This document proposes a new framework for modeling trends and periodic patterns in high-frequency financial data. The framework allows the trend and seasonality to vary over time in a non-parametric way, adapting to changing market conditions. Estimators are provided for extracting an instantaneous trend and seasonality from a time-frequency decomposition of intraday return data. Failing to account for dynamic changes in the seasonality could lead to misestimation of intraday spot volatility, with implications for risk management and other applications.

1. Non-parametric estimation of intraday spot volatility: disentangling
instantaneous trend and seasonality
Thibault Vattera,, Hau-Tieng Wub, Valerie Chavez-Demoulina, Bin Yuc
aFaculty of Business and Economics (HEC), University of Lausanne, Switzerland.
bDepartment of Mathematics, University of Toronto, Canada.
cDepartment of Statistics, University of California, Berkeley, USA.
Abstract
We provide a new framework to model trends and periodic patterns in high-frequency

2. nancial
data. Seeking adaptivity to ever changing market conditions, we enlarge the Fourier Flexible
Form into a richer functional class: both our smooth trend and the seasonality are non-parametrically
time-varying and evolve in real time. We provide the associated estimators
and show with simulations that they behave adequately in the presence of heteroskedastic and
heavy tailed noise and jumps. A study of echange rates returns sampled from 2010 to 2013
suggest that failing to factor in the seasonality's dynamic properties may lead to misestimation
of the intraday spot volatility.
Keywords: Intraday spot volatility, Seasonality, Foreign exchange returns, Time-frequency
analysis, Synchrosqueezing
JEL: C14, C22, C51, C52, C58, G17
Corresponding author: Thibault Vatter. Phone: +41 21 693 61 04. Postal Address: Oce 3016, Anthropole
Building, University of Lausanne, 1015 Lausanne, Switzerland.
Email addresses: thibault.vatter@unil.ch (Thibault Vatter), hauwu@math.toronto.edu (Hau-Tieng
Wu), valerie.chavez@unil.ch (Valerie Chavez-Demoulin), binyu@stat.berkeley.edu (Bin Yu)
Preprint submitted to Elsevier December 6, 2014

3. 1. Introduction
Over the last two decades, increasingly easier access to high frequency data has oered a
magnifying glass to study

4. nancial markets. Analysis of these data poses unprecedented chal-lenges
to econometric modeling and statistical analysis. As for traditional

5. nancial time series
(e.g. daily closing prices), the most critical variable at higher frequencies is arguably the asset
return: from the pricing of derivatives to portfolio allocation and risk-management, it is a
cornerstone both for academic research and practical applications. Because its properties are
of such utmost importance, a vast literature sparked to

6. nd high-frequency models consistent
with the new observed features of the data.
Among all characteristics of the return, its second moment structure is empirically found
as preponderant, partly because of its power to asses market risk. Therefore, its time-varying
nature has received a lot of attention in the literature (see e.g. Andersen et al. 2003). While
most low frequency stylized features

7. nd themselves at higher frequencies, an empirical reg-ularity
is left unaddressed by most models designed to capture heteroskedasticity in

8. nancial
time series. It is now well documented that seasonality, namely intraday patterns due to the
cyclical nature of market activity, represents one of the main sources of misspeci

9. cation for
usual volatility models (see e.g. Guillaume et al. 1994; Andersen and Bollerslev 1997, 1998).
More speci

10. cally, some periodicity in the volatility is inevitable, due, for instance, to mar-ket
openings and closings around the world, but is unaccounted for in the vast majority of
econometrics models. To avoid misspeci

11. cation bias, one possibility is to explicitly incorporate
the seasonality in traditional models, for instance with the periodic-GARCH of Bollerslev and
Ghysels (1996). Alternatively, a pre-

12. ltering step combined with a non-periodic model can
be used as in Andersen and Bollerslev (1997, 1998); Boudt et al. (2011); Muller et al. (2011);
Engle and Sokalska (2012). Even though an explicit inclusion of seasonality is advocated as
more ecient (see Bollerslev and Ghysels 1996), the resulting models are at the same time less
exible and computationally more expensive. While they may potentially propagate errors,
two-step procedures are more convenient and often consistent whenever each step is consistent
2

13. (see Newey and McFadden 1994).
As a result, seasonality pre-

14. ltering has received a considerable amount of attention in
the literature dedicated to high-frequency

15. nancial time series. In this context, an attractive
approach is to generalize the removal of weekends and holidays from daily data and use the
business clock (see Dacorogna et al. (1993)); a new time scale where time goes faster when the
market is inactive and conversely during power hours, at the cost of synchronicity between
assets. The other approach, which allows researchers to work in physical time (removing only
closed market periods), is to model the periodic patterns directly, either non-parametrically
with estimators of scale (e.g. in Martens et al. (2002); Boudt et al. (2011); Engle and Sokalska
(2012)) or using smoothing methods (e.g. splines in Giot (2005)) or parametrically with the
Fourier Flexible Form (FFF) (see Gallant (1981)), introduced by Andersen and Bollerslev
(1997, 1998) in the context of intraday volatility in

16. nancial markets.
Until now, the standard assumption in the literature is to consider a constant seasonality
over the sample period. However, it is seldom veri

17. ed empirically and few studies acknowledge
this issue. Notably, Deo et al. (2006) uses frequency leakage as evidence in favor of a slowly
varying seasonality. Furthermore, Laakkonen (2007) observes that smaller sample periods yield
improved seasonality estimators. In this paper, we argue that the constant assumption can
only hold for arbitrary small sample periods, because the entire shape of the market (and
its periodic patterns) evolves over time. Contrasting with seasonal adjustments considered in
the literature, we relax the assumption of constant seasonality and suggest a non-parametric
framework to obtain instantaneous estimates of trend and seasonality.
From a time-frequency decomposition of the data, the proposed method extracts an instan-taneous
trend and seasonality, estimated respectively from the lowest and highest frequencies.
In fact, it is comparable to a dynamic combination of realized-volatility or bipower vari-ation
(RV/BV) and Fourier Flexible Form (FFF). In this case, the instantaneous trend and
seasonality can be estimated respectively using rolling moving averages (for the RV/BV part)
or rolling regressions (for the FFF part). Nonetheless, our model diers in several aspects.
First, it disentangles in a single step the trend from the seasonality. Second, it yields natu-
3

18. rally smooth pointwise estimates. Third, it is data-adaptive in the sense that there are less
parameters to

19. ne tune. Fourth, the trend estimate is more robust to jumps in the log-price
than traditional realized measures such as the realized volatility of bipower variation.
There is a major reason why dynamic seasonality models are important when modeling
intraday returns, concerning practitioners and academics alike: non-dynamic models may lead
to severe underestimation/overestimation of intraday spot volatility. Hence there are pos-sible
implications whenever seasonality pre-

20. ltering is used as an intermediate step. From
a risk management viewpoint, intraday measures such as intraday Value-at-Risk (VaR) and
Expected-shortfall (ES) under the assumption of constant seasonality may suer from inappro-priate
high quantile estimation. In other words, the assumption may lead to alternate periods
of underestimated (respectively overestimated) VaR and ES when the seasonality is higher
(respectively lower) than suggested by a constant model. In the context of jump detection,
the assumption may induce an underestimation (respectively overestimation) of the number
and size of jumps when the seasonality is higher (respectively lower).
The rest of the paper is organized as follows. In Section 2, we describe our model for the
intraday return, from a continuous-time point of view to the discretized process. To relax the
constant seasonality assumption, we de

21. ne a class of seasonality models which includes the
Fourier Flexible Form (FFF) as a special case. In Section 3, we provide a detailed exposition
of a method aimed at studying the class of models de

22. ned in Section 2. We start by recalling
the link between the FFF and the Fourier transform. We then sketch the theoretical basics
of time-frequency analysis, that was introduced to overcome this limitation (see Daubechies
(1992); Flandrin (1999)). Finally, we introduce the Synchrosqueezing transform, which is
specially aimed at studying the class of models de

23. ned in Section 2, and we provide associated
estimators of instantaneous trend and seasonality. In Section 4, we estimate the model using
four years of high-frequency data on the CHF/USD, EUR/USD, GBP/USD and JPY/USD
exchange rates. To obtain con

24. dence intervals for the estimated trend and seasonality, we
develop tailor-made resampling procedures. In Section 5, we conduct simulations to study
the properties of the estimators from Section 3. We show that they are robust to various
4

25. heteroskedasticity and jumps speci

26. cations. We conclude in Section 6.
2. Intraday seasonality
On a generic

27. ltered probability space, suppose that the log-price of the asset, denoted as p(t),
is determined by the continuous-time jump diusion process
dp(t) = (t)dt + (t)dW(t) + q(t)dI(t); (1)
where (t) is a continuous and locally bounded variation process, (t) 0 is the stochastic
volatility with cadlag sample paths, W(t) is a standard Brownian motion and I(t) is a

28. nite
activity counting process with jump size q(t) = p(t) p(t) independent from W(t).
An example of stochastic volatility popularized in Heston (1993) is the square-root process
from Cox et al. (1985), where 2(t) = h2(t) and
dh2(t) =
h2(t)
dt + h(t)dZ(t); (2)
with Z(t) another Brownian motion such that cov fdW(t); dZ(t)g = dt and 2 2 ensuring
that 2(t) 0 almost surely (Feller's condition). When I(t) is a compound Poisson process
with intensity and q N(J ; 2
J ) (conditional on a jump occurring), then (1) along with (2)
boil down to the model from Bates (1996).
While usual continuous-time models may display realistic features of asset prices behavior
at a daily frequency, seasonality is invariably missing. Contrasting with this observation, the
econometric literature on high-frequency data often decomposes the spot volatility into three
separate components: the

29. rst being slowly time-varying, the second periodic and the third
purely stochastic (see e.g. Andersen and Bollerslev 1997; Engle and Sokalska 2012). As such,
5

30. we may consider a continuous-time model where the volatility satis

31. es
(t) = eT(t)+s(t)h(t); (3)
with T(t) slowly varying, s(t) quickly time-varying and periodic and h(t) the intraday stochas-tic
volatility (e.g. a square-root process (2)). In other words, we decompose the volatility into
three component: T(t) is the trend, s(t) the seasonality, and h(t) is the intraday stochastic
component.
Although trading happens in continuous time, the price process is only collected at discrete
points in time. Ignoring the drift term, equations (1) and (3) suggest a natural discrete-time
model for the return process as
rn = eTn+snhnwn + qnIn; (4)
where n = t= with the sampling interval, as well as
Tn and sn representing the trend and seasonality in the volatility,
hn the intraday volatility component,
wn the white noise,
and qnIn the discretized

32. nite activity counting process.
In Sections 2.1 and 2.2, we describe
exible continuous-time versions of the seasonality s(t)
and trend T(t). In Section 2.3, we put all the pieces back together in an adaptive volatility
model, where seasonality and trend are discretely sampled from the continuous-time versions.
2.1. The adaptive seasonality model
The seasonal behavior inside a time series can be described by repeating oscillation as time
evolves. Due to complicated underlying dynamics, the oscillatory behavior might change from
6

33. time to time. Intuitively, to capture this eect, we consider the adaptive seasonality model
s(t) =
XK
k=1
ak(t) cos f2k(t) + kg ; (5)
where K 2 N, ak(t) 0, 0(t) 0, (0) = 0 and k 2 R. The function ak(t) is called
the amplitude modulation (AM), (t) the phase function, k the phase shift, and 0(t) the
instantaneous frequency (IF).
When the phase function is linear (!t with ! 0) and amplitude modulations are constant
(ak 0 8k), then the model reduces to the Fourier Flexible Form (FFF) (see Gallant (1981)).
When the phase function is non-linear, the IF generalizes the concept of frequency, capturing
the number of oscillations one observes during an in

34. nitesimal time period. As for the AM, it
represents the instantaneous magnitude of the oscillation.
While those time-varying quantities allow to capture momentary behavior, there is in gen-eral
no unique representation for an arbitrary s satisfying (5), even if K = 1. Indeed, there ex-ists
an in

35. nity of smooth pairs of functions (t) and

36. (t) so that cos(t) = f1 + (t)g cos ft +

37. (t)g,
1 + (t) 0 and 1 +

38. 0(t) 0. This is known as the identi

39. ability problem studied in Chen
et al. (2014). To resolve this issue, it is necessary to restrict the functional class and we borrow
the following de

40. nition from Chen et al. (2014):
De

41. nition 1 (Intrinsic Mode Function Class Ac1;c2
). For

42. xed choices of 0 1 and
c1 c2 1, the space Ac1;c2
of Intrinsic Mode Functions (IMFs) consists of functions
f : R ! R, f 2 C1(R) L1(R) having the form
f(t) = a(t) cos f2(t)g ; (6)
7

43. where a : R ! R and : R ! R satisfy the following conditions for all t 2 R:
a 2 C1(R) L1(R); inf
t2R
a(t) c1; sup
t2R
a(t) c2; (7)
2 C2(R); inf
t2R
0(t) c1; sup
t2R
0(t) c2; (8)
ja0(t)j j0(t)j; j00(t)j j0(t)j: (9)
This de

44. nition describes functions, referred to as IMF, mainly satisfying two requirements.
First, the AM and IF are both continuously dierentiable and bounded from above and below.
Second, the rate of variation of both the AM and IF are small compared to the IF itself.
With the extra conditions on the IMF, the identi

45. ability issue can be resolved in the following
way (Chen et al., 2014, Theorem 2.1). Suppose f(t) = a1(t) cos f21(t)g 2 Ac1;c2
can be
represented in a dierent form, for instance f(t) = a2(t) cos f22(t)g, which also satis

46. es the
conditions of Ac1;c2
. De

47. ne (t) = 1(t) 2(t) and

48. (t) = a1(t) a2(t), then 2 C2(R),

49. 2 C1(R) and j0(t)j C, j(t)j C and j

50. (t)j C for all t 2 R for a constant C
depending only on c1. In a nutshell, if a member of Ac1;c2
can be represented in two dierent
forms, then the dierences in phase function, AM and IF between the two forms are controllable
by the small model constant . In this sense, we are able to de

51. ne AM and IF rigorously.
Because these conditions exclude jumps in AM and IF, in this paper, we make the working
assumption that the seasonality component of the market evolves slowly over time. As such,
we do not try to model abrupt changes. Note that the usual tests for structural breaks are
aimed at detecting if and where a break occurs, and they are not the modeling target of the
class Ac1;c2
. To asses its validity when modeling

52. nancial data, we assume that the seasonality
belongs to the Ac1;c2
class and resort to con

53. dence interval estimation.
Note that the model from equation (5) comprises more than one oscillatory components.
To resolve the identi

54. ability problem in this case, further conditions are necessary.
De

55. nition 2 (Adaptive seasonality model Cc1;c2
). The space Cc1;c2
of superpositions of IMFs
8

56. consists of functions f having the form
f(t) =
XK
k=1
fk(t) and fk(t) = ak(t) cos f2k(t) + kg (10)
for some

57. nite K 0 such that for each k = 1; : : : ;K,
k 2 R; ak(t) cos f2k(t) + kg 2 Ac1;c2
and (0) = 0: (11)
Functions in the class Cc1;c2
are composed of more than one IMF satisfying the condition
(11). In this case, an identi

58. ability theorem similar to the one for Ac1;c2
can be proved similarly
as that in (Chen et al., 2014, Theorem 2.2): if a member of Cc1;c2
can be represented in two
dierent forms, then the two forms have the same number of IMFs, and the dierences in their
phase function, AM and FM for each IMF are small. To summarize, with the Cc1;c2
model and
its identi

59. ability theorem, the IF and AM are well de

60. ned up to an uncertainty of order .
A popular member of Cc1;c2
is the Fourier Flexible Form (FFF) (see Gallant (1981)), intro-duced
by Andersen and Bollerslev (1997, 1998) in the context of intraday seasonality. Using
Fourier series to decompose any periodic function into simple oscillating building blocks, a
simpli

61. ed1 version of the FFF reads
s(t) =
XK
k=1
[ak cos (2kt) + bk sin (2kt)] ; (12)
where the sum of sines and cosines with integer frequencies captures the intraday patterns in the
volatility. Thus, we can view s(t) in (12) as a member of Cc1;c2
with (t) = t, k = arctan(ak=bk)
and ak(t) =
p
a2
k + b2
k for k = 1; : : : ;K, that is with a

62. xed daily oscillation.
1Andersen and Bollerslev (1997, 1998) also consider the addition of dummy variables to capture weekday
eects or particular events such as holidays in particular markets, but also unemployment reports, retail sales

63. gures, etc. While the periodic model captures most of the seasonal patterns, their dummy variables allow to
quantify the relative importance of calendar eects and announcement events.
9

64. 2.2. The adaptive trend model
For the remainder of this paper, we use Fy(!) =
R 1
1 y(t)ei2!tdt and Py(!) = jFy(!)j2 for
the Fourier transform and power spectrum of a weakly stationary process y.
Fix a Schwartz function so that suppF [1 ; 1 + ], where 0 1 and F
denotes its Fourier transform.
De

65. nition 3 (Adaptive trend model T c1
). For

66. xed choices of 0 1 and c1 1,
consists of functions T : R ! R, T 2 C1(R) so that FT exists in the distribution
the space T c1
sense, and

70. Z
T(t)
1
p
a
t b
a

74. dt
CT and

78. Z
T0(t)
1
p
a
t b
a

82. dt
CT (13)
for all b 2 R and a 2
0; 1+
c1
i
, for some CT 0.
As we ideally want a trend to be slowly time-varying, the intuition behind (13) is as a bound
on how fast a function oscillates locally. A special case satisfying (13) is a continuous function
T for which its Fourier transform FT exists and is compactly supported in
1
1+c1; 1
1+c1
.
There are two well-known examples of such trend functions; the

83. rst is the polynomial
function T(t) =
PL
l=1 ltl, where l 2 R, which is commonly applied to model trends. By a
direct calculation, its Fourier transform is supported at 0. The second is a harmonic function
T(t) = cos(2!t) with very low frequency (i.e., with j!j 1
1+c1), as its Fourier transform
is supported at !.
More generally, using Plancheral theorem, one can verify that suppFT
1
1+c1; 1
1+c1
implies
R
T(t)p1
a ( tb
a )(t)dt = 0 for all a 2 (0; 1+
c1
] and all b 2 R. In our case however,
the trend is non-parametric by assumption and its Fourier transform might in all generality
be supported everywhere in the Fourier domain. If this is so, then(13) describes a trend that
essentially captures the slowly varying features of those models (i.e., its local behavior is similar
to that of a polynomial or a very low frequency periodic function) but is more general.
10

84. 2.3. The adaptive volatility model
Let the return process be as in (4). We de

85. ne the log-volatility process as
yn = log jrnj :
Furthermore, we assume that the log-volatility process follows an adaptive volatility model,
that is
yn = Tn + sn + zn; (14)
where
sn, discretely sampled from s(t) 2 Cc1;c2
, is the volatility seasonality,
Tn, discretely sampled from T(t) 2 T c1
, is the volatility trend,
and zn = log

87. hnwn + qnIne(Tn+sn)

89. is an additive noise process that satis

90. es E(zn) 1
and var(zn) 1.
It should be noted that boundedness of the mean and variance for the noise is a mild condition
for reasonable econometric models (see the two remarks below).
Remark. Assume

91. rst there are neither jumps (In = 0) nor intraday heteroskedasticity (hn = 1)
and that wn follows a standardized generalized error distribution (GED), wn GED(), with
density given by
f(w) =
21+1=(1=)
ejw=j
2 ;
where =
22=(1=)=(3=)
1=2
and is the gamma function. As noted in Hafner
and Linton (2014), the GED nests the normal distribution when = 2, has fatter (thinner)
tails when 2 ( 2) and is log-concave for 1. Furthermore, because E(log w2n
) =
f2 (1=)= + log (1=) log (3)g and var (log w2n
) = (2=)2 (1=) (see Hafner and Linton
11

92. 2n
2014) with 2n
and the digamma and trigamma functions (i.e., the

93. rst and second derivatives
of log ), then E(zn) E(log w) 1 and var(zn) var (log w) 1 when 0.
Remark. To introduce intraday heteroskedasticity (still without jumps), it is convenient to use
the EGARCH(1,1) of Nelson (1991). De

94. ning vn = log jhnj, the model is obtained by writting
vn =
+

95. vn1 + wn1 + (jwn1j Ejwn1j) ;
where
;

96. ; ; 2 R and j

97. j 1. For this model, it is straightforward that
var
log h2
nw2n
=
2 + 2var (jwnj)
1

98. 2 + var
log w2n
:
Because var jwnj 1 for the GED (see Hafner and Linton (2014)), then var(zn) 1 follows.
For other heteroskedasticity models, note that
E
(log jhnwnj)2
q
E
(log jwnj)2
+
q
E
(log jhnj)2 2
by Cauchy-Schwartz inequality. Hence, if both E
(log jwnj)2
and E
(log jhnj)2
are bounded,
then var(zn) 1 follows. When jumps are added to the model, it is in general not possible
to obtain a closed formula for E(zn) and var(zn). However, boundedness of the mean and
variance is strongly supported by our simulations for cases of practical interest.
Within this framework, every speci

99. c choice of trend T, seasonality s and noise z yields
a dierent model. Due to their non-parametric nature, estimating T, s or the AM and IF
inside s is non-trivial. In practice, the problem would get much easier in the FFF case,
that is in the case of a constant seasonal pattern. However, this assumption is challenged
by empirical evidence. In Laakkonen (2007), the author observes that an FFF estimated
yearly, then quarterly and

100. nally weekly performs better and better at capturing periodicities
in the data. Whereas this sub-sampling is interpretable as a varying seasonality, it is still
piecewise constant. As a piecewise constant function implies that the system under study
12

101. undergoes structural changes (i.e., shocks) at every break point, it is an unlikely candidate for
the market's seasonality. We take the point of view of Deo et al. (2006), where the authors use
frequency leakage in the power spectrum of the absolute return to argue that the seasonality
is not (piecewise) constant but slowly varying. While they allow non-integer frequencies, their
generalized FFF parameters are still constant for the whole sample. In other words, they
correct for frequency leakage, but do not allow for potential slowly varying frequencies and
amplitude modulations (i.e., time-varying FFF parameters). Dynamic alternatives include
either arbitrarily small intervals or a rolling version of their generalized FFF. In the

102. rst
case, an arbitrarily increase in the estimator's variance (as in the issue of testing for a general
smooth member of Ac1;c2
) is expected. In the second case, which we compare to our method
in section 4, it is not clear a priori how one would choose the optimal window size.
In Section 3, we propose an alternative framework to study the time-frequency properties
of the log-volatility process. Allowing the trend and seasonality to evolve dynamically over
time, we are able to obtain T and s in the adaptive volatility model in a single-step. In fact,
if the log-volatility process yn is stationary or close to stationary in some sense, a transform
called the Synchrosqueezing transform (SST) theoretically helps to obtain accurate pointwise
estimates of T 2 T c1
and s 2 Ac1;c2
(Daubechies et al. (2011); Chen et al. (2014)).
3. From Fourier to Synchrosqueezing
In this Section, we start by recalling the relationship between the harmonic model and the
Fourier transform. We then sketch the theoretical basics of the time-frequency analysis. Fi-nally,
we present the Synchrosqueezing transform (SST), which is speci

103. cally aimed at studying
the adaptive volatility model. Its theoretical properties are discussed at the end of the Section.
3.1. Fourier transform and the harmonic model
Oscillatory signals are ubiquitous in many

104. elds. Seasonality is a synonym of oscillation broadly
used in

105. nance, econometrics, public health, biomedicine, etc. There are several character-
13

106. istics one can use to describe an oscillatory signal (see Flandrin (1999)): for example, how
often an oscillation repeats itself on average, how large an oscillation is on average, etc.
To capture these features, the following harmonic model is widely used:
y(t) =
XK
k=1
ak cos(2!kt + k); (15)
where
P
a2
k 1, ak 0, !k 0 and k 2 R. The quantities ak, !k and k are respectively
called the amplitude, the frequency and the initial phase of the k-th component of y. As we
stated in Section 2, both the FFF and its generalized version from Deo et al. (2006) are based
on this model.
When a given signal satis

107. es model (15), the Fourier transform is helpful in determining the
oscillatory features ak and !k. For y(t) de

108. ned in (15), we recall (see (Folland, 1999, Chapter
9)) that its Fourier transform is
Fy(!) =
1
2
XK
k=1
akeik(! !k) +
1
2
XK
k=1
akeik(! + !k);
where is the Dirac measure. Thus, if a signal satis

109. es the harmonic model, the frequency !k
and the amplitude ak can be read directly from its Fourier transform. For more details on the
property of the Fourier transform Fy(!) of a function y(t) with t; ! 2 R, we refer the reader
to (Folland, 1999, Chapter 9).
As useful as the harmonic model can be, it does not

110. t well many real signals. Indeed,
due to their underlying nature, the oscillatory pattern might change over time. For instance,
the signal may oscillate more often in the beginning but less in the end, or oscillate stronger
in the beginning but weaker in the end. We call these time-varying patterns the dynamics of
the seasonality, and it is well known that capturing the dynamics is beyond the scope of the
harmonic model. Unfortunately, when departing from static models such as (15) to consider
dynamic models such as the whole class Cc1;c2
, the usefulness of the Fourier transform is also
limited due to the loss of time localization of events. In other words, because of its incapacity
14

111. to capture momentary behavior, we cannot use the Fourier transform to analyze the dynamics
of seasonality.
3.2. Time-frequency analysis
To overcome the limitation of the Fourier transform, time-frequency analysis was introduced
(see Daubechies (1992); Flandrin (1999) and the references inside for the historical discussion).
Heuristically, to capture the dynamics of the oscillatory signal, we take a small subset of
the given signal y(t) around a given time t0, and analyze its power spectrum. Ideally, this
provides information about the local behavior of y(t) around t0. We repeat the analysis at
all times, and the result is referred to as the time-frequency (TF) representation of the signal.
Mathematically, this idea is implemented by the short-time Fourier transform (STFT) and
continuous wavelet transform (CWT).
The STFT is de

112. ned by
Gy(t; !) =
Z
y(s) g(s t) ei2!sds ; (16)
where ! 0 is frequency, and g is the window function associated with the STFT which
R
extracts the local signal for analysis. g has to be smooth enough and such that
g(t)dt = 1.
The CWT is de

113. ned by
Wy(t; a) =
Z
y(s) a1=2
s t
a
ds ; (17)
where a 0. We call a the wavelet's scale and the mother wavelet which is smooth enough
R
and such that
(t)dt = 0.
In a nutshell, we select a subset of the whole signal by the window function or the mother
wavelet, and study how the signal oscillates locally. From the viewpoint of frame analysis, the
STFT and CWT are de

114. ned as the inner products between the signal to be analyzed and a
pre-assigned family of templates, or atoms { which are fg( t)ei2!g!2R+;t2R in STFT
15

115. and fa1=2
t
a
ga2R+;t2R in CWT. Furthermore, this inner product provides a way to resolve
events simultaneously in time (by translation) and frequency (by modulation and dilatation
respectively), but the properties of their respective templates yield dierent trade-os between
temporal and frequency resolution due to the Heisenberg uncertainty principle. Take the high
frequency region for example. The STFT oers a better frequency resolution in frequency
events but an inferior resolution in time. In contrast, the CWT provides an inferior resolution
in frequency but a better resolution in time.
To further understand the time-frequency (TF) analysis, we consider the continuous wavelet
transform (CWT) of a purely harmonic signal y(t) = Acos(2!t), and the same reasoning
follows for the short-time Fourier transform (STFT). From the Plancheral theorem, the CWT
of y becomes
Wy(t; a) = Aa1=2F (!a)ei2!t: (18)
We observe that if F (!) is concentrated around ! = 1, then Wy(t; a) is concentrated around
a = 1=!.
In the remainder of this section, we focus on the CWT, and assume a continuous time
version of the adaptive volatility model (14). We consider
y(t) = T(t) + s(t) + z(t); (19)
where T 2 T c1
, s 2 Ac1;c2
and z a generalized random process with power spectrum Pz
satisfying
R
(1 + j!j)2ldPz(!) 1 for some l 0. We denote the discretely sampled time-series
by y = fyngn2Z, where yn = y(n) and 0 is the sampling interval.
In Figure 1, we show a simulated toy dataset to be used in the remainder of this section.
The contaminated signal is the sum of a trend, an amplitude modulated-frequency modulated-
16

116. 8
6
4
2
0
−2
0 5 10 15 20
Time (sec)
Signal
0
10
−2
10
−4
0 1 2 3 4 5
10
Frequency (1/t)
Spectrum
6
4
2
0
Trend
2
0
−2
AM / Seasonality
0 5 10 15 20
Time (sec)
Noise
Figure 1: Toy data. Left column, from top to bottom: the trend, the seasonality (thin line) with the amplitude
modulation (thick line) and the noise; Middle column: the contaminated signal, that is the sum of the three
components. Right column: the power spectrum of the contaminated signal.
periodic component and an additive GARCH(1,1) noise. More precisely, we consider
T(t) = 1 + 0:2t + 2e(t20)2
s(t) =
1 + cos ft=(2)g2
cos f2(t)g
(t) = t + t2=40
9=
; ) xn = T(n) + s(n) + zn;
where n 2 f1; ;Ng, N = 2000, = 1=100 and zn is a GARCH(1,1) de

117. ned by
zn = vnwn with
8
:
v2n
=
+ z2 n1 +

118. v2 n1
wn N(0; 1);
with parameters a = 0:1, b = 0:85 and
= 1=(1ab). Note that the instantaneous frequency
is 0(t) = 1 + t=20 which increases linearly with t (i.e., in this case, the number of oscillations
per unit of time is twice higher at the end).
Although one may suspect the existence of a trend and of some periodic component in the
17

119. right column of Figure 1, both the amplitude modulations and the increasing frequency are
rather dicult (if not impossible) to see. Furthermore, although the power spectrum of the
contaminated signal that we display in the right column of Figure 1 suggests events localized
both at very low frequencies and in the [1; 2] interval, the information displayed is insucient
to understand the dynamics of the system.
Using y, we can discretize the CWT for a 0 and n 2 Z as
Wy(n; a) =
X
m2Z
ym
a1=2
m n
a
:
In Figure 2, we show the CWT2 of y based on two dierent mother wavelets: the left panel
is based on the Morlet wavelet and the right panel is based on the Meyer wavelet. Unlike
the Fourier transform, it is now possible to observe the time-varying frequency properties of
the signal: the large increasing band on both CWTs corresponds to the periodic component.
Nonetheless, both pictures are blurred because the resolution in the time-frequency plane is
limited by Heisenberg's uncertainty principle. Furthermore, Figure 2 illustrates the fact that
dierent mother wavelets lead to dierent results. When comparing the left and right panels,
the choice of template colors the representation and therefore in
uences the interpretation
from which properties of the signal are deduced.
3.3. The Synchrosqueezing transform
In Kodera et al. (1978); Auger and Flandrin (1995); Chassande-Mottin et al. (2003, 1997);
Flandrin (1999), the reassignment technique was proposed to improve the time-frequency res-olution
and alleviate the coloration eect of the time-frequency analysis. The synchrosqueez-ing
transform (SST) is a special case of the reassignment technique originally introduced in
the context of audio signal analysis in Daubechies and Maes (1996). Its theoretical properties
are analyzed in Daubechies et al. (2011); Thakur et al. (2013); Chen et al. (2014), where it
2For more details about the matlab implementation of the CWT, available upon request from the authors,
we refer to Appendix A.
18

120. (a))
2
Scale (log
Morlet wavelet
5 10 15 20
Time (t)
10
9
8
7
6
5
4
3
2
Meyer wavelet
5 10 15 20
Time (t)
Figure 2: Continuous wavelet transform of the toy data. Morlet (left) and Meyer (right) wavelets.
is shown that the SST allows to determine T(t), ak(t) and 0(t) uniquely and up to some
pre-assigned accuracy. In brief, by reallocating the CWT coecients, the resolution of the
time-frequency plane is increased so that we are able to accurately extract the trend, the
instantaneous frequencies and the amplitude modulations.
The SST for y satisfying the continuous time adaptive volatility model (19) can be de

121. ned
as follows. Suppose the mother wavelet in equation (17) is such that suppF [1 ; 1 + ],
where 1 and evaluate the CWT of y. According to Daubechies et al. (2011), we extract
information about the IF 0(t) by
!y(t; a) =
8
:
i @tWy(t;a)
2Wy(t;a) Wy(t; a)6= 0
1 otherwise
: (20)
The motivation for (20) can be easily seen from the following example. For the purely harmonic
signal y(t) = cos(2!t), where ! 0, by (18) and the derivative of (18) with respect to time,
one obtains
!x(t; a) =
8
:
! a 2
1
! ; 1+
!
1 otherwise
;
19

122. which is the desired information about its IF. Then the SST is de

123. ned by
Sy(t; !) =
Z
a : jWy(t;a)j
Wy(t; a) a3=2 f !y(t; a) ! g da; (21)
where 0 is the threshold chosen by the user, ! 0 is the frequency and !y(t; a) is de

124. ned
in (20). Keeping in mind that the frequency is reciprocally related to the scale, then (21) can
be understood as follows: based on the IF information !y, at each time t, we collect the CWT
coecients at scale a where a seasonal component with frequency close to ! is detected. This
additional step allows to mitigate the coloration due to the choice of template.
To discretize the SST, !y is

125. rst obtained as
!y(n; a) =
8
:
i@tWy(n; a)
2Wy(n; a)
when jWy(n; a)j6= 0;
1 when jWy(n; a)j = 0;
where @tWy(n; a) is de

126. ned as
@tWy(n; a) =
X
m2Z
ym
a3=2
0m n
a
:
Similarly, Sy(t; !) is discretized as
Sy(n; !) =
Z
a: jWy(n;a)j
fj!y(n; a) !jgWy(n; a)a3=2da:
In Figure 3, we illustrate the analysis by SST3 of the toy data y. Whether we use the Morlet
wavelet or the Meyer wavelet, it is visually obvious that the Synchrosqueezed transform of the
noisy signal, with a darker area that seems to form a straight line from 1 to 2 in both cases, is
close to the instantaneous frequency 0(t) = 1 + t=20. Compared with the CWT, it is visually
obvious that Sy is concentrated in narrower bands around the curves in the TF plane de

127. ned
3For more details about the matlab implementation of the SST, available upon request from the authors,
we refer again to Appendix A.
20

128. Frequency (w)
Morlet wavelet
5 10 15 20
Time (t)
5
4
3
2
1
Meyer wavelet
5 10 15 20
Time (t)
Figure 3: Synchrosqueezed transform of the toy data. On the left: Morlet wavelet. On the right: Meyer
wavelet.
by (t; k0(t)) (here with k = 1).
Using any curve extraction technique, it is possible to estimate the IF with high accuracy
and we denote this quantity as b0. Furthermore, the restriction of Sy to the k-th narrow band
suces to obtain an estimator of the k-th (complex) component of y as
b fC
k (n) =
1
R
Z
jk b 0(n)!j
Sy(n; !)d!; (22)
where R =
R F (!)
! d!. Then the corresponding estimator of the k-th (real) component fk and
amplitude modulations at time n are de

129. ned as
b fk(n) =
n
b fC
o
and bak(n) =
k (n)

135. b fC
k (n)
; (23)
where (z) denotes the real part of z 2 C. Finally, the restriction of Sy under a low-frequency
21

136. 7
6
5
4
3
2
1
0
−1
0 2 4 6 8 10 12 14 16 18 20
Time (sec)
True function vs reconstructed
6
4
2
0
Trend
1.9
1.7
1.5
1.3
1.1
IF
2
0
−2
0 5 10 15 20
Time (sec)
AM / Seasonality
Figure 4: Reconstruction of the components. Left column, from top to bottom: the trend, the instan-
taneous frequency and the seasonality (thin line) with the amplitude modulation (thick line) with the true
underlying function (black) and reconstructed result (red); Right column: the true signal, that is the sum of
the trend and the seasonality, (black) and the reconstructed function (red).
threshold !l suces to obtain an estimator of the trend of y as
b T(n) =
1
R
Z !l
0
Sy(n; !)d!
(24)
= yn
1
R
Z 1
!l
Sy(n; !)d!
:
The theoretical properties of the above estimators have been shown in (Chen et al., 2014,
Theorem 3.2). Suppose y satis

137. es additional technical conditions on the second derivatives of
the amplitude modulations and trend4 in order to handle the discretization. If the sampling
interval satis

138. es 0 1
(1+)c2
, then with high probability b fk, b0
k and bak (respectively b T)
are accurate up to C (respectively C;T ) where C (respectively C;T ) is a constant decreasing
with . We refer the reader to Chen et al. (2014) for the precise statement of the theorem and
more discussions.
In Figure 4, we show the reconstructed component for the toy data using a Morlet mother
wavelet, where integration bands in (22) are the dashed red lines from Figure 3. Because the
4ak(t) 2 C2(R) and supt2R ja00
k(t)j c2 for all k = 1; : : : ;K, and T 2 C2(R) so that

141. R
T00(t)p1
a
tb
a
dt

144. CT for all b 2 R and a 2 (0; 1+
c1
]
22

145. reconstruction is not dependent on the chosen mother wavelets5, it is not shown here for the
Meyer wavelet. Usually, while the instantaneous frequency is recovered precisely, the estimated
amplitude modulations are less ideal. Although this is important when the goal is forecasting,
it can be dampened by further smoothing the amplitude (here with a simple moving average

146. lter of length 1=).
In summary, for the harmonic model (15), the Fourier transform allows us to extract the
features of interest of the oscillation. When it comes to the adaptive volatility model (14), we
count on the synchrosqueezing transform to extract the oscillatory features of a given signal.
4. Application to foreign exchange data
As an empirical application, we study the EUR/USD, EUR/GBP, JPY/USD and USD/CHF
exchange rates from the beginning of 2010 to the end of 2013. We avoid the data preprocessing
by using pre-

147. ltered 5-minute bid-ask pairs obtained from Dukascopy Bank SA6. The log-price
pn is computed as the average between the log bid and log ask prices and the return rn is
de

148. ned as the change between two consecutive log prices.
Because trading activity unquestionably drops during weekends and certain holidays (see
e.g. Dacorogna et al. 1993; Andersen and Bollerslev 1998), we exclude a number of days from
the sample, always from 21:05 GMT the night before until 21:00 GMT that evening. For
instance, weekends start on Friday at 21:05 GMT and last until Sunday at 21:00 GMT. Using
this de

149. nition of the trading day, we remove the

150. xed holidays of Christmas (December 24 -
December 26) and New Year's (December 31 - January 2), as well as the moving holidays of
Martin Luther King's Day, President's Day, Good Friday, Memorial Day, July Fourth (also
when it is observed on July 3 or 5), Labor Day and Thanksgiving. After the cuts, we obtain
a sample of 998 days of data, containing 288 998 = 287424 intraday returns.
5This statement is correct up to the moments of the chosen mother wavelet . The reconstruction error
depends on the

151. rst three moments of the chosen mother wavelet and its derivative but not on the pro

152. le of
(Daubechies et al. (2011); Chen et al. (2014))
6http://www.dukascopy.com/
23

153. In panel 5a of Figure 5, we show the return rn for the four considered exchange rates
for the

154. rst week of trading, that is from January 3, 2010 (Sunday) at 21:05 pm GMT to
January 8, 2010 (Friday) at 21:00 pm GMT (1440 observations). While heteroskedasticity is
obviously present, it is arguably easier to discern its periodicity by looking at the log-volatility
yn = log jrn bj in panel 5b. Note that the smallest observations correspond to the case where
the price does not move during the 5 minutes interval (i.e., rn = 0 and yn = log jbj).
In Figure 6, the autocorrelation (Panel 6a) and the power spectrum (Panel 6b) of the log-volatility
indicate strong (daily) periodicities in the log-volatility. Although the shapes are
dierent for the various exchange rates, they all essentially display the same information: on
average, the seasonality is composed of periodic components of integer frequency. Note that
on average emphasizes the fact that both representations are only part of the picture. As
illustrated in Section 3, this representation misses the dynamics of the signal: for example, some
of the underlying periodic components may strengthen, weaken or even completely disappear
at some point. Another observation is that the JPY/USD curves are fairly dierent from the
others. First, according to the autocorrelation, the overall magnitude of oscillations (i.e., the
importance of the daily periodicity) is smaller. Second, according to the height of the power
spectrum's second peak, the amplitude of the second component is much larger.
In Figure 7, we display jSy(t; !)j, the absolute value of the Synchrosqueezed transform,
in shades of gray for frequencies 0:5 ! 4:5. For all exchange rates, we observe much
darker curves (indicating higher values of jSx(t; !)j) at integer frequencies. As expected from
the power spectrums in Figure 6, the curves at 2 daily oscillations for all exchange rates
are very thin and sometimes fading, except for the JPY/USD. This apparent linearity of the
instantaneous frequency at integer values lends support for an amplitude-modulated version
of the FFF or, in other words, a member of Ac1;c2
such that j(t)j = t. Although this kind
of constant frequency/amplitude modulated model can be estimated in dierent ways, we
carry the analysis within the SST framework. The red lines are the integration bands for the
reconstruction of the k-th periodic component using (22) and (23), that is at ! = k with
k 2 f1; 2; 3; 4g and = 0:02.
24

155. 4
2
0
−2
−4
−3 CHFUSD
x 10
Return
Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08
4
2
0
−2
−4
−3 EURUSD
x 10
Return
Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08
4
2
0
−2
−4
−3 GBPUSD
x 10
Return
Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08
4
2
0
−2
−4
−3 JPYUSD
x 10
Return
Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08
(a) Return rn.
−15
−20
−25
−30
CHFUSD
Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08
Log−volatility
−15
−20
−25
−30
EURUSD
Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08
Log−volatility
−15
−20
−25
−30
GBPUSD
Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08
Log−volatility
−15
−20
−25
−30
JPYUSD
Mon 01−04 Tue 01−05 Wed 01−06 Thu 01−07 Fri 01−08
Log−volatility
(b) Log-volatility yn = 2 log jrn bj.
Figure 5: First week of returns and log-volatilities in January 2010. Top: CHF/USD (left) and
EUR/USD (right); Bottom: GBP/USD (left) and JPY/USD (right).
25

156. 1 2 3 4 5
0.15
0.1
0.05
0
−0.05
CHFUSD
Lag (day)
Autocorrelation
1 2 3 4 5
0.15
0.1
0.05
0
−0.05
EURUSD
Lag (day)
Autocorrelation
1 2 3 4 5
0.15
0.1
0.05
0
−0.05
GBPUSD
Lag (day)
Autocorrelation
1 2 3 4 5
0.15
0.1
0.05
0
−0.05
JPYUSD
Lag (day)
Autocorrelation
(a) Autocorrelation
y(l). The lag l is from 1 to 1440 and the x axis ticks are divided by 288.
1 2 3 4 5
2
10
0
10
−2
10
−4
10
CHFUSD
Frequency (1/day)
Power spectrum
1 2 3 4 5
2
10
0
10
−2
10
−4
10
EURUSD
Frequency (1/day)
Power spectrum
1 2 3 4 5
2
10
0
10
−2
10
−4
10
GBPUSD
Frequency (1/day)
Power spectrum
1 2 3 4 5
2
10
0
10
−2
10
−4
10
JPYUSD
Frequency (1/day)
Power spectrum
(b) Power spectrum Py(!). The frequency ! is from 0 to 5.
Figure 6: Autocorrelation
y(l) and power spectrum Py(!) of the log-volatility.
y(l) =
bE
hn
o n
yn b E(yn)
oi
ynl b E(yn)
and Py(!) =

162. b Fy(!)
2
. Top: CHF/USD (left) and EUR/USD (right);
Bottom: GBP/USD (left) and JPY/USD (right).
26

163. Figure 7: Synchrosqueezed transform Sy(t; !). The Synchrosqueezed transform is displayed in shades of
gray and the red lines are the integration bands selected for the reconstruction.
In Figures 8, 9 and 10, we show the reconstruction results and compare them to more
traditional estimators. Figure 8 displays the results for the trend, that is the slowly time-varying
part of the spot volatility:
b T is estimated using (24) with a frequency threshold !l = 0:95 daily oscillations and
an upper integration bound corresponding to the highest frequency that can be resolved
according to the Nyquist theorem (i.e., half of the sampling frequency 1=2 1= = 144).
The realized volatility and bipower variation (see Barndor-Nielsen (2004)) are simply
computed as rolling moving averages of r2n
and j rn jj rn1 j with 288 and 287 observations
included (except near the borders) and maximal overlap.
In Panel 8a, our reconstructed trend is compared to the usual realized volatility and
bipower variation over the whole 2010-2013 period. Note that the estimates are only
represented between 0 and 103 for the sake of visual clarity, but that a few jumps in
the CHF/USD and JPY/USD exchange rates generate peaks well above the upper limit.
27

164. In Panel 8b, the three trend estimates are compared over a shorter time period in the
summer of 2011, that is in the middle of United States debt-ceiling crisis/European
sovereign debt crisis of 2011. Our reconstructed trend visibly constitutes a smoothed
version of the usual volatility measures, which usually fall inside the 95% con

165. dence
interval except for upward spikes due to jumps. Notice the steep volatility increase
in all exchange rates as stock markets around the worlds started to fall following the
downgrading of America's credit rating on 6 August 2011 by Standard Poor's. This
is especially true for the Swiss Franc, facing increasing pressure from currency markets
because considered a safe investment in times of economic uncertainty.
Figures 9 and 10 display the results for the amplitude modulations and seasonality:
bs is estimated using (22) and (23), as well as bs(n) =
PK
k=1
b fk(n).
The Fourier Flexible Form/rolling Fourier Flexible Form (FFF/rFFF) are estimated us-ing
the ordinary least-square estimator on the whole sample/a four-weeks rolling window.
In Figure 9, we show the amplitude modulations of the

166. rst periodic component. It is
noteworthy that when a unique FFF is estimated for the whole sample, the correspond-ing
amplitude is very close to the mean amplitude obtained with the SST. Furthermore,
we observe that the rFFF closely tracks the SST but is far wigglier. This was expected
because the SST estimator is essentially an instantaneous (and smooth by construction)
version of the FFF. Finally, notice that, as expected from Figure 6, the amplitude mod-ulations
of the

167. rst component in the JPY/USD are much smaller than the three other
exchange rates.
In Panels 10a and 10b of Figure 10, we show the seasonality estimated as a sum of the
four periodic components: the FFF represents the average daily oscillation, the rFFF
is a wiggly estimate of the dynamics and the SST extracts a smooth instantaneous
seasonality. As in Figure 9, we observe that the overall magnitude of oscillation is much
smaller in the JPY/USD exchange rate.
28

168. 0.8
0.6
0.4
0.2
−3
x 10
2010 2011 2012 2013
Trend
CHFUSD
exp(T)
RV
BV
0.8
0.6
0.4
0.2
−3
x 10
2010 2011 2012 2013
Trend
EURUSD
exp(T)
RV
BV
0.8
0.6
0.4
0.2
−3
x 10
2010 2011 2012 2013
Trend
GBPUSD
exp(T)
RV
BV
0.8
0.6
0.4
0.2
−3
x 10
2010 2011 2012 2013
Trend
JPYUSD
exp(T)
RV
BV
(a) Trend reconstruction for 2010-2013.
0.8
0.6
0.4
0.2
−3
x 10
Trend
CHFUSD
exp(T)
RV
BV
Jul 2011 Aug 2011 Sep 2011
0.8
0.6
0.4
0.2
−3
x 10
Trend
EURUSD
exp(T)
RV
BV
Jul 2011 Aug 2011 Sep 2011
0.8
0.6
0.4
0.2
−3
x 10
Trend
GBPUSD
exp(T)
RV
BV
Jul 2011 Aug 2011 Sep 2011
0.8
0.6
0.4
0.2
−3
x 10
Trend
JPYUSD
exp(T)
RV
BV
Jul 2011 Aug 2011 Sep 2011
(b) Zoom during the summer of 2011.
Figure 8: Trend reconstruction results. In each panel: reconstructed trend (black line), realized volatility
(red line) and bipower variation (blue line). In Panel 8b: 95% con

169. dence intervals for the reconstructed trend
(shaded area). Top: CHF/USD (left) and EUR/USD (right); Bottom: GBP/USD (left) and JPY/USD (right).
29

170. 1.4
1
0.6
0.2
2010 2011 2012 2013
Amplitude 1
CHFUSD
SST
rFFF
FFF
1.4
1
0.6
0.2
2010 2011 2012 2013
Amplitude 1
EURUSD
SST
rFFF
FFF
1.4
1
0.6
0.2
2010 2011 2012 2013
Amplitude 1
GBPUSD
SST
rFFF
FFF
1.4
1
0.6
0.2
2010 2011 2012 2013
Amplitude 1
JPYUSD
SST
rFFF
FFF
Figure 9: Amplitude modulations reconstruction results. In each panel: reconstructed amplitude mod-
ulations of the

171. rst component (black line) with 95% con

172. dence intervals (shaded area), Fourier Flexible Form
(dashed line) and rolling Fourier Flexible Form (red line). Top: CHF/USD (left) and EUR/USD (right);
Bottom: GBP/USD (left) and JPY/USD (right).
Remark. In Chen et al. (2014), the authors give theoretical bounds on the error committed
when reconstructing the trend and seasonality with the SST. However, those bounds have
two disadvantages: they are both very rough and impractical to implement. To obtain the
con

173. dence intervals displayed in Figures 8, 9 and 10, we used the following procedure:
1. Use b T(n) and b fk(n) to recover the residuals
bzn = yn b T(n)
XK
k=1
b fk(n)
for n 2 f1; ;Ng.
2. Bootstrap B samples zbn
for b 2 f1; ;Bg and n 2 f1; ;Ng.
3. Add back b T(n) and b fk(n) to zbn
in order to obtain
ybn
= zbn
+ b T(n) +
XK
k=1
b fk(n)
30

174. 1.5
1
0.5
0
−0.5
−1
−1.5
11−07−11 11−07−12 11−07−13 11−07−14 11−07−15
Seasonality
CHFUSD
SST
rFFF
FFF
1.5
1
0.5
0
−0.5
−1
−1.5
11−07−11 11−07−12 11−07−13 11−07−14 11−07−15
Seasonality
EURUSD
SST
rFFF
FFF
1.5
1
0.5
0
−0.5
−1
−1.5
11−07−11 11−07−12 11−07−13 11−07−14 11−07−15
Seasonality
GBPUSD
SST
rFFF
FFF
1.5
1
0.5
0
−0.5
−1
−1.5
11−07−11 11−07−12 11−07−13 11−07−14 11−07−15
Seasonality
JPYUSD
SST
rFFF
FFF
(a) Second week of July 2011.
1.5
1
0.5
0
−0.5
−1
−1.5
11−08−08 11−08−09 11−08−10 11−08−11 11−08−12
Seasonality
CHFUSD
SST
rFFF
FFF
1.5
1
0.5
0
−0.5
−1
−1.5
11−08−08 11−08−09 11−08−10 11−08−11 11−08−12
Seasonality
EURUSD
SST
rFFF
FFF
1.5
1
0.5
0
−0.5
−1
−1.5
11−08−08 11−08−09 11−08−10 11−08−11 11−08−12
Seasonality
GBPUSD
SST
rFFF
FFF
Seasonality JPYUSD
1.5
1
0.5
0
−0.5
−1
−1.5
11−08−08 11−08−09 11−08−10 11−08−11 11−08−12
SST
rFFF
FFF
(b) Second week of August 2011.
Figure 10: Seasonality reconstruction results. In each panel: reconstructed seasonality (black line) with
95% con

175. dence intervals (shaded area) and rolling Fourier Flexible Form (red line). Top: CHF/USD (left) and
EUR/USD (right); Bottom: GBP/USD (left) and JPY/USD (right).
31

176. for b 2 f1; ;Bg and n 2 f1; ;Ng.
4. Estimate b Tb(n) and b fb
k(n) for b 2 f1; ;Bg and n 2 f1; ;Ng to compute pointwise
con

177. dence intervals.
As bzn are found to be autocorrelated, a naive bootstrapping scheme is not appropriate. To
deal with this dependence issue, we use the automatic block-length selection procedure from
Politis and White (2004) along with circular block-bootstrap. In our case, the pointwise boot-strapped
distribution of the estimates is found to be approximately normal. However, a bias
correction proves to be necessary, which is achieved for instance for the trend by substracting
Pthe bootstrapped bias
B
b=1
b Tb(n)=B b T(n) from the estimate b T(n).
In summary, an impressing feature of our estimated trend is its robustness to jumps, which
aect both the realized volatility and, albeit to a lesser extent, the bipower variation. Regarding
the seasonality, Figures 9 and 10 strongly suggest that it evolves dynamically over time. Finally,
although similar estimates of trend and seasonality can be obtained with moving averages and
rolling regressions respectively, the SST provides smooth estimates and does not require an
arbitrary choice of the length of the window or the rolling overlap. One could argue that the
choice of mother wavelet is also arbitrary, but as illustrated in Section 2, the in
uence of this
choice on the SST estimators is negligible. In any case, the main message is that dynamic
methods are necessary to properly understand the seasonality, as static models can lead to
severe underestimation/overestimation of the intraday spot volatility.
5. Simulation study
In this section, we present simulations that use a realistic setting. To study the properties of the
estimators from Section 3, we directly sample 1000 times 150 days of data (i.e., 288150 = 43200
observations) from the adaptive volatility model from equation (14), that is
yn = Tn + sn + zn;
32

178. where Tn and sn are deterministic functions and
zn = log

180. hnwn + qnIne(Tn+sn)

182. with hn, wn and qnIn as described in Section 2. Using the EUR/USD sample (cf. Section 4, for
more details), we collect the

183. rst 150 days of extracted trend and seasonality, the latter either
with constant amplitude (estimated with the FFF) or with amplitude modulations (estimated
with the SST). Furthermore, we use the exponential of the estimated residuals (see Section
4) to obtain realistic parameters to simulate hnwn from either a GED() (with hn = 1), an
EGARCH(1,1) or a GARCH(1,1). Finally, for the sake of simplicity, we assume that the two
components of the jump process are
qn = eTn+snjn, where jn N(0; j) with j 2 f4w; 10wg with w the standard deviation
of the estimated i.i.d. GED(),
and In B(1; ) data) with 2 f1=288; 1=1440g, that is one jump per day or week on
average.
Note that the assumption of proportionality to eTn+sn does not aect the following results.
In Figure 11, we show the results of a preliminary simulation study for both the trend and
seasonality with amplitude modulations when the noise is GED/GARCH with/without jumps,
where the case with is the most extreme (one 10 sigma jump per day). A striking feature
of the two panels is that there is basically no visible dierence between the four kinds of noise
considered. Although not reported here, the pointwise biases and variances for the trend and
seasonality were also studied. While the biases were again indistinguishable (and basically
negligible for the trend) between the GED/GARCH with/without jumps, the variances were
higher for the GARCH with no visible eect from the jumps.
33

184. 30 60 90 120
4
3
2
1
−4
x 10
Days
Trend
w 9 GED, no jumps
estimated exp(trend)
true exp(trend)
30 60 90 120
4
3
2
1
−4
x 10
Days
Trend
w 9 GED, 6j = 1/288, j = 10w
estimated exp(trend)
true exp(trend)
30 60 90 120
4
3
2
1
−4
x 10
Days
Trend
w 9 GARCH, no jumps
estimated exp(trend)
true exp(trend)
30 60 90 120
4
3
2
1
−4
x 10
Days
Trend
w 9 GARCH, 6j = 1/288, j = 10w
estimated exp(trend)
true exp(trend)
(a) Trend for the whole (simulated) data.
1 2 3 4
1
0
−1
Days
Seasonality
w 9 GED, no jumps
estimated seasonality
true seasonality
1 2 3 4
1
0
−1
Days
Seasonality
w 9 GED, 6j = 1/288, j = 10w
estimated seasonality
true seasonality
1 2 3 4
1
0
−1
Days
Seasonality
w 9 GARCH, no jumps
estimated seasonality
true seasonality
1 2 3 4
1
0
−1
Days
Seasonality
w 9 GARCH, 6j = 1/288, j = 10w
estimated seasonality
true seasonality
(b) Seasonality for the

185. rst week of the (simulated) data.
Figure 11: Trend and seasonality preliminary simulation results. In each panel: mean estimate (black
line), true quantity (red line) and estimated 95% con

186. dence intervals (shaded area).
34

187. To compare the results quantitatively, we use two criteria, namely the average integrated
squared error/absolute error (AISE/AIAE), de

188. ned as
AISE = bE
Z 150
0
fbx(t) x(t)g2 dt
and AIAE = bE
Z 150
0
;
j bx(t) x(t) j dt
where the expectation is taken over all simulations for each component x 2 fT; sg. The results
presented in Table 1 can be summarized as follows:
As observed in Figure 11, the jumps do not aect the results, but the choice of process
for hnwn does. Furthermore, the eect of this choice is comparatively more pronounced
on the AISE, which is a measure of the total estimator's variance.
There is no relation between the seasonality dynamics and the trend's estimate; whether
the former is with constant amplitude or with amplitude modulations, the AISE/AIAE
are similar. However, for the seasonality, the AISE/AIAE are higher in the latter case.
Constant Amplitude GED EGARCH(1,1) GARCH(1,1)
No jumps
AISE 1:50=0:30 1:97=0:49 1:83=0:44
AIAE 11:36=5:38 13:25=6:80 12:71=6:47
= 1=288, j = 4w
AISE 1:48=0:30 1:95=0:49 1:81=0:44
AIAE 11:27=5:39 13:16=6:80 12:64=6:47
= 1=1440, j = 4w
AISE 1:49=0:30 1:97=0:49 1:82=0:44
AIAE 11:34=5:38 13:23=6:80 12:69=6:47
= 1=288, j = 10w
AISE 1:47=0:31 1:94=0:49 1:81=0:44
AIAE 11:24=5:41 13:14=6:82 12:63=6:49
= 1=1440, j = 10w
AISE 1:49=0:30 1:96=0:49 1:82=0:44
AIAE 11:33=5:38 13:22=6:80 12:69=6:47
Amplitude modulations GED EGARCH(1,1) GARCH(1,1)
No jumps
AISE 1:45=0:48 1:93=0:68 1:79=0:63
AIAE 11:20=6:66 13:14=7:95 12:60=7:64
= 1=288, j = 4w
AISE 1:43=0:48 1:91=0:68 1:77=0:63
AIAE 11:11=6:67 13:06=7:96 12:53=7:65
= 1=1440, j = 4w
AISE 1:45=0:48 1:93=0:68 1:78=0:63
AIAE 11:18=6:66 13:12=7:95 12:58=7:65
= 1=288, j = 10w
AISE 1:42=0:49 1:90=0:68 1:77=0:63
AIAE 11:09=6:68 13:04=7:97 12:52=7:66
= 1=1440, j = 10w
AISE 1:44=0:48 1:92=0:68 1:78=0:63
AIAE 11:17=6:66 13:12=7:96 12:58=7:65
Table 1: Trend and seasonality complete simulation results. In each cell: trend (left) and seasonality
(right).
35

189. 6. Discussion
Our disentangling of instantaneous trend and seasonality of the intraday spot volatility is an
extension of classical econometric models that provides adaptivity to ever changing markets.
The proposed method suggests a realistic framework for the high frequency time series behav-ior.
First, it lets the realized volatility be modeled as an instantaneous trend which evolves in
real time within the day. Second, it allows the seasonality to be non-constant over the sample.
In a simulation study using a realistic setting, numerical results con

190. rm that the proposed
estimators for the trend and seasonality components have an appropriate behavior. We show
that this result holds even in presence of heteroskedastic and heavy tailed noise and is also
robust to jumps.
Using the CHF/USD, EUR/USD, GBP/USD and USD/JPY exchange rates sampled every
5 minutes between 2010 and 2013, we con

191. rm empirically that the oscillation frequency is
constant, as suggested originally in the Fourier Flexible Form from Andersen and Bollerslev
(1997, 1998). In the four exchange rates, we show that the amplitude modulations of the
periodic components and hence the overall magnitude of oscillation evolve dynamically over
time. As such, neglecting those modulations in the periodic part of the volatility would imply
either an overestimation (when the periodic components are lower than their mean value)
or an underestimation (when the periodic components are higher than their mean value).
Furthermore, we show that the SST estimator gives results comparable to a smooth version of
a rolling Fourier Flexible Form. However, the adaptivity of SST should still be emphasized,
as it does not require ad-hoc choices neither of the rolling overlapping ratio nor of the length
of each window as for a parametric regression.
While we illustrate our model by simultaneously disentangling the low and high frequency
components of the intraday spot volatility in the FX market, it is possible to embed the new
methodology into a forecasting exercise. Although this is beyond the scope of the present paper,
which is rather aimed at presenting the modeling framework, the exercise may be useful in
the context of an investor's optimal portfolio choice or for risk-management purposes. Further
36

192. research directions include the extension of the framework to both the non-homogeneously
sampled (tick-by-tick) as well as multivariate data. We will return to these questions and
related issues in future works.
Acknowledgements
Most of this research was conducted while the corresponding author was visiting the Berkeley
Statistics Department. He is grateful for its support and helpful discussions with the members
both in Bin Yu's research group and the Coleman Fung Risk Management Research Center.
This research is also supported in part by the Center for Science of Information (CSoI), a US
NSF Science and Technology Center, under grant agreement CCF-0939370, and by NSF grants
DMS-1160319 and CDSE-MSS 1228246.
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Appendix A. Implementations details
In this section we provide the numerical SST implementation detail. The Matlab code is
available from the authors upon request and we refer the readers to Thakur et al. (2013) for
more implementation details.
We

198. x a discretely sampled time series y = fyngN n=1, where xn = x(n) with the sampling
40

199. interval and N = 2L for L 2 N+. Note that we use the bold notation to indicate the numerical
implementation (or the discrete sampling) of an otherwise continuous quantity.
Step 1: numerically implement Wy(t; a). We discretize the scale axis a by aj = 2j=nv ,
j = 1; : : : ; Lnv, where nv is the voice number chosen by the user. In practice we choose
nv = 32. We denote the numerical CWT as a N na matrix Wy. This is a well studied step
and our CWT implementation is modi

200. ed from that of wavelab7.
Step 2: numerically implement !y(a; t).
The next step is to calculate the IF information function !y(a; t) (20). The @tWy(a; t) term
is implemented directly by

201. nite dierence at t axis and we denote the result as a N na
matrix @tWy. The !y(a; t) is implemented as a N na matrix wy by the following entry-wise
calculation:
wx(i; j) =
8
:
i@tWy(i;j)
2Wy(i;j) when Wy(i; j)6= 0
NaN when Wy(i; j) = 0:
;
where NaN is the IEEE arithmetic representation for Not-a-Number.
Step 3: numerically implement Sy(t; !).
We now compute the Synchrosqueezing transform Sy (21). We discretize the frequency
domain [ 1
N ; 1
2 ] by equally spaced intervals of length ! = 1
N . Here 1
N and 1
2 are the minimal
and maximal frequencies detectable by the Fourier transform theorem. Denote n! = b
1
2
1
N
!
c,
which is the number of the discretization of the frequency axis. Fix
0, Sy is discretized as
a N n! matrix Sy by the following evaluation
Sy(i; j) =
X
k:jwy(i;k)j!j!=2; jWy(i;j)j
p
aj
log(2)
!nv
Wy(i; k);
where i = 1; : : : ;N and j = 1; : : : ; n!. Notice that the number
is a hard thresholding
parameter, which is chosen to reduce the in
uence of noise and numerical error. In practice we
simply choose
as the 0:1 quantile of j Wy j. If the error is Gaussian white noise, the choice
7http://www-stat.stanford.edu/~wavelab/
41

202. of
is suggested in Thakur et al. (2013). In general, determining how to adaptively choose
is an open problem.
Step 4: Estimate IF, AM and trend from Sy.
We

203. t a discretized curve c 2 ZN
n! , where Zn! = f1; : : : ; n!g is the index set of the
discretized frequency axis, to the dominant area of Sy by maximizing the following functional
over c 2 ZN
n! :
hXN
m=1
log
jSy(m; cm)j
Pn!
i=1
PN
j=1 jSy(j; i)j
!
XN
m=2
jcm cm1j2
i
;
where 0. The

204. rst term is used to capture the maximal value of Sy at each time and
the second term is used to impose regularity of the extracted curve. In other words, the user-de

205. ned parameter determines the smoothness of the resulting curve estimate. In practice
we simply choose = 10. Denote the maximizer of the functional as c 2 RN. In that case,
the estimator of the IF of the k-th component at time t = n is de

206. ned as
0
k(n) := c(n)!;
k 2 RN. With c, the k-th component ak(t) cos(2k(t)) and its AM, ak(t) at time
where 0
t = n are estimated by:
fk(n) :=
2
R !
c(n)+Xb=!c
i=c(n)b=!c
Sy(n; i);
ak(n) :=

212. 2
R !
c(n)+Xb=!c
i=c(n)b=!c

218. Sy(n; i)
;
where is the real part, fk 2 RN and ak 2 RN. Lastly, we estimate the trend T(t) at time
t = n by
T (n) := yn
2
R !
Xn!
i=b!l=!c
Sy(n; i);
where T 2 RN and !l 0 is determined by the model.
42