 0.202 0.283 0.200 0.201 0.203 0.450   .

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 16 / 34

Potential Interpretation Problems

The resulting factor loading vectors of the PCA on the estimated covariance matrix are: vs1 =   0.62 0.48 0.62   vs2 =   0.35 −0.88 0.33   vss =   0.70 0.01 −0.71   . The eigenvalues corresponding to the eigenvectors, rounded to the second decimal place, are λs1 = 0.81, λs2 = 0.13, λss = 0.25

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 17 / 34

Potential Interpretation Problems

The resulting factor loading vectors of the PCA on the estimated covariance matrix are: vs1 =   0.62 0.48 0.62   vs2 =   0.35 −0.88 0.33   vss =   0.70 0.01 −0.71   . The eigenvalues corresponding to the eigenvectors, rounded to the second decimal place, are λs1 = 0.81, λs2 = 0.13, λss = 0.25 For bisymmetric matrices, a skew effect will result even when variables are randomly generated with no associated skew effect. Note also, that the vector vs2 is a typical representative of a curvature eigenvector, which from an intuitive point of view is not present in the original system. The system also indicates the typical order in terms of the explained variance. The first symmetric eigenvector will explain most of the variance, followed by the skew-symmetric eigenvector and the second symmetric eigenvector.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 17 / 34

PCA on a Shift, Skew, Curvature System

Consider the following general system of 3 random variables σ+ = al + as + ac σ0 = al σ− = al − as + ac (10)

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 18 / 34

PCA on a Shift, Skew, Curvature System

Consider the following general system of 3 random variables σ+ = al + as + ac σ0 = al σ− = al − as + ac (10) System represents various parabolic representations of implied volatilities in moneyness space: σ(∆) = bl + bs(∆ − ∆0) + bc(∆ − ∆0)2. Can also be found in FX quotation conventions σ+ = σAT M + 1 2σRR + σST R σ− = σAT M − 1 2σRR + σST R

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 18 / 34

PCA on a Shift, Skew, Curvature System

The system can be standardized with scaled factors of al, as, ac such that the variances of σ+, σ−, σ0 are 1. The resulting system, expressed with variables b, is σ+ = 1 √ 3 bl + 1 √ 2 bs + 1 √ 2 bc σ0 = 1 √ 3 bl σ− = 1 √ 3 bl − 1 √ 2 bs + 1 √ 2 bc

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 19 / 34

PCA on a Shift, Skew, Curvature System

To summarize: The introduced equation system represents a random system which has level, skew and curvature effects. This system is a generalization of parametric forms that have commonly appeared in the financial literature and are used as a volatility quotation mechanism in financial markets. Therefore we will consider PCA analysis of the system previously defined with confidence that these results will proxy interpretation properties of PCA applications in financial markets. Specifically, the next part will address the following set of questions: Do we still observe loading vectors representing level, skew and curvature if we perform a PCA analysis on the covariance and correlation matrix of the system? How much of the variance do the corresponding PCs explain in the original system and in the PCA results?

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 20 / 34

PCA on the Covariance Matrix

The covariance matrix of the ordered variables σ−, σ0, σ+ has the following form   V (σ+)

1 3V (bl)

V (σ+) − V (bs)

1 3V (bl)

V (σ0)

1 3V (bl)

V (σ+) − V (bs)

1 3V (bl)

V (σ+)   . (11) As this is a bisymmetric matrix, we can use the results of the previous sections to calculate the respective eigenvectors/eigenvalues which would represent those from a PCA. By application of Lemma (6), we can calculate the variance of the skew symmetric factor as λss = V (bs). This results in equal variances for both, the principal component representing skew in the original setup and the PCA setup. In this case the principal component representing skew is correctly recovered.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 21 / 34

PCA on the Covariance Matrix

For the other PCs the analysis is somewhat more complicated. The first eigenvector does not always produce a parallel eigenvector, as will be demonstrated in the following simple example. Consider setting the variances of the variables bl, bs, bc in matrix (11) equal to 0.20. The figure below displays the eigenvectors in this case (skew vector not included).

• 1.0

1.5 2.0 2.5 3.0 −0.5 0.0 0.5 Loading Index Loading

• 1. Factor
• 3. Factor

Figure: Loading vectors for the 1. and 3. PC for V (bl) = V (bs) = V (bc) = 0.20.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 22 / 34

PCA on the Covariance Matrix

Is is clear that both loading vectors display some degree of curvature. The dominant curvature effect is clearly seen in the third factor. However, factor one explains both level and some degree of curvature. We will coin this property as bi-explanatory. Therefore, PCA analysis on such a system could potentially not recover a clear parallel shift of the original system. The eigenvalue corresponding to the first symmetric eigenvector of matrix (11) is λs1 = 1 2V (bc) + 1 2V (bl) + 1 2 √ 3

• 3V (bl)2 + 3V (bc)2 + 2V (bc)V (bl).

One can easily show that λs1 is always greater than V (bl). The implication of this is that the first skew symmetric factor will always explain more variance than the

• riginal level component. Similarly, λs2 will always be smaller than V (bc). This

implies that the second skew symmetric factor will always explain less variance than the original curvature component. Therefore the PCA will either over or underestimate the contributed variance of the corresponding factors.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 23 / 34

PCA on the Covariance Matrix

Let us now consider the circumstance where the original system is dominated by a single factor: either skew, shift or curvature. We would expect the corresponding PC to explain 100% of the variance and the loading vectors will display similar shapes as the vectors in the original system. The analysis of the skew symmetric vector is trivial. It has already been shown above that the variance of the skew factor is identical in both systems. Now consider a dominating shift case. Let vs1 again be the first symmetric eigenvector of matrix (11) and λs1 the corresponding eigenvalue. It can be shown, that lim

V (bl)→∞ vs1 =

  

1 √ 3 1 √ 3 1 √ 3

   and lim

V (bl)→∞

λs1 λs1 + λs2 + λss = 1.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 24 / 34

PCA on the Covariance Matrix

In this case, PCA will recover the same loadings vector as in the original system (in the limit). In addition, the corresponding PC will explain 100% of the total variance. Finally, let us consider the curvature. As opposed to the skew and shift results, the curvature result is surprising. It can be shown that lim

V (bc)→∞

λs2 λs1 + λs2 + λss = 0 = lim

V (bl)→∞

λs2 λs1 + λs2 + λss = lim

V (bs)→∞

λs2 λs1 + λs2 + λss . (12) In this case, the second symmetric factor is not the dominating PC. Of greater concern is that, in the limiting case, the second symmetric factor does not explain any variance. It turns out that all the variance is being explained by the first symmetric factor, which we previously identified as a shift. This is due to: lim

V (bc)→∞ vs1 =

 

1 √ 2 1 √ 2

  and lim

V (bc)→∞

λs1 λs1 + λs2 + λss = 1.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 25 / 34

PCA on the Covariance Matrix

Of importance is how this can be corrected. Consider the 3 × 3 case , with three variables and three factors. The first step is to estimate a covariance matrix from data generated by this system. Then, we perform the PCA analysis and obtain eigenvalues λs1, λs2 and λss. While we can estimate the covariance matrix of the original system, we do not know the variances of the individual factors. Using the eigenvalues, however, we can calculate the original variances V (bl), V (bc), V (bs) of the factors. The skew factor variance is correctly recovered and requires no further modification. For the other two variances, the appropriate modification is: V (bl) = 1 2

• λs1 + λs2 ±
• λ2

s1 − 10λs1λs2 + λ2 s2

• ,

(13) V (bc) = λs1 + λs2 − V (bl). (14) Then, both the level and curvature variances are correctly recovered.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 26 / 34

PCA on the Covariance Matrix

Let us consider a simple numerical example, using the following parameter inputs: V (bl) = 0.41, V (bs) = 0.10, V (bc) = 0.11. We generated 50, 000 realizations of the variables σ+, σ0 and σ− using independent, normally distributed random variables bl, bs, bc. The eigenvectors and eigenvalues of the estimated covariance matrix were vs1 =   0.62 0.48 0.62   vs2 =   0.34 −0.88 0.34   vss =   0.71 0.00 −0.71   λs1 = 0.49, λs2 = 0.03, λss = 0.10.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 27 / 34

PCA on the Covariance Matrix

The results appear in table (1). In this table, the left columns represent the original variances of the factors, the middle columns show the PCA variances and the right columns display the corrected factor variances. These are presented both in levels and as a percentage contribution of the total variance.

Table: Comparison of explained variances by level, skew, curvature factors.

Original PCA Corrected Variance % of To- tal Variance % of To- tal Variance % of To- tal Level 0.41 66.1% 0.49 79.0% 0.41 66.1% Skew 0.10 16.1% 0.10 16.1% 0.10 16.1% Curvature 0.11 17.7% 0.03 4.83% 0.11 17.7%

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 28 / 34

PCA on the Covariance Matrix

The PCA indicates that the curvature effect (second symmetric factor) explains the smallest amount (0.03) of the original variance, even though it has the second largest variance (0.11) in the original system. This could lead to the misleading conclusion that curvature was the least important contribution. As we surmised, the first symmetric eigenvector (level vector) explains more variance (0.49) than in the original system (0.41). As was indicated previously, the skew factor was correctly recovered. To correct the other errors, we apply formulas (13) and (14) with λs1 = 0.49 and λs2 = 0.03 which yields the original variances of 0.41 and 0.11 for the level and curvature factor respectively. Consequently, PCA results can be used indirectly to draw conclusions about the

• riginal system.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 29 / 34

PCA on the Correlation Matrix

Finally, consider the following general bisymmetric correlation matrix: A =   1.0 b c b 1.0 b c b 1.0   (15)

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 30 / 34

PCA on the Correlation Matrix

Finally, consider the following general bisymmetric correlation matrix: A =   1.0 b c b 1.0 b c b 1.0   (15) The following can be stated Lemma Let λss be the eigenvalue corresponding to the skew-symmetric eigenvector of a bisymmetric 3 × 3 correlation matrix and λs1, λs2 the eigenvalues corresponding to the symmetric eigenvectors. Then we have λss ∈ [0, 2], (16) λs1 ∈ [1, 3], (17) λs2 ∈ [0, 1]. (18)

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 30 / 34

PCA on the Correlation Matrix

The limit results corresponding to the ones shown for the covariance matrix case are as follows lim

V (bs)→∞ vss =

 

1 √ 2

− 1

√ 2

  and lim

V (bs)→∞

λss 3 = 2 3.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 31 / 34

PCA on the Correlation Matrix

The limit results corresponding to the ones shown for the covariance matrix case are as follows lim

V (bs)→∞ vss =

 

1 √ 2

− 1

√ 2

  and lim

V (bs)→∞

λss 3 = 2 3. For the second symmetric factor we observe lim

V (bc)→∞ vs2 =

  1   . lim

V (bc)→∞

λs2 3 = 1 3.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 31 / 34

PCA on the Correlation Matrix

The analysis of the first symmetric eigenvector yields lim

V (bl)→∞ vs1 =

  

1 √ 3 1 √ 3 1 √ 3

   , lim

V (bc)→∞ vs1 =

 

1 √ 2 1 √ 2

  , lim

V (bs)→∞ vs1 =

  1   .

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 32 / 34

PCA on the Correlation Matrix

The analysis of the first symmetric eigenvector yields lim

V (bl)→∞ vs1 =

  

1 √ 3 1 √ 3 1 √ 3

   , lim

V (bc)→∞ vs1 =

 

1 √ 2 1 √ 2

  , lim

V (bs)→∞ vs1 =

  1   . with lim

V (bl)→∞

λs1 3 = 1, lim

V (bc)→∞

λs1 3 = 2 3, lim

V (bs)→∞

λs1 3 = 1 3.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 32 / 34

Conclusion

We have shown for the case of a bisymmetric covariance/correlation matrix that the structure of eigenvectors is a priori known and that the eigenvectors have the same shape as the empirically observed level, skew, curvature loadings vectors. We show that even if a random system exists, but the covariance matrix is bisymmetric, PCA will indicate the existence of factors which could be interpreted as skew and curvature. This can cause interpretation problems in the analysis of PCA results, when conclusions about the original system are derived.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 33 / 34

Conclusion

We have shown for the case of a bisymmetric covariance/correlation matrix that the structure of eigenvectors is a priori known and that the eigenvectors have the same shape as the empirically observed level, skew, curvature loadings vectors. We show that even if a random system exists, but the covariance matrix is bisymmetric, PCA will indicate the existence of factors which could be interpreted as skew and curvature. This can cause interpretation problems in the analysis of PCA results, when conclusions about the original system are derived. We also find that when a system exists where level, skew and curvature is exogenously given, PCA may not correctly recover these effects. It should be pointed out that not all financial market data displays the bisymmetric property. However, the absence of bisymmetry does not necessarily preclude the potential for PCA interpretation problems. Lord and Pelsser (2007) point this out for other covariance matrix structures.

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 33 / 34

Conclusion

Thank you!

Robert Tompkins, Dimitri Reiswich PCA Interpretation Problems July, 16th 2010 34 / 34

References

Alexander, C., “Common correlation and calibrating the lognormal forward rate model,” Wilmott Magazine, 2003. Alexander, Carol, “Principal Component Analysis of Volatility Smiles and Skews,” SSRN eLibrary, 2001. Cantoni, A. and P. Butler, “Eigenvalues and eigenvectors of symmetric centrosymmetric matrices,” Linear Algebra and its Applications, 1976, 13 (3), 275–288. Cont, R. and J. da Fonseca, “Dynamics of implied volatility surfaces,” Quantitative Finance, 2002, 2 (1), 45–60. Daglish, T., J. Hull, and W. Suo, “Volatility surfaces: theory, rules of thumb, and empirical evidence,” Quantitative Finance, 2007, 7 (5), 507–524. Fengler, M.R., W.K. H¨ ardle, and C. Villa, “The Dynamics of Implied Volatilities: A Common Principal Components Approach,” Review of Derivatives Research, 2003, 6 (3), 179–202. Gantmacher, FR and MG Krein, Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme, Akad 1960. Lord, R. and A.A.J. Pelsser, “Level-Slope-Curvature - Fact or Artefact?,” Applied Mathematical Finance, 2007, 14 (2), 105–130. Meyer, C.D., Matrix Analysis and Applied Linear Algebra, Society for Industrial Mathematics, 2000. P´ erignon, C., D.R. Smith, and C. Villa, “Why common factors in international bond returns are not so common,” Journal of International Money and Finance, 2007, 26 (2), 284–304.

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Conclusion

Schmidt, Peter, Wolfgang K. Haerdle, and Mathias Fengler, “Common Factors Governing VDAX Movements and the Maximum Loss,” Financial Markets and Portfolio Management, 2002, 16 (1), 16–29. Skiadopoulos, G., S. Hodges, and L. Clewlow, “The Dynamics of the S&P 500 Implied Volatility Surface,” Review of Derivatives Research, 2000, 3 (3), 263–282. Zhu, Y. and M. Avellaneda, “An E-ARCH model for the term structure of implied volatility of FX options,” Applied Mathematical Finance, 1997, 4 (2), 81–100.

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