1. Approximation of eigenvalues of spot
cross volatility matrix
with a view towards principal component analysis
Nien-Lin Liu
joint work with Hoang-Long Ngo
The Research Organization of Science and Technology
Ritsumeikan University
Jun. 5th, 2012

2. Introduction
• Interest rate risk management is an important problem in
mathematical finance.
• To decompose the interest rate risks into a few component,
we need to study factor analysis.
• We are interested in the number of eigenvalues of volatility
matrix.
• Principal component analysis (PCA) is a method to analyse
the factors of the term structure of interest rates, commonly
used both in practice and in academics.

3. Introduction
• It is a well known result that three factors are sufficient to
explain most of the spot rate variability:
Figure: The eigenvector of first three factors of Japanese zero rates

4. Introduction
Figure: The eigenvector of first three factors of American zero rates
• We see that the shapes of these factors are of the level
(parallel shift), slope (twist) and curvature (butterfly move).

5. Introduction
• Nevertheless, the empirical results of Liu(2010) show that the
number of factors for the forward rates is much greater than
generally believed:
Table: the proportion of contributions of principle component

6. Introduction
• We introduce another method based on Fourier series, which
is proposed by Malliavin and Mancino(2002, 2009).
• The results reconfirm the observation of Liu:
1
0.5
0 100 200 300 400 500 600 700 800
Percentage of variance explained by the first eigenvalue
1
0.5
0 100 200 300 400 500 600 700 800
Percentage of variance explained by the first two eigenvalues
1
0.8
0 100 200 300 400 500 600 700 800
Percentage of variance explained by the first three eigenvalues
Figure: Percentage of variance explained by the first three eigenvalues as
a function of time for Japanese forward rate.

7. Introduction
1
0.5
0 100 200 300 400 500 600 700 800
Percentage of variance explained by the first eigenvalue
1
0.8
0.6
0 100 200 300 400 500 600 700 800
Percentage of variance explained by the first two eigenvalues
0.9
0.8
0.7
0 100 200 300 400 500 600 700 800
Percentage of variance explained by the first three eigenvalues
Figure: Percentage of variance explained by the first three eigenvalues as
a function of time for American forward rate.
• Three eigenvalues only describe from 70% to 90% for
Japanese forward rate and American forward rate.

8. Introduction
• In the course, we found a problem in MM method that the
estimator of the volatility matrix is not non-negative definite
in general. Therefore some of its eigenvalues may be negative,
which is not expected in practice.
• We alternatively propose an estimation scheme based on the
Quadratic variation method advanced by one of the authors.

9. Numerical Study: Heston Model
• We observe the mean square pathwise errors MSE and mSE
defined as follows:
• ˇ
Suppose that for each k = 0, . . . , N0 , Σ(tk ) is an estimator of
matrix Σ(tk ).
• ˇ ˇ
We denote λ1 (tk ) and λd (tk ) the maximum and minimum
ˇ
eigenvalues of Σ(tk ).
• We also denote λ1 (tk ) and λd (tk ) maximum and minimum
eigenvalues of Σ(tk ).
• Then they are defined as
N0
ˇ 1 ˇ
mSE (Σ, Σ) = |λd (tk ) − λd (tk )|2 ,
N0
k=1
and
N0
ˇ 1 ˇ
MSE (Σ, Σ) = |λ1 (tk ) − λ1 (tk )|2 .
N0
k=1

10. Numerical Study: Heston Model
The means of mSE and MSE of each method are showed in Table
10.
N0 QV FS FS1 FS2 FS3 FS4
MSE 102 20 104 24 23 33 63
mSE 0 7.6 0 0.028 0.056 3.2
MSE 103 6 92 15 9 13 31
mSE 0 7.8 0 0.001 0.08 1
MSE 104 2.1 89 9.2 3.8 5.7 18
mSE 0 7.6 0 0 0.008 0.26
Table: Means of MSE and mSE (×10−4 )

11. Numerical Study: Maximum eigenvalue (N0 = 103 )
0.3 0.3
True
QV
0 0
0 T 0 T
0.3 0.3
FS1
FS
0 0
0 T 0 T
0.3 0.3
FS2
0 FS3 0
0 T 0 T
0.3
FS4
0
0 T

12. Numerical Study: Minimum eigenvalue(N0 = 103 )
−16 −16
x 10 x 10
1 1
True
QV
0 0
−1 −1
0 T 0 −16 T
x 10
0 2
FS1
FS
0
−0.1 −2
0 T 0 T
0 0
FS2
FS3
−0.003 −0.03
0 T 0 T
0
FS4
−0.1
0 T

13. Numerical Study: Maximum eigenvalue(N0 = 104 )
0.3 0.3
True
QV
0 0
0 T 0 T
0.3 0.3
FS1
FS
0 0
0 T 0 T
0.3 0.3
FS2
FS3
0 0
0.30 T 0 T
FS4
0
0 T

14. Numerical Study: Minimum eigenvalue(N0 = 104 )
−17 −16
x 10 x 10
5 1
True
QV
0 0
−5 −1
0 T 0 −16 T
x 10
0.2 2
FS1
FS
0 0
−0.2 −2
0 −16 T 0 −3 T
x 10 x 10
2 2
FS3
FS2
0 0
−2 −2
0 T 0 T
0.05
FS4
0
−0.05
0 T

15. Numerical Study: Euro swap rates and Euribor rates
• We reconsider the empirical study in Malliavin, Mancino and
Recchioni(2007):
-3 -4
x 10 x 10
1
0.5 1
0 0
0 200 400 600 800 0 200 400 600 800
First eigenvalue Second eigenvalue
-5 -5
x 10 x 10
3 0
2
-10
1
0 -20
0 200 400 600 800 0 200 400 600 800
Third eigenvalue 13th eigenvalue
Figure: Estimated eigenvalues using Fourier series method (dotted line)
and Quadratic variation method (solid line)

16. Numerical Study: Euro swap rates and Euribor rates
1
0.5
0 100 200 300 400 500 600 700 800
Percentage of variance explained by the first eigenvalue
1
0.9
0.8
0 100 200 300 400 500 600 700 800
Percentage of variance explained by the first two eigenvalues
1
0.9
0 100 200 300 400 500 600 700 800
Percentage of variance explained by the first three eigenvalues
Figure: Percentage of variance explained by the first three eigenvalues:
Fourier series method (dotted line) and Quadratic variation method (solid
line)

17. Numerical Study: American spot rate
• We use Quadratic variation method to analyze the structure
of American spot interest rates
60
1st eigenvalue
2nd eigenvalue
50
3rd eigenvalue
40
30
20
10
0
0 T
Figure: Estimated values of the first three eigenvalues

18. Numerical Study: American spot rate
1
0.98
0.96
0.94
0.92
0.9
1st
0.88 2nd
3rd
0.86
10th
0.84
0 T
Figure: Percentage of variance explained by the first one, two, three and
ten eigenvectors

19. Summary
1 We studied two methods to estimate the eigenvalues of spot
cross volatility matrix.
2 The estimated covariance matrix by using Fourier series
method is not non-negative definite hence it contains negative
eigenvalues.
3 The empirical studies show that Quadratic variation method is
easier to implement, much faster and able to avoid negative
eigenvalue problem.
4 Quadratic variation is also applicable to diffusion processes
with jumps while Fourier series method is not suitable.

20. References
1 P. Malliavin and M.E. Mancino: Fourier series method for
measurement of multivariate volatilities, Finance Stoch., 6,
pp.49–61, 2002.
2 P. Malliavin, M.E. Mancino and M.C. Recchioni: A non-parametric
calibration of the HJM geometry: an application of Itˆ calculus to
o
financial statistics, Japan. J. Math., 2, pp.55–77, 2007.
3 P. Malliavin and M.E. Mancino: A Fourier transform method for
nonparametric estimation of multivariate volatility, Ann. Statist.,
37(4), pp.1983–2010, 2009.
4 H.L. Ngo and S. Ogawa: A central limit theorem for the functional
estimation of the spot volatility, Monte Carlo Methods Appl., 15,
pp.353–380, 2009.
5 S. Ogawa and K. Wakayama: On a real-time scheme for the
estimation of volatility, Monte Carlo Methods Appl., 13, pp.99–116,
2007.

21. Thank you for listening.