Comput Econ DOI 10.1007/s10614-015-9533-4

On the Historical Exchange Rates Euro/US Dollar Fernando Vadillo1

Accepted: 6 October 2015 © Springer Science+Business Media New York 2015

Abstract In this paper we study the exchange rate between the Euro and the US Dollar between January 3, 2003 until September 18, 2014. First we research if these exchanges follow a geometric Brownian motion, i.e. it is that future exchange rates are independent of past movements; our results are consistent with this assumption and then we make several estimates of the volatility. In the second part of the paper, we study possible periodic motions: first using discrete Fourier transform and later wavelet analysis. The techniques used are easy to understand and express, and can be implement in a transparent way by means of a few lines of code in Matlab. They can be used to understand behaviors that would seem chaotic. Keywords Exchange rate Euro-Dollar · Black–Scholes model · Volatility · Fourier analysis · Wavelet analysis Mathematics Subject Classification

92B05

1 Introduction With day 0 defined to be 3 January 2000, let Rn , n = 0, . . . , 3701 denote the exchange rate between the Euro (EUR) and the US Dollar (USD), the last day is the 19 September 2014. The values of Rn are plotted in Fig. 1, this picture seem to suggest that the points lie on a continuous jagged curve, remembering the Brownian motions of

B 1

Fernando Vadillo [email protected] Department of Applied Mathematics and Statistics and Operations Research, University of the Basque Country (UPV/EHU), Leioa, Spain

123

F. Vadillo 1.6 1.5 1.4

Rates

1.3 1.2 1.1 1 0.9 0.8 Jan 2000

Dec 2001

Dec 2003

Dec 2005

Dec 2007

Dec 2009

Dec 2011

Dec 2013

Fig. 1 Historical Exchange Rates EUR/USD

the literature Ross (1999), Allen (2007), Klebaner (2005), Øksendal (2010), Higham (2000), Roberts (2009). This paper seeks to explain this exchange rate dynamic. We do not know many studies on this subject in recent years, perhaps the most interesting and original paper is Federici and Gandolfo (2012), where the authors reject the possibility of chaotic dynamics, while for example Rabanal and Tuesta (2010) tries to adapt the exchange rate between two countries using a newer generation of models known as the new open economy macroeconomics (NOEM). This paper is organized as two parts: the first in Sect. 2 describes a brief review of the GBM, and we study if the sequence Rn follow this type of model testing   the contingency table for the successive differences of the logarithms Dn = log Rn /Rn−1 ; because our results are consistent with this assumption, we computer the historial volatility. In second part in Sect. 3 we present the discrete Fourier transform and computer the periodigram for the sequence Rn in order detecting possible periods; the bad results suggest use wavelet analysis in Sect. 3.2, despite the wavelets still remain largely unfamiliar to students of economics and finance, in the past decade considerable progress has been made in finance (see for example Gallegati and Semmler 2014), in particular the techniques used is that used in Gallegati et al. (2014) to study the effect of increased productivity on unemployment for US or more recently Shahbaza et al. (2015) to analyze the time-frequency relationship between oil price and exchange rate in Pakistan. The final section summarizes the results that we consider most important. Our numerical methods were implemented in Matlab© (see for example Higham and Higham 2000; Moler 2004 for a simple introduction); the experiments were carried out in an Intel(R) Core(TM)2 Duo CPU E6850 @ 3.00GHz. The codes for the numerical tests and for this example are available on request.

2 The Geometric Brownian Motion Model The famous Black-Scholes model was the starting point of a new financial industry, and it has been a very important pillar of options trading since then. Let S(t) denote the

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On the Historical Exchange Rates Euro/US Dollar

value of an asset at time t that may take any non-negative value. In the Black-Scholes model for asset price behavior (see Black and Scholes 1973, Chap. 3 of Ross 1999 and Chaps. 6 and 7 of Higham 2000), S(t) follows a Geometric Brownian Motion (GBM) satisfying a linear SDE: d S(t) = μS(t)dt + σ S(t)dW (t),

(1)

where μ and σ are two non-negative constants, μ is called the drift, and σ is the volatility that quantifies the risk. It is possible to solve exactly (1) by means of Itô’s formula with F(S, t) = ln(S) (see for instance Allen 2007; Øksendal 2010; Klebaner 2005; Higham 2000). Its solution is S(t) = S(0) exp (βt + σ W (t)) ,

(2)

where β = μ − σ 2 /2, and its expectation is E(S(t)) = S(0) eβt .

(3)

2.1 Testing the Contingency Table In the GBM model (1), the ratios S(t + dt)/S(t) is independent of the prices up to time t, and in addition the successive differences of the logarithms  log

 S(t + dt) , S(t)

is a normal variable with mean β dt and variance σ 2 dt. Let define the successive differences of the logarithms  Rn , for n > 1. Dn = log Rn−1 

In order to know if Rn follow a GBM, let us classify each day as being in one of the four possible states as follows: State 1: State 2: State 3: State 4:

If If If If

Dn ≤ −0.01, −0.01 < Dn ≤ 0, 0 < Dn ≤ 0.01, 0.01 < Dn .

Because Dn ≤ −0.01 ⇔

Rn ≤ e−0.01 ≈ 0.9900, Rn−1

nth is in State 1 if represents a loss of more than 1 per cent; it is in State 2 if percentage loss is less than 1 per cent; in the State 3 if percentage gain less than 1 per cent (e0.01 ≈ 1.0101); in 4 gain more than 1 per cent.

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F. Vadillo Table 1 Contingency tables i

j

Total

1

2

3

i

4

j 1

2

3

4

1

27

82

92

12

213

1

12.68

38.50

43.19

5.63

2

100

678

764

93

1635

2

6.12

41.46

46.72

5.70

3

69

787

712

84

1652

3

4.18

47.64

43.09

5.09

4

18

88

83

12

201

4

8.96

43.78

41.29

5.97

The Table 1 on the left shows the tomorrow’s state depending on today’s state, for instant, 764 of the 1635 days in the State 2 were followed by a State 3, and so on. The table on right express the data in terms of percentages. Thus, for instance, a large drop (more than 1 %) wa followed 12.68 % of the time by another large drop, 38.50 % of the time by a small drop, 43.19 % of the time by a small increase, and 5.63 % of the time by a large increase. It is interesting to note that the tomorrow’change would be affected by today’change slightly: in any State the more likely it is a small increase or drop with similar probability, although when has dropped is slightly more likely growth and vice versa. Moreover, this conclusion is confirmed because using a standard statistical procedure testing for independence in a contingency table Everitt (2006), the p value is equal to 1. These results are consistent with the assumption that these exchanges follow a geometric Brownian motion. 2.2 Volatility Estimates From the above result, we assume that {Dn } are independent and normal random variables with mean (μ − 21 σ 2 )dt and variance σ 2 dt. Hence we can use a Monte Carlo approach to estimate the mean and variance. Suppose that t = tn is the current M are available, the sample mean y variance are time and {Dn+1−i }i=1 aM =

M 1  Dn+1−i , M i=1

1  (Dn+1−i − a M )2 , M −1 M

b2M =

i=1

we may therefore estimate the unknown parameter σ comparing a M with (μ− 21 σ 2 )dt √ and b2M with σ 2 dt, and hence we let σ ≈ σ1 = b M / dt, in our case the mean is about 6.3283 × 10−5 and σ1 ≈ 0.0063542, that is a volatility of 0.63 percent. Assuming that Dn has zero mean, this variance var (Dn ) = E(Dn2 ) and σ 2 dt ≈

M 1  2 Dn+1−i , M i=1

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On the Historical Exchange Rates Euro/US Dollar 0.02

Volatility

0.015

0.01

0.005

0 Dec 2001

Dec 2003

Dec 2005

Dec 2007

Dec 2009

Dec 2011

Dec 2013

Fig. 2 The blue dashed line is the constant volatility σ ≈ 0.0063542 and the red the sequence of σn computing (4). (Color figure online)

but considering that Dn correspond to different times, might be more appropriate to give extra weight to the most recent values, the

σ dt ≈ 2

M  i=1

2 αi Dn+1−i ,

with

M 

αi = 1, and α1 > α2 > · · · > α M > 0.

i=1

Usually, it use a geometric progression declining weights s.t. αi+1 = ωαi , for some 0 < ω < 1. Then M i 2 i=1 ω Dn+1−i σ 2 dt ≈ . M i i=1 ω A popular choice is ω = 0.94, because 0.94200 ≈ 4.2225 × 10−6 and 0.94300 ≈ 8.6767 × 10−9 are considered only a few hundred steps, and a new volatility estimate M is needed at each time, suppose σn2 dt is our estimate at time tn using {Dn+1−i }i=1 then new estimate at time tn+1 can be computed as 2 2 σn+1 dt = ω σn2 dt + (1 − ω)Dn+1 .

(4)

Obviously, this time-variable estimate of the volatility it is against to the underlying assumption of constant volatility in the Black–Scholes model (1), however, it would seem reasonable to expect that the sequence σn from (4) should not stray too far from the constant volatility. In Fig. 2 we have plotted in blue dashed line the constant volatility σ ≈ 0.0063542 and in red the sequence of σn computing (4). The most important differences between the constant and σn ≈ 0.01688 is around December 2008 coinciding with the serious fluctuations in Fig. 1. In my opinion, the most important consequence of this result is that the stochastic differential equations (1) is going on small noise regime, an active challenging field of research; recent contribution such as Anderson et al. (2015) is only a small example.

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F. Vadillo Periodicity rate EUR/USD 250

200

power

150

100

50

0

0

0.1

0.2

0.3

0.4

0.5

cycle/month

Fig. 3 Periodograma of the Historical Exchange Rates EUR/USD

3 Fourier and Waves Analysis of the Historical Exchange Rates EUR/USD 3.1 The Fourier Analysis of the Historical Exchange Rates EUR/USD Assuming that the time series Rn , for n = 0, 1, . . . , M is equal time spacing dt, the Fourier analysis allows us to estimate its periodic behaviors, at least approximately. As is well known (see for example Moler 2004; Trefethen 2000), the discrete Fourier transform (DFT) of Rn is k = R

1  Rn e−2πikn/(M+1) , M +1 M

(5)

j=0

where k = 0, 1, . . . , M is the frequency index. In Fig. 3 we have plotted the vecn | against the cycles per moth, we have considered that a month has 20 daily tor | R observations. The maximum power 249.0456 occur for frequency 0.0054 cycles/moth which corresponding to 185.10 moths/cycle, that mean 15.4250 years/cycle which is greater than the overall time, therefore we can conclude that there are no periodic fluctuations in these exchange rates EUR/USD. 3.2 The Wavelet Transform Fourier analysis provides frequency information but loses time information, unlike the wavelets use a time-scale and not a time–frequency view and they can be scaled as appropriate for data. The idea of the wavelet happened in the 1980s, a geophysicist J. Morlet wished to analysis a particular class of signals associated with seismic exploration. With the collaboration of de theoretical physic A. Grossman and P. Goupillaud, they wrote a famous paper Goupillaud et al. (1984) in 1984 which the beginning of this

123

On the Historical Exchange Rates Euro/US Dollar 1.5

1

0.5

0

−0.5

−1

−1.5 −3

−2

−1

0

1

2

3

Fig. 4 Real part of the Morlet wavelet for ω0 = 6, b = 0 and a = 0 (blue) a = 0.5 (red) a = 1.5 (green). (Color figure online)

technique and also a productive collaboration between mathematics (see for example Meyer 1992) scientists and technicians from other areas. The basis idea start with a function ψ, called the analysing wavelet or mother wavelet, and then it construct the bi-parametric family 1 ψab (t) = √ ψ a



t −b , b ∈ R, a > 0, a

(6)

where b is the shift parameter while a dilated or contracted the function. An simple example it is the Morlet wavelet ψ0 (η) = π −1/4 eiω0 η e−η

2 /2

,

(7)

consisting of a plane wave with the non-dimensional frequency ω0 modulated by a Gaussian. In Fig. 4 we shown real part of three Morlet’s wavelets for ω0 = 6, b = 0 and a = 0, 0.5, 1.5, with the wavelet one sees the action of a accordion. The wavelet theory proof the following result (42.2.1 Theorem, pp. 397 from Gasquel and Witonski 1999). Theorem 3.1 Suppose that the function ψ ∈ L1 (R) ∩ L2 (R) such that  +∞ (k)| (k) it is the Fourier transform of ψ. dk = K < +∞, where ψ (i) −∞ |ψ|k| (ii) ||ψ||2 = 1. For any f ∈ L 2 (R) there exist the wavelet coefficients C f (a, b) =

∞ −∞

∗ f (t) · ψab (t) · dt,

where ψ ∗ indique the complex conjugate function. Moreover, we have the following results:

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F. Vadillo

(a) Conservation of energy 1 K





C f (a, b) 2 · da db = a2 R2



∞

−∞

| f (t)|2 · dt.

(b) Reconstruction formula which define the wavelet transform of f 1 f (x) = K

R2

da db . a2

C f (a, b) · ψab (x) ·

3.3 The Wavelet Transform of the Historical Exchange Rates EUR/USD Following Torrence and Compo (1998) we use the continuous wavelet transform of a discrete sequence Rn , for n = 0, 1, . . . , M and a mother wavelet ψ(η) depending on a non-dimensional time parameter η. This wavelet transform is defined as the convolution of Dn with a scaled and translated version of ψ(η) Wn (s) =

M 

R j ψ∗

j=0



( j − n)dt s

,

(8)

and varying the wavelet scale s and translating along the time index n, we can construct a picture showing both the amplitude and its variation with time. Although it is possible to calculate the wavelet transform using (8), it is faster to do the calculations in Fourier space by the convolution theorem, the wavelet transform is the inverse Fourier transform of the product Wn (s) =

M 

k ψ ∗ (sωk )eiωk ndt , R

(9)

k=0

where the angular frequency is defined as

ωk =

⎧ 2π k ⎪ ⎨ (M+1)dt , k ≤ ⎪ ⎩

−2π k (M+1)dt ,

M+1 2 ,

(10) k>

M+1 2 .

Using (9) and the Morlet wavelet we have obtained the Fig. 5 where the bottom axis is time, vertical axis is the period in days logarithmically scaled in base-two, and the contours corresponding to the amplitudes |Wn (s)|2 with the colorbar on the right. Finally, the magenta line is the cone of influence where edge effects become important. A magenta contour line testing the wavelet power 5 % significance level against a white noise null is displayed as is the cone of influence, represented by a shaded area corresponding to the region affected by edge effects.

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On the Historical Exchange Rates Euro/US Dollar 0

.585

1

1

7.5

0

1.8

5

1

1.6

0

0

1

2.585 0 1

0

1.4

1

1

1.2

0

6.5

1

0

0.8

1

1

0

6

2

0

1.51 8

1

7

Period

0

2

1. 58 5

0

2

8

0.6 0.4

5.5

0.2 5 Jan 2000

0 Dec 2001

Dec 2003

Dec 2005

Dec 2007

Dec 2009

Dec 2011

Dec 2013

Fig. 5 The bottom axis is time, vertical axis is the period in days logarithmically scaled in base-two, and the contours corresponding to the amplitudes |Wn (s)|2 with the colorbar on the right. (Color figure online)

In view of this Fig. 5, we can make some comments: (1) The first thing that stands out it is that all periods are between 27 = 128 and 28 = 256 days, i.e. about half to one full year, shorter periods there exist from December 2007 probably justified by the financial crisis. (2) Between January 2000 and December 2002, the amplitudes of these movements is low, later beginning to observe higher amplitudes: from 1.585 red curve to 2 dark red curve, this behavior continues until almost December 2005. (3) In the following time, until autumn 2007, very little activity was recorded. (4) Throughout 2008 we can observe many different periods and amplitudes, we can say many different periods even highlight the longest, this time corresponding the he most sharp oscillations in Fig. 1 and bigger volatilities in Fig. 2, this time coincides with the mayor financial crisis. (5) During 2009 we observe small amplitudes (blue lines). (6) In 2010 and 2011 we can observe more activity, again we have high periods with significant amplitudes, and finally in 2012 and 2013 the activity is very low. The choice of the another wavelet basis function have given similar results.

4 Conclusions In this paper we have studied the exchange rate between the Euro (EUR) and the US Dollar (USD) between January 3, 2003 until September 18, 2014 with the following conclusions: 1. Testing the contingency table for the successive differences Dn , our results are consistent with the assumption that these exchanges follow a geometric Brownian motion. 2. He have used a Monte Carlo approach to estimate the volatility σ . Our approach shown its value is quite small. This means we have a small noise with the consequent numerical difficulties.

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F. Vadillo

3. In order to estimate some periodic behaviors, we have used the classical Fourier analysis with poor results. 4. Finally, using the wavelet analysis we have discovered that, roughly speaking, the major activities have been detected in periods between six months and a full year corresponding to the top half of the Fig. 5. Moreover, throughout 2008, 2009, 2010 and the first half of 2011, we can observe much higher activity probably justified by the financial crisis of 2007–2008. Acknowledgments This work was supported by the Spanish Ministry of Economy and Competitiveness, with the project MTM2014-53145, and by the Basque Government, with the project IT-641-13.

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