• First observation of stock default and unlisted in the Vietnamese stock market • The question of stock trading after credit shock • The continuing low liquidity of the Vietnamese stock
Trang 2MỤC LỤC
TÓM TẮT (Tối đa một trang A4)
ABSTRACT (Tối đa một trang A4)
DANH MỤC CÁC KÝ HIỆU, CÁC CHỮ VIẾT TẮT
Chương KẾT LUẬN VÀ KIẾN NGHỊ
TÀI LIỆU THAM KHẢO
Phụ lục b: Minh chứng đăng ký sở hữu trí tuệ, sản phẩm ứng dụng (gồm báo cáo về kết quả thử nghiệm hoặc ứng dụng các sản phẩm KH&CN của đề tài/dự án (thiết bị, công nghệ, quy trình công nghệ, ), ý kiến nhận xét của người sử dụng và các tài liệu về đo đạc, kiểm tra, đánh giá thử nghiệm các sản phẩm KH&CN của đề tài/dự án của các Cơ quan đo lường thử nghiệm có thẩm quyền (của các Phòng thí nghiệm chuyên ngành, các Trung tâm kỹ thuật, Trung tâm đo lường, Cơ quan giám định kỹ thuật, ); bản vẽ thiết kế (đối với sản phẩm là thiết bị), các số liệu điều tra, khảo sát gốc, sổ nhật ký hoặc sổ số liệu gốc của đề tài/dự án
Phụ lục c: Minh chứng kết quả đào tạo
3 Phụ lục quản lý gồm:
− Xác nhận quyết toán tài chính của cơ quan chủ trì;
− Phiếu gia hạn, điều chỉnh nội dung, điều chỉnh hạng mục kinh phí (nếu có);
− Biên bản đánh giá, kiểm tra giữa kỳ (nếu có);
− Quyết định phê duyệt kinh phí;
− Hợp đồng;
− Thuyết minh đề cương đã được phê duyệt
Trang 3Quantitative Modelling of Liquidity and Credit Risk in
Post-Financial Crisis
***Final report***
QCF group December 2013
Contents
2.1 Fundamental of Credit risk 5
2.2 Indifference pricing of defautable claims 6
2.3 Option price in function of jump’s characteristics 9
2.4 Credit contagion phenomena analysis using Random matrix theory 10
2.5 Distribution and dynamics of correlation coefficient 12
2.6 Eigenvalue spectrum 14
2.7 One-factor model 16
3 Liquidity risk 19 3.1 Measuring liquidity risk at the microstructure level 20
3.2 Measuring liquidation cost with different market impact model 22
3.2.1 Transient market impact model and optimal problem 23
3.2.2 Numerical results 24
3.3 Study of Vietnamese stock market liquidity 26
3.3.1 Liquidity measures 26
3.3.2 Correlation and dependency 28
3.3.3 Seasonality of liquidity measures 29
Trang 4E Optimal Execution with Transient Impact 34
Trang 51 Summary
4 years after the financial crisis, risk management system worldwide has been considerablemodified in order to adapt to the increasing financial risk environment, including systematicrisk (market risk), credit risk, liquidity risk and operational risk However, major problemsstill exist and have not been scientifically analyzed In particular, failure of most credit riskmodels which is the root of collapse many large financial firms resulting a global economicdownturn, raise the necessity of new quantitative research The popular correlation hy-pothesis between credit-linked event is no more relevant As we can see, default contagionbetween stock and other complex product is far more complicated than it had been mod-eled Secondly, liquidity risk is also a major concern during the crisis time especially for asmall market like the Vietnamese stock market Liquidity constraint is always an obstacle
in any investment decision Our research project has objective of improving the existingquantitative models in credit and liquidity risk in the framework of the post-financial crisischanges Those researches also have potential application in the local market of Vietnam.Since the beginning of the project, some event related to our research in the local markethave happened:
• The nearly default of Vinashin and its consequence of credit downgrading on thecountry economy, banks and corporate
• First observation of stock default and unlisted in the Vietnamese stock market
• The question of stock trading after credit shock
• The continuing low liquidity of the Vietnamese stock market
Such events raise the necessity of high quality research and development in the area of riskmanagement in Vietnam Our project will focus on the topic of credit and liquidity risksusing quantitative method and numerical analysis which is one of the major strength ofJohn von Neumann team This report summarize our work over the past 2 year on thesubjects aboves which have resulted in the following publications:
Journal with review:
- Q Nguyen, One-factor model for the cross-correlation matrix in the Vietnamese stockmarket, Physica A, Volume 392, Issue 13, 1 July 2013, Pages 29152923
- M M Ngo, Q Nguyen, H Pham, Indifference pricing with counterparty risk, submitted
to Applied Mathematics and Computation
- Pietro Fodra, Huyn Pham, Semi Markov model for market microstructure, accepted toApplied Mathematical Finance
- N M Dang, Optimal Execution with Transient Impact, submitted to Applied Mathematicsand Computation
- N M Dang, Yin Chen, Dynamic Execution of VWAP Orders with Short-Term tions, submitted to Applied Mathematics and Computation
Trang 6Predic-International Conference:
- Q Nguyen, Financial market risk analysis through cross-correlations eigenvector ponents distribution, Financial Networks and Systemic Risk (FNET), Kyoto University,Kyoto, Japan, July 2013
com N M Dang, Optimal Execution with Transient Impact, NUScom UTokyo Workshop on Quancom titative Finance, September 2013, Singapore
Quan-In the next two sections we present our work on the two main topics: credit risk andliquidity risk
Trang 72 Credit risk
Credit risk is refers to the risk associated to sudden change of credit behavior of a counterparty such as: changes in credit quality (downgrades or upgrades in credit ratings), vari-ations of credit spreads, and the default event Managing of credit risk is traditionally adifficult task, and it appeared that most models of credit risk have failed to measure thecredit risk in the context of the global financial crisis 2008-2010 One of the most importantlack of those models is the inflexibility in the treating of correlation of the factors that drivecredit risk Or, in hindsight, these approaches have severely underestimated contagion phe-nomena across sectors and the resulting systemic risk The researches on the contagionphenomena have been carried out, particularly for the credit intensities and credit deriva-tives in the literature (see for example Jarrow & Yu (2001), Giesecke & Weber (2003), Frey
& Backhaus (2008), Brigo and al (2008), Crepey, Jeanblanc & Zagrari (2008)) However,the wide extent of the financial crisis and all these lessons call for a more thorough analysis
of these risks in all aspects of financial markets
In this first half of the report, we present some fundamental research on the credit riskthen present our working result We firstly continue to analyze the problem of pricing
a credit risky asset with different method: indifference pricing, and secondely, propose amodel for identifying the credit contagion phenomena:
The main result concerning credit risk are:
• We price a defautable claim (an option) of an asset with the presence of counterpartyrisks using indifference method (which is invented by John von Neumann in the1950s) We also combine with novel techniques recently developed by one member ofour group, Prof Huyen Pham, the decomposition of the pricing process into 2 classicalprocess and the solvation of each problem by the Backward Stochastic DifferentialEquation (BSDE) approach
• We characterize the price behavior of the claim in funciton of each parameter, such
as default intensity, loss’s size, etc and obtain
• For large credit portfolios (containing in general more than one hundred names), how
to model the contagion phenomena and to describe the counterparty risks becomesmuch more complicated because of the tractability difficulties It will be a challengingproject to generalize our work to such a large scale
2.1 Fundamental of Credit risk
Fundamental of credit risk modelling (see Appendix A for more details): we analyzedtraditional models and studied why they have been failed We obtained important result
on the foundation of density approach of credit modeling Several works are concentrated
on default risks related to the underlying firm of a credit derivative, the main approachesbeing the structural and reduced-form (intensity) ones Once the underlying name defaults,the credit derivatives ceases to exist For this reason, most works concern only what goes onuntil the default time and neglect somehow the default impact afterwards Our approach
Trang 8explore the behavior of assets price event after the defaul time and find a global optimalcontrol strategy (see next sub-section for more details) We also study the behavior of thecredit default swap or CDS (see Appendix B for more details), the most tradable creditderivative product.
2.2 Indifference pricing of defautable claims
We present briefly the method and result of our work For more detail, reader could seethe Appendix ??
Option pricing was one of the most important application of mathematics into financialmarket.This problem has been solved completely by Black and Scholes in the 1970s byassuming that the underlying asset follow a log-normal process:
where Dt= Iτ ≤t, 0 ≤ t ≤ T is the jump process and τ is the
We also decompose the asset price process into two part: before and after default:
St= SF
tIτ >t+ Std(τ )Iτ ≤t, t ∈ [0, T ]where SFis F-adapted process that represented the discount price process in the default-free market, governed by:
Trang 9With the presence of credit default, the market become in complete in the sense thatthere exist no perfect hedging strategy for a derivative asset and therefore one can not pricesuch instrument Instead of pricing in a risk-neutreal probability space, the infifferencemethod solve this problem in a completely difference angle: the seller of the asset withpresence of credit risk may invest in the underlying asset and money At any time, hewill try to balance his portfolio in order to achieve the highest utility (defined by a convexutility function) at the maturity of the product Suppose that at time zero, he must decide
to sell or not sell the product The indifference method suggest that by always using anoptimal strategy, he would be able to obtain an indifferent final wealth in both cases Theequality of the two final wealth will determine the product’s price
Mathematically, we have the following indifference pricing equation:
where
in which V (x) denotes the optimal wealth if he does not sell the option and start toinvest (optimally) with x dollar while which VB(x + P r) denotes the optimal wealth if hedoes sell the option for price P r and start to invest with x + P r Thus in order to solve 2.3
we have to solve two optimization problems
Our main contribution in this part is to applying the Minimal Entropy method in tosolving 2.3 in one optimization step This method suggested by [32, proposition 3] andusing the Exponential utility function of:
and with the deterministic conditional density (Remark ??)
Trang 10Finally, our pricing problem is reduced to solving an unique optimal control problem of
supπ∈AEP∗[U (XTπ− B)|F0]Where A and also AF, Ad(θ) are sets of admissible trading strategies (wealth process X)defined in [27]
This approach is more intuitive and closer to the practical behavior than the traditionalpricing method which based on a rigorous mathematical treatment
In order to derive the global price of the claim, we then apply the BSDE method propose
by Jiao and Pham ([27]) to the above decomposition framework
Numerical: We calculate the price of an option susceptible to jump with parameters:Current price=112, strike price=110, interest rate (r)=0.03, volatility(σF)=0.2, T=1, in-tensity (λF)=0.1, Constant absolute risk aversion p = 3
The reference price : 14.0750 ( call option)
Figure 1: ”Indifference price 1” : The volatility is the same between before and after default.
”Indifference price 2” : The relation between before default volatility and after default volatility is
Trang 112.3 Option price in function of jump’s characteristics
In this section we characterize the behavior of option price in function of jump’s parameters.This study is of great important for practitioners/traders because they can estimate the
“prime” of their products regards the potential jump of the underlying
• Option price in function of the jumps’s intensity
Figure 2: ”Indifference price 1” : The volatility is the same between before and after default.
”Indifference price 2” : The relation between before default volatility and after default volatility is
σd(θ) = σF(2 −Tθ) ”Indifference price 3” : The relation between before default volatility and after default volatility is σ F = 0.2, σdt(θ) = σF + 0.1e−3(t−θ) And black-dot is the reference price that was compute by Merton-jump model.
In this figure 2 we found that option price increase almost linearly with respect tothe intensity of the jump as expected
• Option price in function of the trader’s risk aversion
Trang 12Figure 3: We consider the case the volatility is the same between before and after default ence price 1” : Constant absolute risk aversion p = 1 ”Indifference price 2” : Constant absolute risk aversion p = 4.5 ”Indifference price 3” : Constant absolute risk aversion p = 8.5 And black-dot is the reference price that was compute by Merton-jump model.
”Indiffer-In this figure 3 we found that option price increase with respect to the risk aversion
of the seller as expected
2.4 Credit contagion phenomena analysis using Random matrix theory
On the third issue, the correlation structure is often proposed on the fundamental assetprocesses (structural approach) or on the credit intensity processes (intensity approach).There are furthermore some simplified but more tractable models, which are popular amongthe practitioners, such as the factor models or the ”top down” models for large credit port-folios However, recent failure of such model in the context of the crisis also demonstratesthat the correlations of the factors that drive credit risk in these models have broken down.Not surprisingly, most models of credit risk have then provided little guidance for the man-agement of credit risk A novel contagion risk model is therefore necessary to improve themodeling of credit risk To this end, researchers have begun to change or innovate keyassumptions and elements of the modeling of credit risk They have taken steps to look
at changes or innovations to both structural models, which consider that a default occurswhenever the value of the assets underlying the liabilities falls below some threshold, andreduced-form models, which depend on a random default time whose distribution depends
Trang 13on economic variables They have stressed the need to include jump terms based on son distributions in the modeling of credit risk, while incorporating a robust assumption offiltration, or the observed developments of the factors that drive credit risk Emphasis alsohas been given to the benefits of including variable recovery rates in the modeling of creditrisks.
Pois-Cross-correlation of financial time series is of great important in managing investmentrisk and in constructing an efficient portfolio [35] When the number of studying timeseries is small and if non-stationary exists between time series, which is usually the case infinance, one may use the detrended cross-correlation approach [44, 56, 45, 25] In contrast,when one studies the cross-correlation matrix of the whole stock market which composes ofseveral hundreds of individual stocks, it becomes a high-dimensional and complex systemwhich is not easy to study Recently, some method developed in statistical physics havebeen employed [?, ?], in particular the random matrix theory (RMT) [42, ?] These authorscompare the eigenvalue spectrum of the financial correlation matrix with that of a randommatrix and found these important remarks:
• Most of eigenvalue λ of the financial correlation matrix fall within a tight rangepredicted by random matrix theory λ− < λ < λ+, where the RMT bounds λ− and
λ+ are positive and close to 1 (explicit formula is given in the next section) Inconsequence, it was argued that most of the structure of cross-correlation matrix aredue to noise This clustered group of eigenvalue is called the bulk
• There is one largest eigenvalue λmaxwhich is ×10 − 30 higher than the maximumexpected value predicted by RMT λ+ and its corresponding eigenvector is assigned
to the market portfolio
• There are several other eigenvalue slightly greater than λ+ which reflect the sectorbehavior
• There are a number of eigenvalues smaller than λ−, the lower bound of eigenvaluepredicted by RMT, correspond to a specifically high correlated pair of stocks.[?]Other studies on developed stock market (US, European, Japan) showed accordancewith these findings [42, ?, 19, 43, 51, ?, 9, 10, 52, 55, ?] In addition, a few of studies onemerging markets [?, ?, 40] showed some slight different properties compare to the abovedeveloped markets, such as:
1 The average value of non-diagonal elements of C is higher and fluctuates more namically [?, ?, 40]
dy-2 The largest eigenvalue λmax is significantly higher (×50) than λ+ [?, 40]
3 Excepting the largest eigenvalue, there are fewer number eigenvalue greater than
λ+[?, ?, 40]
4 In contrast, there are a large proportion of eigenvalues smaller than λ− In quence, the total ratio of eigenvalues which fall within RMT bounds is far smaller[?,40]
Trang 14conse-The point 1 is quite evident because in emerging markets, stocks move more in dem due to the low-diversification level of companies and the important impact of commonmacro economic factors In consequence, the average correlation coefficient in emergingmarkets is usually higher than those in developed markets Furthermore, as the marketparticipants are less professional, they create more volatility and higher fluctuation of cor-relation between stocks.
tan-In investigating the largest eigenvalue time evolution, Kulkarni et al [?] found positivecorrelation between the largest eigenvalue and the market volatility Because it is commonlyknown that correlation tends to increase during volatile period, the correlation between thelargest eigenvalue and the correlation level is implicitly mentioned In subsequence section,
we will show numerically that the largest eigenvalue is proportioned to the average value ofcorrelation matrix elements Together with the point 1, the largest eigenvalue in emergingmarkets turns out to be higher than that in developed markets, or the point 2 In [?],the author mentioned a temporal opposite movement between the largest eigenvalue andthe bulk of small eigenvalues due to the propulsion effect: as the sum of all eigenvalues isalways constant, a change in value of the largest eigenvalue must be compensated by anopposite change of the others, or the shift of the bulk This remark explained the point 4because a high value of the largest eigenvalue in emerging markets will result in a shift ofthe eigenvalue bulk out of the lower RMT bound λ−
A part from these deviations from the conventional RMT, there are more subtle butsystematic differences also in the bulk of eigenspectrum as discussed in [29, 11] Thesedifferences, masked by noise, may contain useful information on the cross-correlation struc-ture In addition, RMT is also used in order to study time-lag cross correlations in multipletime series of finance [53] as well as biology and atmospheric geophysics [46] These au-thors found long-range power-law cross correlations in the absolute values of returns thatquantify risk and find that they decay much more slowly than cross correlations betweenthe returns
In this paper, we investigate the statistical properties of cross-correlation matrix of
N = 90 stocks traded in the Ho Chi Minh city stock exchange (HCMCSE) from 1 January
2007 to 2 May 2012 using RMT method Our studies are accorded to the previous findings
in both developed and emerging markets In addition, we try to quantify the magnitude ofthe largest eigenvalue by analyzing its dependencies on the stock number and the averagecross-correlation coefficient We employ the one-factor model similar to [?] in order todemonstrate these behaviors
Important result have been found as follow:
2.5 Distribution and dynamics of correlation coefficient
In this section, we analyze the statistical properties of the elements of cross-correlationmatrix C Figure 4(a) shows the distribution P (Cij) of 90 stocks over the whole analyzingperiod from 1 January 2007 to 2 May 2012 Other descriptive statistics are presented intable 1 We found that the average value hCiji of 0.3663 is relatively large in comparison
to others studies, suggesting that stocks in Vietnamese market are strongly correlated.Furthermore, the whole period correlation matrix elements are all positive, suggesting that
Trang 15Figure 4: (a) Probability density of Cij for the whole period 2007-2012 (b) Temporalevolution of cross-correlation distribution using a sliding window of 250 days with five dayslag time One notices strong fluctuation of the average value hCiji as well as the distributionshape (c) Summary statistics of the distributions in (b) plotted in function of times
the diversification level within the Vietnamese stock market is quite low In consequence,its systematic risk is rather high We also found that over the whole period the distribution
is relatively closed to normal, with a small positive skewness of 0.13 (there is a few number
of high elements higher than 0.6) and kurtosis of 2.7 (3.0 is considered as normal) Thesefindings support our above assumptions in the one-factor model
In figure 4(b) we plot the time-varying distribution of cross-correlation matrices using asliding window of 250 days with five days lag time We found that this distribution changesconsiderably over time Figure 4(c) shows the temporal dynamic of the average value hCijiand others descriptive statistics of elements of C using the same slide windows The averagecorrelation coefficients peaked at the crisis period of end 2008, then gradually decreases.Until August 2010, its distribution standard deviation is relatively low, its skewness isnegative and kurtosis is closed to the corresponding normal value After August 2010, thedistribution became positively skew and its shape deviated from that of normal distribution
Table 1: Statistics for cross-correlation of Vietnamese stock between Jan 2007 - May 2012
Trang 16Figure 5: (a) Probability density of λiin comparison with RMT density (the red solid line).Remark that the largest eigenvalue λmax is 21 times higher than λ+ Insert graph show azoom into the bulk We found 2 other eigenvalues larger than λ+ The bulk is significantlydeviated from RMT, with only half fall within the RMT bounds (b) Inverse participantratio of eigenvalues (c) The time-varying comparision of hCiji, the largest eigenvalue, per-centage of deviating eigenvalue and the average of 80 small eigenvalues using the slidingwindows of 250 days with 5 days lag time.
2.6 Eigenvalue spectrum
In this section, we decompose the cross-correlation matrix and calculate its eigenvaluesand eigenvectors The eigenvalue spectrum is showed in 5(a) together with the spectrumpredicted by RMT theory We find similar characteristics as most other studies: the largesteigenvalue λmax = 34.4, while N × hCiji = 90 × 0.3664 = 32.9 The similarity between
λmax and N × hCiji again support arguments in our simple one-factor model In our study,the theoretical λ− and λ+ are 0.546 and 1.589, respectively In the insert graph, we foundthat apart from the largest eigenvalue, there are two other eigenvalues which are beyond
to other studies in developed markets [42, ?], but in line with those in emerging markets[?] These eigenvalues are assigned to the sector group in the correlation matrix [?] Thatmeans stocks in emerging markets are less sector-specific than in developed markets
In addition, we found that only about half of small eigenvalue bulk lies between RMTbounds, considerably lower than results in developed markets [?] We account this fact
to the repulsion effect between the largest eigenvalue and the small eigenvalues [?] as thesum of all eigenvalues remains constant: as λ+ is higher in emerging markets as discuss
Trang 17Nb Groups Market L N Q hCiji λmax λ+
Table 2: Data summary and largest eigenvalue statistics of some previous studies; (˜):estimated from publication; (*): calculated using formula ?? Remark that in studies 1and 5, N are closed but as hCiji is 2× higher in 5, λmax is also 2× higher in 5 In 4 and 8,
hCiji are similar, but N is 2.3× higher in 4 resulting in λmax of 2.4× higher
previously, the small eigenvalues bulk is repulse further to the left resulting in a high ratio
of RMT off-limit We demonstrate this effect in figure 5(c) where the movements of thelargest eigenvalue and the average of 80 smallest eigenvalues are opposite with correlationfactor of -0.97 Combined with the previous findings, we deduce that the ratio of deviatingeigenvalue from RMT bounds, mainly due to the shift of small eigenvalues bulk beyond λ−,
is caused principally by the repulsion effect when there exists a positive average correlation
in C The correlation between the ratio of deviating eigenvalue (which are considered noise eigenvalue in some studies) and the average correlation coefficient is 0.97 as showed
λmax to λ+ and there is no magnification effect of the stock number N in λ+ (though itstill depends on N through Q, but to a certain limit) In reality when this assumption is
in general no longer true, i.e hCiji is different from zero, the value of λmax will linearlydepend on the stock number N We examined this relation by plotting the time-varyingevolution of λmax and the average of the cross-correlation matrix using a sliding window of
250 days with five days lag time in figure 5(c) We found almost identical dynamic betweenthese two quantities as found in [?, ?], with correlation of 0.99
In figure 6, we show the component distribution of some eigenvectors correspondingto: the largest eigenvalue λ1, the second largest deviating eigenvalue λ2 (larger than λ+),one eigenvalue of RMT bulk λ45 and the smallest eigenvalue λ90 We find similar result
to other studies: the components of the eigenvector correspond to the largest eigenvalueare almost equal This vector is argued to represent the market portfolio In contrast, the
Trang 18eigenvector corresponding to the smallest eigenvector λ90has only 3 significant components.
We identified that these 3 stocks [?] are the most active stocks during the studying periodand has the highest cross-correlation coefficients in the market Portfolios constructed fromthese smallest eigenvectors will have smallest volatility and therefore, could be used todetermine statistical arbitrage strategy The component spectrum of λ2 and λ45, however,
do not reveal any remarkable characteristics We further investigate the eigenvector bycalculating the Inversed participation ratio (IPR) defined as
in them, as we have seen in the previous paragraph These small eigenvalues are the results
of some particular high correlated pair or triple stocks[?] It is interesting to note that the
Ik is higher than that of the average bulk at the high and low ends of the small eigenvaluebulk (see also [?]), suggesting that there are some particular correlation structure of smallgroup of stocks yet to be identified It is remarkable that the IPR at the high end of thebulk is higher than the average bulk while their corresponding eigenvalues are around 1and are still inside the range predicted by RMT One may suspect that the eigenvalues fallwithin the RMT bound do not need to be pure noise [?]
2.7 One-factor model
In this section, we present simulation result of the one factor model for the cross-correlationmatrix described earlier Our model takes N , L and ρ0 as input, and generates the cross-correlation matrix ˜C and its eigenvalue spectrum as output Firstly, we simulate the empir-ical matrix ˜C by taking the same number of N , L and let ρ0 equal to hCiji We present theeigenvalue spectrum of ˜C in 7 (a) and the inverse participation ratio of random eigenvalue
in 7 (b) As N , L are the same as in Section 2.6, the RMT theoretical spectrum is thesame and is plotted altogether Figure 8 displays components of some eigenvectors Theeigenvalue statistics of simulated and empirical data are showed in table 3
Table 3: Eigenvalue statistics of real and simulated data
Trang 19Figure 6: Components of eigenvectors u1, u2, u45and u90 In u1the components of all stocksare positive and relatively uniform, this eigenvector characterizes the market portfolio Inthe smallest u90 there are only 3 significant components These three stocks are found tohave strong correlation with each other Other two eigenvectors do not show remarkablecharacteristic.
Figure 7: The largest eigenvalue and the small eigenvalue bulk could be well approximated
to “empirical” data by eigenvalue spectrum (a) and by the inverse participant ratio (b)
Trang 20Figure 8: Components of eigenvectors u1, u2, u45 of random generated data Components
of largest u1are identical and equal to 1/√N , while components of the others are randomlydistributed
We found that the largest eigenvalue of simulated data is closed to that of empiricaldata, suggesting that the high magnitude of λmax, the most important deviation from RMT,could be explained by an unique factor: the positive average correlation between stocks.The high value of stock number N will enhanced this effect but is not the original cause
On the other hand, we found that the small eigenvalue bulk also shifts to the left of theRMT bounds, as a consequence of the repulsion effect by the largest eigenvalue The modelpredicts about 70% of eigenvalues out the RMT low limit λ− The model also explainsthe IPR value of the largest and the bulk eigenvector However, it does not explain theappearance of few eigenvalues higher than the RMT high λ+; the component spectrum ofeigenvector corresponding to the smallest eigenvalue (where there is a few high components
as discussed in 2.6; the high IPR number of some eigenvalues at both ends of the smalleigenvalue bulk as shown in figure 5(b) Addition features are needed to include into themodel in order to demonstrate these behaviors
Finally, we simulate the dependence of λmax on two factors, hCiji and the stock number
N using our model On each simulation, we kept 2 fixed parameters as that of empiricaldata and varied the rest Figure 9 shows the dependence of λmax on hCiji (a) and on N(b) We found that when hCiji varies from 0 to 1, the λmaxgoes from the RMT high bound
λ+ to N The dependence is positive but sublinear and an explicit analytical function is
on our future research plan In contrast, the dependence on N is almost linear This resultsupports our discussion in the previous section
The result of correlation analysis will help us to study the credit contagion phenomenaand propose new model for pricing an derivative on a basket of stock This work is planfor the future project