Kyoto University,
Graduate School of Economics
Research Project Center Discussion Paper Series

An Analysis of CDS Market Liquidity by the Hawkes Process

Masahiko Egami
Yasuyuki Kato
Tomochika Sawaki

Discussion Paper No. E-13-001

Research Project Center
Graduate School of Economics
Kyoto University
Yoshida-Hommachi, Sakyo-ku
Kyoto City, 606-8501, Japan

June 2013

An Analysis of CDS Market Liquidity by the Hawkes
Process
Masahiko Egami

Yasuyuki Kato

Tomochika Sawaki

Abstract
We study the credit default swap (CDS) markets in the U.S. and Japan, focusing on
bid-ask spreads which are closely related to the liquidity of the markets. Since bid-ask
spreads dramatically surged during the ﬁnancial crisis (2008-2009) and the market becamevery illiquid, it is crucially important to investigate how bid-ask spreads ﬂuctuate. In thispaper, not only do we make dynamic analysis of the bid-ask spreads in both countries but
propose a model to predict bid-ask spreads via the self-exciting intensity process (the Hawkes
process).
Key words: CDS contract; liquidity; bid-ask spread; the Hawkes process; self-exciting processes, ﬁnancial crisis, credit risk.
JEL Classification: G10, G17, G21

1Introduction

The credit default swap (CDS) contract is a major single-name credit derivative and serves as the building
block for many other credit derivative instruments. We refer to O’Kane [13] for the deﬁnition of the CDS
contract: A CDS is a bilateral over-the-counter contract whose purpose is to protect one party, theprotection buyer, from the loss from par on a speciﬁed face value of bonds, or loans following the defaultof their issuer. The CDS markets expanded rapidly during the ﬁrst half of the last decade. We havewitnessed, however, a huge pike of bid-ask spreads in the U.S. CDS market during the ﬁnancial crisis
period. It is said that the liquidity evaporated and the market participants had diﬃculty in completing1Electronic copy available at: http://ssrn.com/abstract=2200307

their transactions. See the bid-ask spreads (“BAS” hereafter) and the number of quotes in the U.S. andJapan CDS markets in Figure 1.

Figure 1: Daily averages of BAS and Quote numbers
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the crisis period, BAS increased drastically in both U.S. and Japan. It is obvious that the variability of
BAS is closely related to the liquidity of the CDS markets. In this sense, we believe that it is crucial
to understand in what way the spreads ﬂuctuate, sometimes vastly, and to know mechanism behind it.
Second, in view of Longstaﬀ et al. [10], the CDS premia are largely explained by credit component, so
the information about liquidity of the market is mainly contained in BAS. Of course, however, market
makers often bear underling credit risk during their holding period until they sell the contract. Thus,
if the default probability of the underling loan/bond goes up, they may end up with holding an open
position, so that BAS also contains, to some extent, credit risk information. But again, this increased
credit risk level is promptly reﬂected in the rise of CDS premia, and hence the credit part of BAS seems
to vanish soon or later, as long as there is enough liquidity is provided in the market. Also, it is worth
mentioning that Roll [15] claims in the study of the stock market that BAS are eﬃciently determined bymarket makers through their functioning as liquidity providers to the market. We therefore investigate

2Electronic copy available at: http://ssrn.com/abstract=2200307

Table 1: Comparison of the CDS indices between U.S. and Japan(million $, 2009)
gross notional net notional # contracts
CDX.NA.IG SERIES 9

1,324,894

75,213

22,334

iTraxx Japan SERIES 11

13,032

1,448

1,050

(million $, 2010)

gross notionalJapan

1,116,900

World30,261,000

the CDS markets in the U.S. and Japan and ﬁt some models to explain ﬂuctuations of the BAS.
Studying both the U.S. and Japan markets is beneﬁcial because the degree of market eﬃciency diﬀer.
Table 1 is a comparison of the contracts of the CDS indices, indicating a big diﬀerence in size.
In particular, an observation of time series data of CDS premia reveals the following: Larger and more
frequent price swings have been recorded in the U.S. market. Lower-grade names are more frequently
traded than investment-grade names in the U.S., while trade volumes in higher-grade names are far
greater (than lower-grade) in Japan. Since the Japanese market is in its early period, a full utilizationof the market to trade credit risks may have not been accomplished. It seems instructive to analyze the“developed” and “developing” markets simultaneously since one can understand the process of market
evolution as well as the similarity and diﬀerence of the two markets.
Now let us take a look at Figure 1 again for the time series data of BAS and the numbers of quotes.
First, the number of quotes are far greater in the U.S. market compared to Japan. In both countries, as
we know, there are high peaks in the quote numbers and BAS during the ﬁnancial crisis (2008 ∼ 2009)1 .
The relationship between these two variables needs more investigation: we shall also see the non-crisis
periods in developing a model that predicts BAS.
We shall brieﬂy mention main results of this paper:(1) In Section 1.1, by simple regressions, changes in BAS of U.S. companies’ CDS contracts can be
explained both by their respective bid prices and the economic factors that surround the companies.
This is not the case in Japan, where the CDS market does not fully incorporate general and/or
company-speciﬁc economic factors. The diﬀerence in the market eﬃciency may be attributable tothis contrasting phenomenon.
1

A special mention is necessary for a huge upsurge in the bid-ask spread of Softbank contract: it is due to its

announced merger of the struggling mobile provider, Willcom.

3Electronic copy available at: http://ssrn.com/abstract=2200307

(2) We observe time-series data of bid price and BAS in Section 2.1. By the Grander causality test, we
ﬁnd that the two variables have causality in both directions: Bid price is a cause of BAS changes
and vice versa. Moreover, there exists a “reversal” phenomenon in BAS changes. If one observes
an increase in BAS at some point, then in the next period BAS has a tendency to decline.
(3) Over a long period (Jan 2002 through 2010) there is a large variability in correlation coeﬃcients
between bid prices and BAS (in Section 2.2).
(4) We construct a model to explain BAS movements based on quote numbers. The main purpose ofthis modeling is to predict near-future BAS, which we believe is of practical use. We observe ashort-run surge and fall in the correlation coeﬃcient between the quote numbers and BAS. Morespeciﬁcally, we attempt to model the number of quotes by a self-exciting intensity process (or the
Hawkes process). The term “self-exciting” implies that the intensity of quote arrivals to the market
makers increases as quote numbers themselves increase. We then use the time-changed Brownian
motion to model BAS ﬂuctuations. The new clock here is arrival times of quotes. See (3.1)∼(3.3).
The prediction results shown in Section 3 are very promising. From what we observe in the two
performance tests, the Hawkes process-based prediction does pretty well, whether it is used for the
U.S. or Japanese market, and whether it is used for higher or lower credit rating companies.
There are numerous empirical studies on how CDS prices are determined. Blanco et al.

[2] ﬁndthe impact of ﬁrm-speciﬁc stock returns is stronger on CDS price changes than on corporate bondspread changes. Norden and Weber [11] report, among other things, that the lower company’s credit
rating, the more sensitive CDS prices to its stock price, and that the CDS market contributes more to
price discovery than the bond market and this eﬀect is stronger for U.S. than for European companies.
Acharya and Johnson [1] provide empirical evidence that there is an information ﬂow from the credit
default swap markets to equity markets and the ﬂow is concentrated on days with negative credit news.
More speciﬁc to liquidity, Longstaﬀ et al.

[10] report that the majority of the corporate spread isdue to default risk (rather than liquidity risk), which result holds for all rating categories. Moreover,they ﬁnd the nondefault component is time varying and strongly related to measures of bond-speciﬁc
illiquidity as well as to macroeconomic measures of bond market liquidity. In this vein, more recently,
Bühler and Trapp [3] propose a reduced-form model to decompose bond spreads and CDS premia into
three components (pure credit risk, liquidity risk and a component measuring the reltion between credit
and liquidity risk components). Ericsson and Renault [6] report that in ﬁnite maturity debt markets,
there exist decreasing and convex term structures of liquidity. The Hawkes process we shall employ inthis paper is proposed by Hawkes [8]. In the ﬁnance literature, there are a number of papers that model4

certain economic variables by using the Hawkes processes. For example, Hewlett [9] uses for order arrivals
in the stock market and Errais et al. [7] use for default events in a credit-related portfolio of large size.
We mention the data we use in this paper. For the regression analysis in section 1.1, vector autoregression in section 2.1 and DCC-GARCH in section 2.2.1, we take all the information including bid
prices and bid-ask spreads from DATASTREAM by Thomson Reuters. The period is January 2003 to
September 2010. Since the quote numbers are not available in DATASTREAM we subscribe, we obtain
quote numbers and bid-ask spreads from Markit Group Limited. The period is January 2008 to August
2012. (Note that the data of quote numbers are available only after January 2008.) Quote numbers areused (along with bid-ask spreads) for DCC-GARCH in section 2.2.2 and for the analysis based on the
Hawkes process in sections 3.1 ∼ 3.2. Note that all the tables from the statistical analysis are placed inAppendix.

1.1

Preliminary Regression Analysis

Bid vs Economic Variables: We shall take a look at “Bid” price of the CDS via a simple regression. Thisis because the BAS may be inﬂuenced by bid prices.
∆(Bid)t = a0 + b1 ∆(Interest)t +b2 ∆(VI)t + b3 ∆(Volume)(1.1)+ c1 ∆(StockPrice)t + c2 ∆(StockSigma)t + c3 ∆(StockVol)t + d1 (Crisis)t + ϵt .

The objective variable is ∆(Bid)t is the diﬀerence in bid prices at time t. The ﬁrst three valuables
are, in essence, related to macro-economic situations: ∆(Interest)t is the change in yield rates of 10year government bonds, (VI)t is the change in VIX (U.S.) or Nikkei Volatility Index, ∆(Volume)t is thechange in trading volumes in the Dow Jones Index or TOPIX. The next three valuables are speciﬁc tothe underlying loan/bond issuer: ∆(StockPrice)t is the change in the issuer’s stock price, ∆(StockSigma)
is the change in the stock price volatility from the last 4 prior weeks, and ∆(StockVolume) is the change
in trading volume of the issuer’s stock. Finally, we attach a dummy variable (Crisis) taking a value ofzero if t is prior to September 15, 2009 and of unity if t is after that date. The result is summarized inTable 3 in the Appendix.
The third and fourth columns of the upper and lower panel are the adjusted R2 of regression (1.1)
for the U.S. and Japanese markets, respectively. In the U.S. market, the average (over 20 names) R2 ’s
are 0.1308 in the one-year CDS and 0.1988 in the ﬁve-year CDS. Several variables such as ∆(VI)t ,∆(StockPrice)t ∆(StockSigma)t have signiﬁcant non-zero coeﬃcients. (See Table 4.) In contrast, inJapan, the average (over 20 names) R2 are 0.01324 in one year CDS and 0.03966 in ﬁve year contract, far

5below the U.S. counterparts. The three valuables that seem to have signiﬁcant impact in the U.S. market
are not the cases in Japan. To get more information, we separate Japanese corporate names into two
categories: high credit rating (A- or greater by the Japanese rating agency R & I) and low credit rating(BBB+ or lower). The average R2 ’s are improved in the high rating category: one year CDS, 0.02471
and ﬁve year 0.05504, while the low rating category become even worse: one year CDS, 0.001767 and ﬁve
year, 0.02429.
It is clear that part of variations of the bid prices in the U.S. market can be explained by the microand macroeconomic factors. But in Japan, bid prices are not so well explained by these factors, especiallyin the low rating category. In both markets, the R2 are better in ﬁve year contracts than one year.

BAS vs Bid: The next question is to what extent BAS can be explained by bid prices:(1.2)∆(BAS)t = a0 + a1 ∆(Bid)t + d1 (Crisis) + ϵt .

The results are shown in the ﬁfth and sixth columns of Table 3. The averages (over 20 names) in Japanare 0.09647 (1 year) and 0.1093 (5 year) with signiﬁcant non-zero estimates of a1 in almost all names.
In the U.S. market, the numbers are 0.08024 (one year) and 0.16369 (ﬁve year) with signiﬁcant non-zeroestimates, too.

BAS vs (Bid + Economic Variables): A natural question is then if we add six variables in (1.1) to theestimation (1.2), can we get any improvement?
∆(BAS)t = a0 + a1 ∆(Bid)t +b1 ∆(Interest)t + b2 ∆(VI)t + b3 ∆(Volume)(1.3)+ c1 ∆(StockPrice)t + c2 ∆(StockSigma)t + c3 ∆(StockVol)t + d1 (Crisis)t + ϵt .

To answer the question, we conduct the partial F-test: See the last two columns of Table 3. The marginalcontribution of the six variables to R2 is tested. In the U.S. market, the six additional variables havestatistically signiﬁcant contribution in 9 names out of 20 names at 5 % signiﬁcant level. Especially, 7
names (in 1 year contracts) and 6 names (in 5 year) out of 10 names in low rating category show signiﬁcant
improvement. Since, in the U.S., the CDS market is more liquid in the lower grade category and is used
for hedging credit risk of these companies, the level of BAS incorporates many factors. Norden andWeber [11] report that the higher the credit risk, the more sensitive to stock return ﬂuctuations the CDS
price becomes. Our result here is consistent with this: Since a lower-graded company is more exposed to“ﬂight to liquidity”, market makers may promptly respond to changes in general and/or company-speciﬁcfactors by widening and shrinking BAS. In Japan, in contrast, only in 4 names (mostly higher grade) out

6

of 20 names do we observe marginal improvements. The latter result is consistent with the regression
result of (1.1).
As a consequence of this test, we may say that in the U.S. market, BAS is explained both by bid
prices and by micro-and macroeconomic factors, while in the Japanese market, BAS is explained only bybid prices. Since the CDS market in Japan is in its incipient stage, BAS are charged for some reasonsthat do not necessarily reﬂect the general economic or company speciﬁc factors. In particular, in thelower grade category, this phenomenon is more apparent.

2Time Series Analysis

2.1

Autoregressive Model

To make a further comparison between the U.S. and Japanese markets and to better understand theinteraction between bid prices and BAS, we consider the following vector autoregression (VAR): We
tested 20 names in each market, using the data from Jan-2003 to Sep-2010.
∆(Bid)t = a1 + b11 ∆(Bid)t−1 + b12 ∆(BAS)2,t−1 + ϵ1,t ,(2.1)
∆(BAS)t = a2 + b21 ∆(Bid)t−1 + b22 ∆(BAS)2,t−1 + ϵ2,t







σ12

ϵ1,t

ρσ1 σ2 has the variance-covariance matrix . The results are shown in Table
ρσ1 σ2σ22ϵ2,t5. Let us ﬁrst see the signs of each coeﬃcient. The most striking feature is that the sign of b22 (∆BAS→

where ∆BAS) is negative in both markets and in both 1 year and 5 year contracts, while the sign of b21 (∆Bid→∆BAS) is positive in most of the cases. Hence if one observes an increase in CDS price at some period,
the next period is likely to witness an increase in BAS. Regarding this point, it is interesting to observe
5 out of 20 U.S. companies have negative signs on ∆(Bid) variables in regression (1.3), while 9 out of 20
U.S. companies have positive signs on the variable. See Table 4.
On the other hand, if one observes an increase in BAS at some point, then one may see a decline
in BAS in the next period. This “reversal” property checks BAS not to keep rising, so BAS remains inreasonable stable levels.
Another point to make is that there exists strong Granger causality from ∆BAS to ∆Bid in 11 names(6 names) out of 20 in the US 1 year (resp. 5 year) CDS market. There are also 9 names in 1 year
contracts and 8 names in 5 year contracts in Japanese market. This causality is a bit counterintuitive.
A possible explanation is that the market participants may take BAS into consideration when they7

determine absolute levels of credit risk (i.e., bid price). To get more insights into this phenomenon,
we conducted more Grander causality tests including quote numbers (Quotes, hereafter): See Table 6
where we test ∆(Bid)∆(Quotes) and ∆(BAS)∆(Quotes) and report the F -statistics. The causality
observed here is less signiﬁcant than the causality from ∆(BAS) to ∆(Bid). Hence if we look at a longtime period, the relationship between the bid prices and BAS is stronger than that between Quotes and
BAS.

2.2

DCC-GARCH Model

2.2.1

BAS-Bid

Next, we study dynamic correlation models with GARCH(1, 1), following the method proposed by Engel[5]. A brief explanation of this model is as follows: Deﬁning an appropriate ﬁltration (Ft )t≥0 , the n1/2

dimensional vector at time t, yt is modeled by yt = µt + at where µt = E(yt |Ft−1 ) and at = Ht

vt withvt ∼ i.i.dN (0, In ). We then have Ht = Cov(at |Ft−1 ). Now Ht is in turn modeled by
Ht = Dt Rt Dt√√
where Rt = (ρij,t )n×n is a positive deﬁnite matrix, Dt = diag{ h11,t , . . . , hnn,t }. Moreover, Rt is
Rt = diag(Qt )− 2 Qt diag(Qt )− 2
11

where Qt satisﬁes
Qt = (1 − θ1 − θ2 )Q̄ + θ1 ϵt−1 ϵ′t−1 + θ2 Qt−1 .
Here ϵt is the standardized innovation vector with ϵt = Dt−1 at and Q̄ is the unconditional correlation
matrix of ϵt . For Qt to be positive deﬁnite, a suﬃcient condition is that θ1 , θ2 > 0, θ1 + θ2 ≤ 1.
We ﬁt the above DCC-GARCH(1, 1) to 10 names (5 each in the U.S. and Japan)(1) Bid - BAS: from Jan-2003 to Sep-2010 (Figure 2) and(2) BAS - Quotes: from Jan-2008 to Aug-2012 (Figure 3).2
First, from Figure 2, we observe large variability in correlation coeﬃcients (between Bid price andBAS) in general. This is consistent with the causal relationship between the two variables conﬁrmed
in Table 5. In the U.S. market, however, the ﬂuctuations seem greater as the credit rating goes worse.
Compare AT&T with credit rating A- and Chesapeake Energy (BB-). In contrast, in the Japanese2

The quote numbers are available only after January 2008.

8

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!$%(#

!$%(#

!$%'#

!$%'#

!$%&#

!$%&#

!"#

!"#

Figure 3: Estimated dynamic correlations between BAS and Quotes
10

market, irrespective of credit ratings, there exists a large variability across the names. This big swing ofcorrelation coeﬃcients indicate that it is unlikely that there exists a stable (over time) linear relationshipbetween the two variables.

2.2.2

BAS-Quotes

Next, from Figure 3, we see that the variability in correlation coeﬃcients (between Quote numbers andBAS) is not as large as in the previous pair. This may explain weaker causality relationship between thetwo variables. (Recall the results in Table 6.) But there are sharp pikes occasionally. See for example
Boeing (A): the correlation coeﬃcient is very low around 0.2 in most of the times but it jumped up to
0.8 from time to time.
For the purpose of predicting BAS, we believe that short-run analytical tool is of more practical use.
As experienced in the recent ﬁnancial crisis, liquidity crisis would break out all of sudden. The cost to
hedge credit risks could surge in the short run. Thus, it would be helpful if one could foresee near-future
BAS levels when trying to load and/or unload credit risks. From this point of view, although the causality
relationship is stronger (at least in the long run) between Bid price and BAS, we shall concentrate on
the Quote numbers to explain BAS. The occasionally observed high level of correlation between Quote
numbers and BAS may provide us with a useful tool to predict liquidity levels in the short run.

3

The Hawkes Process

3.1

Specification

We build models to estimate and predict bid-ask spreads (BAS) in two steps: Let M be a standard Poisson
random measure on R+ × R+ deﬁned on a probability space (Ω, F, P) where (Ft )t≥0 is the augmented
ﬁltration generated by M . Let B be a standard Browinian motion independent of M . Deﬁne λ and N
by setting N0 = 0, λ0 = a and
(3.1)
(3.2)

−ct

λt : = ae∫
Nt : =∫+[0,t]×R+be−c(t−s) dNs ,t > 0,(0,t)

M (ds, dz)1(0,λs ] (z),t > 0,

where a, b, c ∈ (0, ∞) are constants. Refer this formulation to Çinlar[4]. It is called self-exciting (or the
Hawkes process) in the sense that λt is aﬀected by the path of N over (0, t). The impact of N decays
exponentially over time with rate c and the reversion level is a. The sensitivity parameter b takes care of11the impact on λ when N increases. In our context, N is for the number of quotes and λ is the intensityof arrivals of these quotes. Moreover, we model the evolution of BAS, X = {Xt , t ≥ 0}, by the Brownian
subordination of N :

((

(3.3)

Xt := X0 exp

β2
α−
2

)

)
Nt + βBNt .

Since N is an increasing Lévy process and B is a Brownian motion in R independent of N , X is also a Lévy
process (Theorem VII.6.2.[4]). The simple idea is the following: In Section 2.2 we observe when there
are enough numbers of quotes, the correlation between BAS and quote numbers becomes instantaneously
higher and the changes in BAS are not uniform in the real time. Accordingly, this time-change seems
natural. That is, BAS changes when there arrives a quote. Figure 4 show some sample paths of theintensity process λt and the corresponding counting process Nt . Figure 5 are some sample paths of theintensity process λt with the corresponding Brownian subordination Xt .
87intensity processintensity process

6
5
4

6

4

2

3
0

0

1

2

3

4

5

6

7

8

9

10

40

20

30

15
QuotesQuotes

2

20

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

10

5

10

0

0

0

0

1

2

3

4

5

6

7

8

9

10

Figure 4: Hawkes process
Top: intensity process, Bottom: accumulating Quotes; left: t=10, a=2.0, b=0.5, c=1.0, right:
t=10, a=0.5, b=2.0, c=2.0

Our estimation of the parameters in (3.1) is based on Ozaki [14]. Let t1 , t2 , . . . , tn be the times ofoccurrence, the log-likelihood function of Nt is written∫ T∫ T
log L(t1 , t2 , . . . , tn |θ) = −λ(t|θ)dt +log λ(t|θ)dNt
00

where [0, T ] is the observation period. In the speciﬁcation of (3.1), it becomes


n
n
i
∑
∑
∑
b −c(tn −ti )(3.4)
log L(t1 , t2 , . . . , tn |θ) = −atn +(e− 1) +log a +e−c(ti −tj )  .
c
i=1

i=1

12

j

30

7

25
intensity processintensity process

6
5
4
3

15
10
5

0

1

2

3

4

5
Days

6

7

8

9

0

10

10.4

10.2

10.3

10.1
Bid−Ask Spread

Bid−Ask Spread

2

20

10.2
10.1
10
9.9

0

1

2

3

4

5
Days

6

7

8

9

10

0

1

2

3

4

5
Days

6

7

8

9

10

10
9.9
9.8

0

1

2

3

4

5
Days

6

7

8

9

9.7

10

Figure 5: Brownian Subordination
Top: intensity process, bottom: Bid-Ask Spread (α = 0.0005, β = 0.003), left: t=10, a=2.5,
b=0.5, c=1.0, right: t=10, a=0.5, b=2.0, c=1.5

For the parameters α and β in (3.3), we use the standard formula for geometric Brownian motion:()
1 2
log Xt+∆t − log Xt = α − σ ∆Nt + β 2 ∆Nt .
2
As for simulation, we use Algorithm 2 in Ogata[12] and we reproduce it for the record. By assumingthat the minimum value of the intensity function is µ, the jump size at each point is not larger than B,
and the Λi are values, of the piecewise constant function such that λ(t|t1 , · · · , tn ) ≤ Λi for tn ≤ si ≤ t <si+1 ≤ tn+1 .
1) Set Λ0 = µ and s0 = 0.
2) Generate a random variable U0 uniform in (0, 1) and put u0 = − log(U0 /Λ0 ).
3) If u0 ≤ T , then put t1 = u0 . Otherwise stop.
4) Set i = j = k = 0 and n = 1.
5) Set k equal to k + 1 and put Λk = λ(tn |t1 , . . . , tn−1 ) + B.
6) Set j equal to j + 1 and generate Uj .
7) Set i equal to i + 1 and put ui = − log(Uj /Λk ).
8) Put si = si−1 + ui . If si > T , stop.

139) Set j = j + 1 and generate Uj .
10) If Uj ≤ λ(si |t1 , · · · , tn−1 )/Λk , set n equal to n + 1, put tn = si and go to step 5. Otherwise go on.
11) Set k equal to k + 1, put Λk = λ(tn |t1 , . . . , tn−1 ) and go to step 6.
Just as a preliminary test, we use the samples of one month data June 2012 ∼ July 2012 and estimate
the parameters a, b and c. (See Table 2.) Note that when feeding data into (3.4), we assume that quotes
in one day are uniformly distributed within the day. By using the estimates, we simulate quote numbers
and compare a sample path of Nt with the actual quotes. See Figure 6. The result is encouraging and
we proceed with some full-ﬂedged estimations in the next subsection.

Table 2: Parameter Estimation for the Preliminary Testing
Quotes
Bid-Ask Spreadlog-likelihood

â

b̂

ĉ

α̂

β̂

Toyota Motor

722.2295.7061711.5061712.13881-0.0004221298

0.005755804

MetLife3165.603

6.230013

2.830023

3.191574

0.0006556054

0.0034706

3.2

Prediction

Let us move on to predictions of BAS via the Hawkes model. We shall conduct two types of performance
tests.

3.2.1

First Test

We take 5 year CDS contracts and prepare three sets of in-sample period (two-and-half years) of Quotes
and BAS:(a) January 2008 to June 2010(b) January 2009 to June 2011(c) January 2010 to June 2012
with the respective out-of-sample period (with the length of one month) July 2010, July 2011, and July
2012. The ﬁrst two periods (a) and (b) include the ﬁnancial crisis within their range. By using thesein-sample period data, we estimate the model parameters in (3.1) and (3.3). Then we conduct Monte

14

+!"

%#"

*!"
%!"

)!"
(!"

$#"

'!"
&!"

$!"

%!"
$!"

#"

#!"
!"

!"
$&'()*" %$'()*" %+'()*" %,'()*"-'().",'()."$%'()."#*,-./" $#,-./" $(,-./" $+,-./"

25',-.0"#!,-.0"#%,-.0"

90

80

20
70

60

QuotesQuotes

15
50

40

10
30

20

5

10

0

0

2

4

6

8

10
Days

12

14

16

18

0

20

0

2

4

6

8

10

12

14

16

18

20

22

Days

Figure 6: Comparison of observed (top) and simulated (bottom) quote numbers
Left: Toyota Motor, right: MetLife

50
45

40

intensity process

35

30

25
20

15
10

5

0

0

100

200

300
Days

400

500

600

Figure 7: Intensity simulation
T=650, a=1.105, b=0.7034, c=0.7504

15

Carlo simulations to predict the BAS’s for the out-of-sample period (for the next one month), which arecompared with the actual data.
In this experiment, note that we have to take into account the following fact: If we use the Quote
data of Toyota from Jan 2010 to June 2012 (two and half years), the parameters in (3.1) are estimated as(â, b̂, ĉ) = (1.105, 0.7034, 0.7504). A typical sample path of the Monte Carlo simulations of the intensity
process is shown in Figure 7. It starts with â = λ0 = 1.105, but (λt )t>0 spends throughout the period
above the level 1.105 and at this end of this period (two and half years), λT is above 20. Hence it is by
no means appropriate to use â = 1.105 as the initial level of λ for the out-of-sample period that follows.
In view of this, we did the following two-step estimation:(1) Use the whole two-and-half year sample data to estimate (â, b̂, ĉ).
(2) Use only the ﬁnal one month period, re-estimate a, while ﬁxing (b̂, ĉ) at the values in step (1). The
result is, say, (â′ , b̂, ĉ).
(3) Simulate (N = 10, 000), with (â′ , b̂, ĉ), the intensity (λt )t>0 for a one-month period.
This procedure is justiﬁed also by our observation that the values of (b, c) are not so much varying within
the two-and-half year periods. These parameters are rather characteristic of CDS reference (=company)
names. The results are shown in Table 7. The reference names are Dell, AT&T, AK Steel, Belo, Valero
from the U.S. market and Toyota, Sumitomo, Softbank, Acom, and Kintetsu from Japan. We compare
the actual data and simulated data in Table 8. The ﬁrst (without counting the column where we put
company names) column shows the observed average number of daily Quotes. The second column isthe 95 % conﬁdence interval of the simulated mean. The third column is the observed average BAS
in the respective one-month period, and the fourth column is the 95% conﬁdence interval of the meanof simulated daily BAS’s. The conﬁdence intervals obtained by the Hawkes model for BAS are very
narrow and generally well capture the observed BAS average within their ranges. In the periods (b) and(c), it is very rare that the estimated intervals diﬀer from the observed spreads by more than 3 basis
points. The exceptions are AK Steel and Kintetsu in period (c). Considering the facts that the period(b) contains the ﬁnancial crisis in its range and the names contain both high and low credit ratings, theHawkes model consistently well performs. Turning to the period (a), we see that the model is good in
the U.S. market. But in this period, the model fails to capture the BAS for low credit names in theJapanese market (Softbank, Acom, and Kintetsu). Based on the regression results in Section 1.1, thisis due to the underdeveloped state of the CDS market in Japan, especially low transaction volumes inlower credit contracts. The sixth column is the standard deviations of the observed BAS’s during the

16

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()*+''",-.+''"/01+''"(23+'#"

425+'#"

426+'#"()*+'#"'#"")*+,((#-./,((#

012,((#)34,($#

536,($#

537,($#)*+,($#$!"!##

789-51-:"'!""<3=>.:#

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'"!##

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()*+''",-.+''"*!"":;<0840=")!"">5?@0<"/01+''"(23+'#"

425+'#"

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012,((#)34,($#

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234.##"+56.#$"

758.#$"

759.#$"+,-.#$"*+,-$$#

&#""

+!""

678,40,9"&!"":1;<,8"%#""./0-$$#

123-$$#*45-$%#

647-$%#

648-$%#*+,-$%#

345/##",67/#$"

869/#$"

86:/#$",-./#$"

123-##"*45-#$"

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648-#$"*+,-#$";<=1951>"*!""?6@A1="

)!""
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315*$%",-./##"'()*$%"

789-51-:";2<=-9"'!""012/##")!""

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()*+''",-.+''"/01+''"(23+'#"

425+'#"

426+'#"()*+'#"*+,-##"./0-##"

Figure 8: Comparison of 14 month bid-ask spreads (actual vs predicted)
Left: U.S.(from the top, Dell, AT&T, AK STEEL, BELO, VALERO), Right: Japan(from thetop, TOYOTA, SUMITOMO, ACOM, SOFTBANK, KINTETSU)

17

one month (out-of-sample) periods. For the seventh column, we compute standard deviations of eachpredicted sample path of BAS’s and take the average over N = 10, 000 simulations. The left end valuesof the conﬁdence intervals are in general greater than the observed standard deviations. The ﬁfth andeight columns show the 95% conﬁdence interval obtained from ARMA-GARCH (1, 1). As is clearly seen,
these values are by far worse than the results from the Hawkes process.

3.2.2

Second Test:

Again, we take 5 year CDS contracts. The next prediction experiment is done in the following way:(a) Take fourteen sets of two-and-half year in-sample periods; (1) Jan 2009 ∼ June 2011, (2) Feb 2009
∼ July 2011, · · · , and (14) Feb 2010 ∼ July 2012, so that each adjacent set is lagged by one month.
(b) For each set, we estimate the model parameters and use these values to predict the out-of-sample
periods; (1) July 2011, (2) August 2011, · · · , and (14) August 2012, respectively.
(c) These predicted BAS are compared with the actual data for the fourteen months from July 2011
to August 2012. We did this procedure to the same ten companies as in the First Test.
We plot the BAS numbers (actual vs. predicted) in Figure 8. For the full results, see Tables 9 and
10 that show the estimated parameters for the two countries, and Tables 11 through 15 include actualand predicted Quotes and BAS numbers. Of course not perfect, the simulated prediction ﬁts the actual
observation pretty well. At least, the up and down trends are nicely tracked in all of the cases: both fairly
stable BAS’s and volatile ones. From what we observe, the Hawkes process-based prediction performs
well even though there are diﬀerence between the U.S. and Japanese markets, and between higher andlower rating grade companies.

4

Conclusion

We have done static and dynamic analysis for both the U.S and Japanese CDS markets focusing on thebid-ask spreads. We indicate similarities and diﬀerences between the two markets and also between thehigh credit and low credit ratings in the above sections. In particular, in the dynamic part, we treat
both long run (i.e., DCC GARCH) and short run analysis (i.e., the Hawkes process ﬁtting). The Hawkes
process based model for bid-ask spreads ﬁts the data well. We believe that this model can be a useful
tool for liquidity risk management as well. Investors with a large portfolio of credit risk exposures may

18

be able to estimate transaction costs and, more importantly, foresee possible surge of bid-ask spreads
ahead of time. The latter could lead to a signiﬁcant amount of cost savings.
In doing prediction, one could use the most recent actual values available in the market for a in (3.1),
instead of using the estimated values â′ we did above. This parameter is very important because theinitial level of the intensity process λ, and may improve the accuracy of predictions.

A

Tables19

Table 3: Adjusted R2 in Regression and partial F-test
RatingRegres. (1.1)

Regres. (1.2)

F-statistics

S& P

one-yearﬁve-year

one-yearﬁve-year

one-yearﬁve-year

GENERAL ELECTRIC

AA+-0.0041-0.01410.48370.43090.8926

0.4893

WALMART STORES

AA

0.07360.13780.0614-0.0057

1.264

1.067

BOEINGA

0.08280.15830.03590.06101.450

0.5942

McDONALDS

A

0.02360.12830.0808-0.00631

0.88001.205

METLIFE

A-

0.4153

0.48130.28750.48522.633*

4.740**

DELL

A-

0.00500.1734-0.0056-0.0055

3.504**

2.201*

AT & TA-

0.03470.02020.00400.3993

1.112

4.301**AMERICAN ELECTRIC POWER

BBB

0.04930.0096-0.0056

0.012

1.014

1.108

VALERO ENERGY CORP

BBB

0.1416

0.0656-0.0047-0.0060

0.74550.9623

DOW CHEMICAL

BBB

0.21160.28450.0891

0.02190.81081.32010.10330.15320.10260.1387average in Investment Grade
INTERNATIONAL LEASE FINANCE

BBB-

0.24480.31830.04130.2742

5.071**

13.67**FORD MOTOR

BB+

0.1717

0.19560.22890.5616

1.036

3.213**AK STEEL

BB-

0.03740.19000.02800.00650.59100.8333

BELO CORP

BB-

0.02490.08480.02440.14965.092**

1.514

FOREST OIL

BB-

0.10090.25790.04590.10172.2331*2.414*UNISYSBB-

0.20130.2590-0.0055

0.11978.763**

17.30**IRON MOUNTAIN

BB-

0.05260.0446-0.00440.0728

1.960

1.559

CHESAPEAKE ENERGY

BB-

0.07500.2905-0.00010.08922.548*

7.405**

B+

0.62910.61210.18100.20544.611**

9.877**

CCC+

0.04520.1906

0.03890.2472

14.63**

1.830

average in High Yield

0.15830.24430.05780.1887average in U.S.

0.13080.19880.08020.1637

ALLY FINANCIAL
MGIC INVESTMENT

RatingRegres. (1.1)

Regres. (1.2)

F-statistics

R& I

one-yearﬁve-year

one-yearﬁve-year

one-year

TOYOTA MOTOR

AAA

0.01840.05860.01140.00671.120

ﬁve-year
1.844

CHUBU ELECTRIC POWER

AA

0.0044-0.0030-0.00280.02910.93782.163*NIPPON STEEL

AA

0.06890.1650-0.0031-0.00470.79720.8567ASAHI GROUP

AA--0.0001

0.0037-0.0031-0.0047

0.5756

0.3447

NOMURA HOLDINGS

AA--0.0050-0.0133

0.0033-0.0059

0.9868

1.311

SUMITOMO CORP

AA-

0.01270.01420.0466

0.06873.412**

3.670**HITACHIAA-

0.05240.16470.07350.16051.807

2.367*SOFTBANKA-0.0046

0.0071-0.00140.11642.862**

2.022

ACOM

A

0.08500.11140.02630.0157

1.394

1.574

AEON

A

0.01520.04200.01790.088

2.324*

3.942**

0.02470.05500.01880.047

MAZDA MOTOR

BBB+-0.0039-0.01290.15070.1403

1.266

1.658

MITSUBISHI MATERIALS

BBB+-0.0125-0.0036

0.0959-0.0062

1.298

1.061

KINTETSU

BBB+

0.04940.08760.11530.01030.64450.3521SAPPORO HOLDINGS

BBB+-0.0128-0.00970.23000.3034

2.100

1.012

IHI

BBB+

-0.0133-0.0155-0.00490.0208

1.387

1.873

MAEDA

BBB+

-0.02480.0117-0.0067

0.11660.32020.3023

ANRITSUBBB+

0.0271-0.0324

0.94850.7406

3.127**

1.524

SHINSEI BANK

BBB

0.0002-0.00130.25370.04670.64480.6355average in investmant grade

PIONEER

BBB

0.051

0.2000-0.00490.02080.4197

1.479

AIFUL

CCC

0.01120.01890.17080.34880.5414

0.5314average in High Yield

0.00180.02430.17410.1716average in Japan

0.01320.0397

0.09650.1093p-value less than **:1 percent, *:5 percent

20

Table 4: Coeﬃcients to regress Bid-Ask Spreads in one-year contract CDS∆(Bid)∆(Interest)∆(VI)∆(Volume)∆(StockPrice)GE

0.2907**

WALMART-0.2075**∆(StockSigma)∆(StockVol)

1.4353-0.0523-1.24170.39700.1183*0.7001∆(Crisis)10.1862

11.9972

1.1781

5.7987-74.0718

0.5392-0.0041

BOEING

0.0945**0.4920-0.0594-0.6176

15.8032

50.7611-0.8783

0.0916MCDONALDS-0.2545**-1.28620.0246-1.0882-7.9366

0.13410.8426

0.0544

METLIFE

0.2064**-6.5181-0.1835-6.4302118.0829

191.4951

9.97640.5749DELL

0.0225-0.3574-0.1762**

0.4616-24.7058*-86.5426**

3.3039**-0.0636

AT& T-0.0667*

0.26540.07420.594519.1169

8.70331.81400.0819AMERICAN ELEC

0.0058-1.4441-0.0414-0.4360-4.9118

41.7283

0.02460.0354

VALERO-0.0025-2.3010-0.0199-0.287716.0494

1.75870.75880.1094

DOW CHEM

0.0709**-2.2646-0.0547-0.0598

8.679328.0496

1.67090.0262FORD

0.1271***60.4911-0.4105-9.4144-182.8653

111.2600

16.0730

0.8918AK STEEL

0.0659**3.92420.0223-4.1301-44.3807-111.5615

3.1360-0.5274

BELO

0.0571**

6.51472.0444**17.6603-93.0524-373.1725**-14.3063

0.6033

FOREST OIL

0.0949**-12.7739-0.8435**-2.2583-36.3112-105.9673

0.5321-0.0171

UNISYS-0.0575*

53.3839-5.9294*

121.096*-648.0591**

2431.2284**-153.4435**

0.70194

IRON MOUNT-0.000765-5.2244*0.12962.6758-1.3485

7.3423-1.2660-0.1488

CHESAPEAKE

0.0874-13.5363-2.4360**

18.6688

34.2353292.7386

5.88170.2696ALLY FIN

0.121**

55.5200.277

65.886249.358

1424.196**

10.371-1.233

INTER LEASE

0.0431-45.2685-1.34628.5344-200.9290**

258.4313*-9.8886-1.7914-0.1772**

0.3973-0.8202-57.8877*-443.8731**

854.4722**

1.2105-2.9807∆(Bid)∆(Interest)∆(VI)∆(Volume)∆(StockPrice)∆(StockSigma)∆(StockVol)∆(Crisis)

TOYOTA

0.0750*-1.7605

0.0215-0.6001-30.2484*-19.3537

0.6859-0.1581

CHUBU

0.0483-0.4397

0.0960*-0.6300

13.2446-34.4850

0.65740.1404NIPPON STEEL

0.03980.09090.1136-3.2692

5.686134.0362-0.2207-0.0197

MGIC

ASAHI

-0.1743**

0.1162-3.6418-0.8729-12.7071-57.8543

0.8834-0.0608

NOMURA

0.05351.29484-0.0089-4.9527-72.4630*-59.0562-1.0178-0.4256

SUMITOMO

0.2001**-9.5908-0.0704-4.9918

90.6216**40.8240

5.0567-0.7049

HITACHI

0.2784**-7.6758

0.19622.938824.424461.2449-5.4337*0.0014SOFTBANK

0.065327.1565-0.4313-69.9410**-30.2923-26.3252-10.9522-0.7359ACOM-0.2270*0.82491.666063.1870

20.3758

342.1862-39.6392-0.4889AEON-0.1710**22.1799-0.4404-2.9913-154.3648**-230.6126

9.9578-0.0267

MAZDA

0.3077**6.9161-0.47888.3572

1.0218-7.8888

3.7957-0.0118

MATERIALS-0.2066**-2.40280.12231.7258-15.3096-67.9243

1.69731.1109KINTETSU-0.3615**-12.0857

0.0533-8.5618-7.2398-121.4281

4.1523-0.2595

SAPPORO

0.4274**6.7441-0.5101-7.5348-117.4324*

255.8141-0.5101

IHI-0.0030-17.8202-0.5927-48.1890*

6.6656180.2065

22.9154

MAEDA-0.0281-10.6764-0.0324

2.7386-2.0083-47.6794-0.9447-0.3996

ANRITSU0.1504**-0.0388

0.00040.0574-0.1979-1.9310**-0.0332SHINSEI-0.4533**-9.1703-1.2079

5.0721-226.8239-79.9928-13.5235-4.1858

PIONEER

0.0119-0.51420.0710-6.13420.0738-6.0303

1.83190.0658AIFUL

0.1181**

94.4605

4.6687-178.1841

309.5854

505.0846

41.071019.0328

p-value less than **:1 percent, *:5 percent

21

Table 5: Coeﬃcients of VAR and F statistics in the Granger causality testb21∆(BAS)t−1 → ∆(Bid)tb22

one-yearﬁve-year

one-yearﬁve-year

one-yearﬁve-yearGE

0.11120.0013-0.6327-0.37113.9086*0.0281

WALMART

0.10270.0420-0.3586-0.4183

6.5872*0.0284BOEING

0.09730.0393-0.4157-0.32700.04431.6247MCDONALDS

0.08210.0122-0.2437-0.5034

4.3776*0.0505

METLIFE

0.13590.0077-0.2222-0.18100.65200.0021DELL

0.18440.05668-0.2214-0.384213.2521**

13.2715**

AT& T

0.1388-0.1948-0.3243-0.5985

2.391979.4863**

AMERICAN ELEC

0.04810.02366-0.4518-0.4895

2.5624

1.5244

VALERO

0.02660.0072-0.4184-0.49296.8563**0.91080.0932

DOW CHEM

0.02610.0152-0.5347-0.4633

2.7214

FORD-0.03480.0424-0.2581-0.414946.7527**

2.2168AK STEEL

0.03750.0194-0.4400-0.5855

0.60980.0654

BELO CORP

0.09970.0241-0.3302-0.0891

32.7864**4.4147*

FOREST OIL

0.00630.0243-0.3302-0.38538.7364**0.2129UNISYS

0.03420.0242-0.2074-0.029610.3175**

2.162

IRON MOUNT-0.0336-0.0111-0.3493-0.5323

1.12236.7821**CHESAPEAKE

0.20380.0645-0.4258-0.3973

5.3495*0.0777

ALLY FIN

0.09070.2201-0.2989-0.626424.0233**

0.695

INTER LEASE

0.02210.0112-0.4850-0.4874

1.5611

6.3805*MGIC-0.0077

0.0320-0.3865-0.4915

1.626913.1742**b21
one-year∆(BAS)t−1 → ∆(Bid)tb22
ﬁve-year

one-yearﬁve-year

one-yearﬁve-year

TOYOTA

0.2656

0.0985-0.3140-0.360713.2953**8.0016**CHUBU ELEC

-0.0087

0.0519-0.0764-0.15933.8622*

1.3519NIPPON STEEL

0.29160.1413-0.3812-0.38460.094

2.1191ASAHI

0.2981

0.0983-0.1464-0.2458

20.413**12.8373**

NOMURA

0.00130.0013-0.2645-0.3155

0.66990.4889SUMITOMO

0.12180.0668-0.3071-0.3627

0.0183

3.9927*HITACHI

0.27130.1992-0.3545-0.3038

1.935111.9009**

SOFTBANK

0.11220.1495-0.4015-0.4195

2.69720.5928ACOM

0.02530.0164-0.3948-0.3982

34.4122**5.4584*AEON

0.2897

0.0972-0.1583-0.2863

55.3525**

43.4629**

MAZDA

0.05190.0293-0.1650-0.187717.302**

6.8298**MATERIALS

0.18880.0338-0.3032-0.3713

7.2391**0.0021KINTETSU

0.34800.2253-0.3133-0.3750

7.0442**

0.0974SAPPORO-0.0815-0.0057

0.1466-0.0046

13.1345**

23.5656**SHINSEI

0.00327

0.0238-0.0980-0.12201.58470.0025

PIONEER

0.03450.0328-0.2008-0.3238

0.00340.3243

AIFUL-0.0823-0.0734-0.1692-0.1296

2.9913

1.1188IHI

0.11270.0314-0.0173-0.0565

2.10311.1289MAEDA

0.0190-0.0059-0.0120-0.0586

0.0032

3.1155ANRITSU-0.0275

0.0781-0.0063-0.152400.0365p-value less than **:1 percent, *:5 percent

22

Table 6: Granger causality test between Bid and Quotes, BAS and Quotes∆(Bid)t−1 → ∆(Quotes)t∆(Quotes)t−1 → ∆(Bid)t∆(BAS)t−1 → ∆(Quotes)t∆(Quotes)t−1 → ∆(BAS)tGE

0.01260.14670.488

0.0018WALMART

0.32110.34951.36081.6334BOEING

0.61420.59632.85844.1077*MCDONALDS

0.77440.02830.00820.4433

METLIFE

9.1977**0.9902

1.81254.1901*DELL

0.08882.6586

1.139226.8677**

AT& T

0.03160.01980.76730.589311.0412**0.16191.81722.1367

VALERO

0.83760.00536.7073**0.3051DOW CHEM

2.28260.11500.92032.8238FORD

0.056514.2146**

0.10620.0788AK STEEL

0.9162

3.8678*0.433

0.3012

BELO CORP

1.38480.07190.25862.0608FOREST OIL

2.29360.13591.69040.9324UNISYS

0.74870.42230.29031.3612IRON MOUNT

1.12461.386369.8123**

1.8085CHESAPEAKE

9.2459**0.0202

10.7715**

3.9106*ALLY FIN

0.07660.13470.01250.0495

ARAMARK

00.07750.00480.41880.04240.16720.05150.952

AMERICAN ELECSPRINGLEAF∆(Bid)t−1 → ∆(Quotes)t∆(Quotes)t−1 → ∆(Bid)t∆(BAS)t−1 → ∆(Quotes)t∆(Quotes)t−1 → ∆(BAS)t

TOYOTA

0.21230.91890.02060.3146CHUBU ELEC

2.1346

0.2582

0.39880.5898NIPPON STEEL

1.40480.5576

0.46710.1592

NOMURA

0.97710.20910.24730.3415

SUMITOMO

0.00040.022100.0331HITACHI0.08050.02940.92032.4511SOFTBANK

5.578*

0.4172

0.53810.0765ACOM0.74540.08470.00032.0202

AEON

0.05481.76790.06630.2014SHARP

2.86631.00180.52451.5966

TOKELP

0.43570.01431.13310.2273MAZDA

1.26182.80380.968

6.2024*MATERIALS

1.134

0.05931.68052.4058KINTETSU

1.36640.01970.6691

6.0978*SAPPORO

0.001

0.0076

0.22270.9759SOJITZ

0.00161.24450.95771.0536SHINSEI

0.02423.00590.3434

0.094

PIONEER

1.408614.2741**

34.2881**8.9154**

AIFUL

1.22280.07360.10740p-value less than **:1 percent, *:5 percent

23

Table 7: Estimated Parameters from the Hawkes Model
July-2010

Brownian Subordination Xt

Hawkes process Nt

α

β

a

b

c

DELL

0.00006651

0.01076

5.3136

1.96402.0275

AT&T

0.0004781

0.029423.05250.73559

0.77562

AK STEEL

0.00001813

0.005699

1.96974.42004.4961

BELO

0.00001244

0.005040

4.67985.42635.4675VALERO

0.00007151

0.01100

3.53993.1716

3.2918TOYOTA

0.0002054

0.019544.24570.582860.62786

SUMITOMO

0.0003156

0.023142.43380.59184

0.66534

SOFTBANK

0.0007965

0.03557

1.25200.707810.74581ACOM0.0005500

0.02761

2.06780.57667

0.63167

KINTETSU

0.0007635

0.03358

2.88600.287720.30472

July-2011

Brownian Subordination Xt

Hawkes process Nt

α

β

a

b

c

DELL

0.00001504

0.007841

5.60422.39832.4643

AT&T

0.0002281

0.02174

3.3623

1.0487

1.1244

AK STEEL

0.000004735

0.005419

4.37264.53644.6039BELO

0.00000003761

0.004458

5.26184.8533

4.9274

VALERO

0.00002346

0.009111

5.22343.36093.4746TOYOTA

0.00001659

0.01356

2.62240.74000

0.78350

SUMITOMO

0.00005132

0.01686

2.83300.68333

0.73233SOFTBANK

0.0003955

0.03520

1.63530.59403

0.64523

ACOM

0.0002321

0.02443

3.12800.644360.70577

KINTETSU

0.00005990

0.01726

3.46860.501630.55862

July-2012

Brownian Subordination Xt

Hawkes process Nt

α

β

a

b

c

DELL

0.00002012

0.005093

5.86582.54372.6073

AT&T

0.0001426

0.01595

2.5936

1.19451.2901

AK STEEL

0.00006131

0.009857

4.97992.10032.1417BELO

0.00001767

0.006032

4.16132.41112.4738VALERO

0.00001138

0.005074

7.80702.24092.4085TOYOTA

0.00007224

0.012023.80040.703370.75037

SUMITOMO

0.00006464

0.01380

3.38970.69198

0.75398

SOFTBANK

0.0002347

0.02638

2.96050.41663

0.46977ACOM

0.0001325

0.017072.43900.643870.72749

KINTETSU

0.0001388

0.013643.94400.558440.62944

24

25

730

1431

884

405

333

148

198

182

AK STEEL

BELO

VALEROTOYOTASUMITOMOSOFTBANKACOMKINTETSU

317

KINTETSU

185

1366

892

1195

395

350

227

184

326

AT&T

AK STEEL

BELO

VALEROTOYOTASUMITOMOSOFTBANKACOMKINTETSU

1391

DELL

Observed

282

ACOM

July-2012

175

SOFTBANK986

VALERO

300

1192

BELO

300

714

AK STEELSUMITOMO

432

AT&T

TOYOTA

1206

DELL

Observed

305

AT&T

July-2011

759

Observed

DELL

July-2010[428.2, 433.3][280.8, 285.1][279.0, 282.5][459.4, 465.8][562.4, 569.9][1838.1, 1911.0][1663.1, 1777.9][2027.2, 2066.4][500.6, 530.4][2489.6, 2645.0]

HawkesQuotes[343.7, 347.9][372.49, 377.73][195.1, 198.8][376.5, 382.3][385.6, 392.0][2101.2, 2239.9][1020.7, 1196.3][1832.8, 1896.0][2912.7, 2995.3][2046.2, 2172.0]

HawkesQuotes[246.01, 248.85][252.6, 256.8][200.9, 205.6][275.8, 280.0][538.8, 545.2][1396.7, 1506.9][4143.3, 4496.2][1370.3, 1426.3][473.0, 496.8][959.0, 1016.6]

HawkesQuotesAverage

Average

47.48[58.33, 58.65][37.03, 37.24][25.57, 25.79]
37.08[10.10, 10.16]

23.27[9.096, 9.144]

10.72[9.021, 9.145]

8.691[17.59, 17.85]

16.74
9.573

[40.52, 40.86][8.864, 9.067]

7.765
55.00

[10.13, 10.28]

HawkesAverage[12.58, 12.66][20.64, 20.83][10.23, 10.33][7.633, 7.680][5.595, 5.623][5.056, 5.195][9.94, 10.02][18.77, 18.87][8.787, 9.081][4.953, 5.057]

Hawkes12.51

Observed

10.81

21.25

10.62

7.160

5.417

4.992

10.11

17.73

6.912

4.892

Observed[35.64, 36.01][79.35, 80.06]

27.39[123.8, 125.1]

44.46[9.262, 9.333][6.592, 6.650][10.312, 10.62][12.47, 12.67][31.35, 31.48][8.119, 8.463][8.010, 8.089]

Hawkes

79.25

8.620

7.361

10.842

13.33

32.13

6.706

6.631

Observed[34.96, 35.93][-187.3, 229.2][-33.13, 503.5][-173770, 64040][-478.3, 195.4][-5597, 10179][-133.0, 378.7][-168.1, 254.7][5.634, 7.011][5.974, 22.15]

ARMA[19.16, 19.26][-8095, 13426][3.822, 12.81][7.729, 8.129][-2261, 2171][-13333, 4568][0.698, 23.16][-165.4, 277.5][-25.52, 75.85][-308.4, 804.4]

ARMA[67.47, 68.60][17.85, 197.9][100.6, 112.8][-10.39, 22.41][-26.01, 16.51][-1197300, 409600][12.67, 12.91][13.06, 34.78][0.673, 10.10][3.082, 14.14]

ARMA

1.524

2.446

2.504

1.053

0.4322

0.363

2.098

10.92

0.562

2.854

Observed

0.65621.391

0.85710.2853

0.3462

0.037

0.341

1.965

0.946

0.116

Observed

2.588

16.89

22.9280.55740.8788

1.457

1.696

3.975

0.832

1.086

Observed

Table 8: Conﬁdence Interval

HawkesHawkes

[6.573, 6.711][4.167, 4.255][4.470, 4.578][1.174, 1.199][1.022, 1.043][0.7758, 0.8270][1.663, 1.774][7.215, 7.414][1.227, 1.331][1.028, 1.101][9.774, 11.72][1003, 3364][39.20, 1795.9][-140110, 671280][612.9, 2872.6][18174, 95901][77.09, 695.3][18.30, 2267][0.437, 4.097][1.790, 23.60]

GARCH[2.109, 2.175][48720, 189600][28.49, 54.26][2.829, 4.160][-8940, 116610][-11257, 85716][14.72, 77.34][22.45, 761.27][18.75, 174.6][41.07, 243.8]

GARCH[10.22, 13.59][591.9, 1311][39.60, 95.48][32.62, 140.91][10.25, 39.20][-325700, 4594300][505.1, 2195.1][17.51, 43.35][4.051, 20.10]

Standard Deviation[1.594, 1.627][3.880, 3.981][1.978, 2.031][0.9803, 1.0013][0.5841, 0.5961][0.8375, 0.9127][0.5053, 0.5505][2.120, 2.170][1.826, 1.999][0.6997, 0.7508]

GARCH[4.873, 18.95]

Standard Deviation[7.629, 7.844][14.20, 14.61][25.26, 26.06][1.425, 1.460][1.197, 1.227][1.721, 1.953][1.582, 1.709][2.489, 2.559][2.112, 2.363][0.5467, 0.5841]

HawkesStandard Deviation

26

0.012979

0.012976
0.012493

0.000042086

0.000043858

0.000054719

0.000068604

0.000057343

0.000062346

0.000066158

0.000073166

0.000066513

0.000072239

0.000072655

Oct-11

Nov-11

Dec-11

Jan-12

Feb-12Mar-12

Apr-12

May-12

Jun-12

Jul-12

Aug-120.012010

0.012020

0.012066

0.012329

0.013059

0.0132200.013196

0.013200

0.015522

0.014533

0.000071113

0.000085632

0.000071408

0.000075297

0.000085411

0.000064635

0.000084683

Brownian Subordination Xt

Feb-12Mar-12

Apr-12

May-12

Jun-12

Jul-12

Aug-12ACOM

0.00023239

0.00019655

0.00021702

0.00020381

0.00016544

0.00016888

0.00019918

0.00015557

0.00016493

0.00007625

0.00007631

0.00011889

0.00013246

0.00013933

Jul-11

Aug-11

Sep-11

Oct-11

Nov-11

Dec-11

Jan-12

Feb-12Mar-12

Apr-12

May-12

Jun-12

Jul-12

Aug-12α0.015147

0.000093725

0.017203

0.017073

0.017193

0.017024

0.018931

0.020007

0.020452

0.020533

0.020536

0.020913

0.021404

0.021843

0.022699

0.024453

β0.013748

0.013799

0.013960

0.014088

0.015433

0.015486

0.015401

Jan-120.015478

0.000095597

0.000057152

Oct-110.015808

Dec-110.000018569

Sep-110.0162090.000068349

0.000035829

Aug-110.016860βNov-110.000051322

αBrownian Subordination Xt

Jul-11SUMITOMO

0.013076

0.000020989

Sep-110.013069

0.000018071

Aug-110.013558

β0.000016592

α2.81890.78632

0.70337

0.782220.75400

0.754120.77056

0.747430.75474

0.73500

0.72895

0.760100.810370.65726

0.69198

0.69274

0.65393

0.65625

0.693130.675130.675120.66188

0.653700.65591

0.656670.672390.68333b

2.02882.43902.01182.21712.30752.7094

1.67311.72222.38732.29262.31422.6692

3.07620.61761

0.64387

0.64653

0.63196

0.633700.67155

0.65868

0.673640.657620.651360.65357

0.647620.65200

0.64436b0.689630.72749

0.71886

0.70366

0.705710.764020.73104

0.74606

0.72976

0.722020.72386

0.716850.720630.70577c

0.715130.753980.770900.70860

0.713240.75098

0.732410.732860.719530.71001

0.71287

0.71357

0.72968

0.73233c

0.847300.75037

0.84308

0.81430

0.814510.830630.80799

0.81537

0.79560

0.788310.820440.80966

Hawkes process Nt

3.1280a3.42273.38973.06093.52373.53223.74552.24082.31902.80062.64752.3621

3.11573.01782.8330a0.750530.74952c
0.78350

Hawkes process Nt

3.43283.80042.8812

3.94443.56103.76102.61302.2421

3.10702.75192.55633.07240.74000b

Hawkes process Nt

2.6224a0.00030875
0.00037748
0.00041580
0.00032639
0.00035196
0.00033239
0.00030165

Sep-11
Oct-11
Nov-11
Dec-11
Jan-12
Feb-12Mar-12α0.000107611
0.000131801
0.000143724
0.000141769
0.000130508
0.000133300

Sep-11
Oct-11
Nov-11
Dec-11
Jan-12
Feb-12Mar-12

Aug-12

Jul-12

Jun-12

May-120.000125669

0.000138771

0.000149173

0.000135597

0.000126775

0.000055501
0.000062030

Aug-110.000059995

Apr-120.024911

0.026377

0.026789

0.026846

0.026916

0.027340

0.027831

0.028725

0.028730

0.030709

0.031750

0.032377

0.034311

0.035201

β

0.012694

0.013638

0.013938

0.014079

0.014528

0.014685

0.015114

0.015265

0.015710

0.015883

0.015805

0.015879

0.016011

0.017274

βBrownian Subordination Xt

0.00024638

0.00023471

0.00030011

0.00029868

Jul-11KINTETSU

Aug-12

Jul-12

Jun-12

May-120.00025177

0.00039551

Aug-11

Apr-120.00039551

αBrownian Subordination Xt

Jul-11SOFTBANK

Table 9: Estimated parameters for the ﬁve Japanese companies

Brownian Subordination Xt

Jul-11TOYOTA

0.44226

0.416630.505130.49694

0.523310.544120.55843

0.529640.517190.502810.52635

0.53676

0.56029

3.8032

3.94403.1583

4.95994.2262

4.94663.01622.4297

3.16542.19353.27413.90443.8646b

0.52788

0.558440.53379

0.509430.505760.51266

0.50758

0.50676

0.49152

0.488650.52735

0.514440.526180.50163c

0.588620.629440.59452

0.56909

0.56538

0.57194

0.56666

0.566450.55087

0.54689

0.59695

0.58360

0.595210.55863c

0.49533

0.46977

0.56085

0.55194

0.57889

0.59579

0.615160.58589

0.57353

0.556800.58004

0.590190.61450

0.64523

Hawkes process Nt

3.4686a2.74812.96052.4164

3.10402.96753.42892.24311.99062.76142.72962.07752.07822.06470.59403b

Hawkes process Nt

1.6353a27

0.0056130
0.0053604

0.000020935
0.000020056
0.000020427
0.000020072
0.000024623
0.000020222
0.000020652

Sep-11

Oct-11

Nov-11

Dec-11

Jan-12

Feb-12Mar-120.000020123
0.000019657
Brownian Subordination Xt

Jul-12

Aug-12

AT&T

0.00012698

Aug-120.015991

0.015947

0.016051

0.016092

0.017972

0.018419

0.0074576

0.000023701
0.000026799
0.000025131

Jan-12

Feb-12Mar-120.000055330
0.000060977
0.000069379

Jun-12

Jul-12

Aug-120.000027452

0.000022800

Dec-110.000038632

0.000020861

Nov-11

Apr-120.0065889
0.0069322

0.000018222

Oct-11

May-120.0064216

0.000016477

Sep-110.0099172

0.0098303

0.0095676

0.0082294

0.0062442

0.0061243

0.0059747

0.0057674

0.0055195

0.000006972

Aug-110.0054188

0.000004735

βJul-11αBrownian Subordination Xt

0.00014261

Jul-12

AK STEEL

0.00015646

Jun-120.016743

0.00016139
0.00015024

0.00015687

Mar-12

Apr-120.00016713

Feb-12

May-120.017657

0.00018088

0.018819

Jan-120.019367

0.019786

0.00018296

0.00019574

Oct-110.019966

0.021099

Dec-110.00020046

Sep-110.00019276

0.00020493

Aug-11

0.021741

β0.0052551

0.0050931

0.0052700

Nov-110.00022809

Jul-11α0.000019792

Jun-120.0052944

0.000017746
0.000017630

Apr-12

May-120.0058944

0.0061908

0.0063359

0.0063918

0.0074039

0.0075556

0.0076707

0.000013632

Aug-110.0078406

β0.000015041

αb2.51322.54382.53822.53942.52422.49102.51542.46712.45922.31192.47602.45542.47112.3983c2.59512.60732.62002.62202.60632.57392.59862.54972.54172.39292.55762.53702.55402.46431.2122

1.19451.18651.20151.20191.21191.17951.1406

1.13781.0915

1.0913

1.0554

1.0774

1.0487b

1.31791.29011.27581.29131.28481.3033

1.26761.2256

1.22201.1683

1.17071.1307

1.15401.1244c

4.95934.97993.31723.64053.3876

3.71132.29942.71442.6876

3.46712.6743

1.27672.88734.3726a2.17682.10032.30932.89413.28573.62983.6981

3.7682

3.83194.00434.16814.30444.43334.5364b

2.21882.14172.36812.95603.34873.70983.77473.84373.90504.07754.24164.37254.50224.6039c

Hawkes process Nt

1.60562.59362.29133.69823.91153.34412.3476

1.15922.19702.75692.17152.96342.80253.3623a

Hawkes process Nt

5.25965.86586.28587.3452

7.33636.11083.8172

3.58654.62506.35914.72605.63284.52685.6042a

Hawkes process Nt

0.000003964
0.000005936
0.000009107
0.000008521
0.000011502
0.000013896
0.000012508

Sep-11
Oct-11
Nov-11
Dec-11
Jan-12
Feb-12Mar-120.000019070

0.000033224
0.000037788
0.000026189
0.000022476
0.000018998
0.000007534
0.000010019

Sep-11
Oct-11
Nov-11
Dec-11
Jan-12
Feb-12Mar-12

Aug-12

Jul-12

Jun-12

May-120.000011390

0.000011380

0.000012933

0.000010671

0.000007558

0.000024063

Aug-11α0.000023464

Apr-120.0059390

0.0060318

0.0060524

0.0064420

0.0064015

0.0062960

0.0058756

0.0055705

0.0052743

0.0050463

0.0048189

0.0046206

0.0044389

0.0044582

β0.0050759

0.0050737

0.0050859

0.0051140

0.0050642

0.0050507

0.0054632

0.0073780

0.0075005

0.0077191

0.0088179

0.0089289

0.0090561

0.0091114

βBrownian Subordination Xt

0.000020190

0.000017672

0.000019655

Jul-11VALERO

Aug-12

Jul-12

Jun-12

May-120.000018243

0.000000624

Aug-11

Apr-120.000000038

αBrownian Subordination Xt

Jul-11BELO

Table 10: Estimated parameters in the ﬁve U.S. companies
Brownian Subordination Xt

Jul-11DELL

2.53422.41112.60322.64302.64223.22463.62323.87163.93244.15074.42004.5316

5.11124.8533c

2.57132.47382.66632.70692.70603.27333.6926

3.93973.99824.24024.50914.61885.19774.92747.71317.80708.49596.28316.03165.2422

3.95103.2156

4.1386

5.19584.33165.89084.64715.2234a2.10592.24092.17942.1784

2.19822.27562.20673.20403.19403.22293.27373.26383.28003.3609b2.32272.40852.49422.39392.41432.49322.42353.46793.4314

3.39983.45113.44103.45853.4746cHawkes process Nt

3.42634.1613

4.12215.50705.3993

4.19232.43932.89272.98133.64872.8387

3.89733.35325.2618b

Hawkes process Nt
a

28

387.9
418.2
454.1

300
368
286
280

Jul-11
Aug-11
Sep-11
Oct-11

488.0

400
406
381
408
403
395
389
359.1

Feb-12Mar-12
Apr-12
May-12
Jun-12
Jul-12
Aug-12Average

370
316
383
368

Mar-12
Apr-12
May-12
Jun-12323.8

390

Feb-12Average

429.7

319

Jan-12

350

259

Dec-11

333

296

Nov-11

Jul-12259

Oct-11

Aug-12

361.9

247

Sep-11

381.0

401.5

487.9

507.3

440.2

520.5

287.1

318.3

377.8

355.5

296.0

417.7

440.1

300
343

Jul-11

HawkesObserved

Quotes

456.0

559.1

564.0

407.8

622.9

381.6

336.6

Aug-11SUMITOMO

564.2

396

Jan-12395.8

Dec-11

447.7

343
273

Nov-11

356.3

HawkesQuotes

25.58

46.53

22.79

-1.65

32.45

39.31

40.68

-26.39

-0.22

45.89

20.09

14.27

69.12

28.31

26.99

Error(%)

28.30

43.72

42.77

1.20

52.68

28.09

38.97

-4.61

-15.00

64.01

15.39

27.26

58.79

13.64

29.31

Error(%)

11.28910.23810.72010.69410.84810.00010.08310.21912.17114.08715.86017.634

9.948

8.385

7.160

Observed

8.938

9.379

8.691

8.682

9.114

9.399

8.770

8.736

9.552

10.151

9.998

11.488

8.669

7.085

5.417

Observed

11.07410.167

10.10513.89410.15910.12810.18510.06813.91214.43215.13712.419

9.232

7.531

7.661

HawkesAverage

8.957

9.448

9.159

8.579

8.999

9.304

8.918

8.746

10.08910.11812.557

10.566

7.522

5.781

5.617

HawkesAverage

-0.70

-0.69

-5.74

29.92

-6.35

1.28

1.01

-1.48

14.30

2.45

-4.56

-29.57-7.20-10.197.01

Error(%)

-0.04

0.74

5.38

-1.19

-1.27

-1.00

1.70

0.11

5.63

-0.33

25.60

-8.03

-13.23-18.41

3.69

Error(%)

0.95891.06871.03300.8284

1.10161.0146

1.09310.87830.9596

1.1132

1.29861.02300.82150.6057

0.5861

Hawkes

91.91

145.12139.02110.74177.53220.81

193.0477.37

108.96

56.80

88.09

-19.36-30.09-50.5669.31

Error(%)

1.00901.12271.05340.81411.23110.00000.27050.4094

1.39301.4872

1.52502.26551.20471.06440.2853Observed

1.301

1.2226

1.152

1.451

1.264

1.230

1.396

1.027

1.5200

1.715

1.746

1.320

1.176

1.003

0.9949Hawkes

69.54

8.90

9.40

78.18

2.69

416.18150.85

9.12

15.33

14.52

-41.73

-2.41

-5.74

248.70Error(%)

Standard Deviation

0.62250.43600.4322

0.39310.39690.31630.37300.49520.45920.71000.6904

1.2686

1.17511.22300.3462

Observed

Standard Deviation

Table 11: The Second Test in TOYOTA and SUMITOMO
Observed

TOYOTA

29
430.6
445.4

283
229
274
225
335
431
378
391
336
331
343
362
327.6

Sep-11
Oct-11
Nov-11
Dec-11
Jan-12
Feb-12Mar-12
Apr-12
May-12
Jun-12
Jul-12
Aug-12Average

382.3

328.0

566.4

418.0

531.7

303.0

260.9

330.6

232.0

321.5

404.1

434.9

317
352

Jul-11
Aug-11345.7

Quotes277.8

Hawkes233.3

Average

291.5

280.2

263.3

334.0

246.8

225.3

306.3

299.9

Observed245

Aug-12KINTETSU

227

Jul-12

272

Mar-12

218

312

Feb-12

Jun-12

310.9

267

Jan-12

243

198

Dec-11

238

403.7

258

Apr-12222.5

230Oct-11
Nov-11

May-12242.6

182

Sep-11

196.9
264.9

175
201

Jul-11

HawkesObserved

Quotes

18.53

23.03

25.54

-0.91

68.57

6.91

40.67

-29.71-22.11

46.95-15.33

40.41

42.77

23.56

9.06

Error(%)

20.61

18.97

23.43

20.76

40.33

27.96

48.43

-20.90-15.63

54.68

16.24

-3.27

33.30

31.79

12.51

Error(%)

43.95

43.39

47.32

58.52

54.26

44.53

47.16

40.56

61.09

61.17

60.37

50.79

22.28

13.08

10.81

Observed

33.03

22.26

23.26

27.60

34.30

26.26

23.02

29.30

45.85

50.08

59.80

65.26

29.56

15.30

10.62

Observed

43.74

44.96

58.45

61.15

47.63

51.09

41.02

47.47

62.92

65.09

55.65

35.52

17.69

11.12

12.60

HawkesAverage

34.86

25.75

25.77

30.84

31.24

24.74

27.30

41.32

48.06

51.92

76.38

61.93

22.34

10.20

10.31

HawkesAverage

-0.67

3.61

23.52

4.49

-12.2214.73-13.03

17.04

3.00

6.41

-7.82

-30.07-20.59-14.95

16.57

Error(%)

3.83

15.66

10.76

11.73

-8.92

-5.77

18.64

41.01

4.83

3.69

27.73

-5.10

-24.43-33.31-2.93

Error(%)

6.704

4.383

4.555

5.313

6.126

4.709

6.030

7.238

8.186

10.40

16.45

11.70

4.462

2.288

2.017

Hawkes100.88139.24

81.91

68.75

58.80

166.27494.51

23.92

25.44

217.96

59.00

45.47

-59.30-44.94135.30Error(%)

4.637

1.732

1.524

6.328

5.786

3.794

5.977

3.009

8.749

5.205

5.641

8.658

5.136

2.722

0.6562

Observed

4.928

4.843

6.709

6.157

6.465

6.056

5.591

4.939

6.177

7.423

5.310

3.998

2.259

1.473

1.599

Hawkes

45.82

179.64340.21

-2.71

11.74

59.61

-6.47

64.16

-29.40

42.62

-5.87

-53.83-56.02-45.87143.61

Error(%)

Standard Deviation

4.580

1.832

2.504

3.148

3.858

1.769

1.014

5.841

6.526

3.271

10.34

8.042

10.9644.156

0.8571

Observed

Standard Deviation

Table 12: The Second Test in SOFTBANK and KINTETSU

Aug-11SOFTBANK30

Quotes

315.0
262.6

287.5
257.2

241
173
205
234
219
177
209
209
184
180
216.4

Nov-11
Dec-11
Jan-12
Feb-12Mar-12
Apr-12
May-12
Jun-12
Jul-12
Aug-12Average

1470.7

Average2545.51391
1560

Jul-12

1610

Jun-12
Aug-122451.2

1856

May-122127.42163.93060.92664.42327.4

1779
1639

Apr-121359.41365.91760.82318.5Mar-12

1439

1012

Dec-11
1487

1319

Feb-121685.81492

Oct-11
Nov-11
Jan-122114.61426

Sep-11

1865.31374

Aug-112099.91206

HawkesQuotes

289.0

233.1

288.2

301.5

286.1

271.4

333.5

Jul-11

Observed

197.3

200

Oct-11

DELL

221.9

240

Sep-11412.1

277

Aug-11

378.2

Hawkes282

Observed

45.68

38.71

83.00

52.25

64.92

62.56

30.83

-8.58

-5.08

73.99

75.78

12.99

48.29

35.76

74.12

Error(%)

34.56

42.89

56.27

11.53

37.88

48.34

43.82

-15.66

8.25

74.29

18.73

35.70

38.95

48.76

34.13

Error(%)

9.614

11.14312.50510.08910.547

9.164

8.874

9.607

9.586

9.987

9.715

9.928

9.960

8.607

4.892

Observed

30.41431.59337.07736.59728.64823.30520.45523.65921.74723.65129.88342.68252.96232.28621.246

Observed

9.136

12.34510.23810.183

8.204

6.872

10.1659.093

10.11910.19510.21010.08610.132

5.015

5.053

HawkesAverage29.67832.93237.21132.83823.53620.17624.53520.26622.84322.06940.73751.19143.29723.17920.688

HawkesAverage

-4.77

10.79

-18.140.94-22.22-25.01

14.55

-5.35

5.56

2.09

5.10

1.60

1.74

-41.74

3.29

Error(%)

-1.84

4.24

0.36

-10.27-17.84-13.43

19.95-14.34

5.04

-6.69

36.32

19.94

-18.25-28.21-2.62

Error(%)

4.000

3.5625

4.265

3.401

2.678

2.446

3.459

2.310

2.7348

3.120

5.705

7.170

6.983

4.252

3.908

Hawkes43.06-25.26

74.35

41.04

-61.6240.48

303.21-12.95115.4455.69-14.28-2.08-34.19-57.90180.91

Error(%)

0.8954

0.94612.8544

0.39671.69040.93391.08160.5446

0.6165

0.06180.71620.15450.31932.10390.1158Observed

1.008

1.2159

1.069

1.069

0.966

0.736

1.090

0.761

0.9259

1.066

1.260

1.195

1.392

0.658

0.712

Hawkes236.8728.52-62.54169.44-42.84-21.16

0.74

39.73

50.18

1623.3275.96

673.19335.85-68.73514.53

Error(%)

Standard Deviation

4.37184.76642.44592.41166.97681.74150.85792.65371.26942.00376.65547.3223

10.6106

10.0987

1.3913Observed

Standard Deviation

Table 13: The Second Test in ACOM (Japan) and DELL(U.S)

Jul-11

ACOM

31

322
318
188
360
440
541
467
343
316
185
159
337.9

Oct-11
Nov-11
Dec-11
Jan-12
Feb-12Mar-12
Apr-12
May-12
Jun-12
Jul-12
Aug-12Average2954.02215.5888.4

1635.81267.51262.62046.8

714
253
578
916
796
746
929
843
738
747
767
1579
1366
1101
862.4

Jul-11
Aug-11
Sep-11
Oct-11
Nov-11
Dec-11
Jan-12
Feb-12Mar-12
Apr-12
May-12
Jun-12
Jul-12
Aug-12Average1845.12255.1

1951.01760.12082.2

1628.62229.21655.2

HawkesQuotes

527.1

328.9

518.2

436.0

803.0

771.7

680.8

452.0

236.2

445.9

550.6

412.3

587.2

Observed

AK STEEL265

Sep-11

545.3

610.8

432
394

Jul-11

HawkesObserved

Quotes161.10104.8249.84-20.04154.37135.62

182.14

50.36

76.08

118.31180.05

80.70

53.70

775.69313.73

Error(%)

68.45

106.88180.0837.98

134.12

65.26

25.83

2.72

-34.39137.18

73.15

28.04

121.60

38.39

41.40

Error(%)

51.41

49.82

55.00

50.48

41.95

43.89

48.05

45.86

56.97

65.84

65.70

68.18

64.15

46.11

17.73

Observed

8.034

7.580

7.765

8.603

9.325

8.768

8.801

7.762

7.649

8.181

7.816

8.292

8.387

6.638

6.912

Observed

50.01

67.16

40.69

46.34

39.29

47.15

41.29

52.67

61.10

66.52

64.63

73.15

61.30

20.08

18.82

HawkesAverage

8.502

7.598

8.980

10.318

8.685

8.637

8.119

8.780

8.618

8.869

8.718

7.736

7.998

6.945

9.022

HawkesAverage

-2.74

34.82

-26.01

-8.22

-6.36

7.41

-14.08

14.83

7.25

1.04

-1.62

7.29

-4.45

-56.45

6.19

Error(%)

6.37

0.24

15.65

19.93

-6.86

-1.49

-7.76

13.11

12.67

8.41

11.53

-6.71

-4.64

4.63

30.52

Error(%)

1.405

0.8639

1.310

1.385

1.574

1.617

1.563

1.323

0.9373

1.401

1.579

1.235

1.537

1.363

1.976

Hawkes99.39

226.17132.88

68.11

91.48

97.59

81.90

151.2575.74

162.74

72.03

37.03

46.45

39.22

108.85

Error(%)

6.796

9.625

10.916

7.436

5.967

5.974

7.559

5.761

4.656

3.977

5.043

3.965

5.073

17.233

1.965

Observed

5.877

12.863

7.314

6.209

5.685

5.759

5.138

4.735

6.157

6.414

7.242

6.825

3.808

1.984

2.145

Hawkes

2.21

33.64

-32.99-16.51

-4.74

-3.60

-32.03-17.81

32.23

61.29

43.59

72.13

-24.95-88.48

9.18

Error(%)

Standard Deviation

0.75270.26490.56250.82400.82200.81860.85930.5266

0.53330.53330.91760.9012

1.04920.9786

0.9463Observed

Standard Deviation

Table 14: The Second Test in the AT&T and AK STEEL

Aug-11

AT&T

32

Quotes

1341
1237
1168
994
892
1015

Mar-12
Apr-12
May-12
Jun-12
Jul-12
Aug-12

Quotes1969.41346.8802.3
755.6
1076.61153.8

904
1119
850
604
797
782
890

Sep-11
Oct-11
Nov-11
Dec-11
Jan-12
Feb-12Mar-12

1836.61570.2

1264
1195
1115
988.0

Jun-12
Jul-12
Aug-12Average

1348.61236.31311.1

886
1225

Apr-12
May-121101.81683.81613.3

1215

Aug-111422.9Hawkes986

Observed

2171.81873.21744.2

1713.02658.72228.12437.81315.31800.2

1963.82165.7

Jul-11VALERO

1027.81114

Feb-12Average

808
1045

896

Jan-121685.6908

Oct-11
Nov-11
Dec-112451.3859

Sep-112740.5920

Aug-113627.7

Hawkes1192

Observed

38.83

40.83

53.69

-2.19

7.03

30.23

20.97

-3.37

0.67

82.42

98.09

20.36

117.85

32.78

44.31

Error(%)

113.59

84.55

95.54

72.33

127.63

80.12

81.79

18.07

72.27

143.04141.71

85.64

185.37

197.88204.34

Error(%)

9.074

8.761

9.573

9.410

9.357

7.935

7.905

9.530

9.800

10.3559.912

10.45210.085

8.973

4.992

Observed

19.283

14.63016.73915.53615.98916.20518.03419.60521.08123.62124.56826.71423.81823.31510.107

Observed

8.786

9.041

9.090

10.029

8.518

7.001

9.056

9.620

10.11610.091

9.823

10.27910.129

5.130

5.089

HawkesAverage18.992

16.561

17.69516.29915.257

16.58619.17918.83123.54321.35832.09623.38025.03910.095

9.969

HawkesAverage

-3.06

3.19

-5.05

6.58

-8.96

-11.77

14.55

0.94

3.22

-2.55

-0.90

-1.65

0.43

-42.83

1.95

Error(%)

-0.62

13.19

5.71

4.91

-4.58

2.35

6.35

-3.95

11.68

-9.58

30.64

-12.48

5.13-56.70-1.36

Error(%)

1.834

1.6743

1.747

1.581

1.971

1.969

2.306

1.573

2.1072

1.879

2.914

1.725

2.250

0.918

1.061

Hawkes

18.84

32.30

-16.74

7.92

-1.95

23.93

38.28

11.43

20.32

-13.47

53.65

-7.42

-11.10-84.60211.28Error(%)

0.78620.96430.36310.65530.80061.08660.89550.42180.29581.02710.71991.2902

0.53381.91600.03710Observed

0.847

0.7389

0.796

0.749

0.630

0.487

0.607

0.561

0.8151

1.009

1.208

1.296

1.540

0.750

0.675

Hawkes

151.66-23.38119.13

14.34-21.32-55.17-32.2633.04

175.53

-1.72

67.85

0.46

188.50-60.841719.01

Error(%)

Standard Deviation

2.00141.26552.09801.46482.0106

1.58881.66781.41171.75142.17191.8964

1.86292.53105.95880.3407

Observed

Standard Deviation

Table 15: The Second Test in BELO and VALERO
Jul-11BELO

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34