Wavelet Estimation of Integrated Volatility∗
Esben Høg
Asger Lunde
Email:
[email protected]
Email:
[email protected]
The Aarhus School of Business, Department of Information Science Fuglesangs All´e 4, DK-8210 Aarhus V, Denmark First version: November 2002, This version: January 2003
Abstract
In this paper we derive an estimator of integrated volatility using wavelet methods. We demonstrate that the performance of this estimator is indeed as good as a Fourier based method by Malliavin & Mancino (2002). Computationally the wavelet based method is preferable. This result emerges in a series of simulations of a stochastic volatility commonly used in financial economics. With respect to computational time, the wavelet based method is much faster than the comparable method based on Fourier analysis.
∗ Preliminary and very incomplete draft. Comments are very welcome.
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Høg & Lunde: Wavelet Estimation of Integrated Volatility
1. Introduction The conditional variance of financial time-series is important when pricing derivatives, calculating measures of risk, and hedging against portfolio risk. Therefore, there has been an enormous interest amongst researchers and practitioners to model the conditional variance. As a result, a large number of volatility models have been developed since the seminal paper of Engle (1982). A new approach is to estimate the integrated volatility using ultra-high frequency data. This approach was encouraged by the observation by Andersen & Bollerslev (1998a) that the forecasts of ARCH type model should be evaluated by comparison with integrated volatility estimated by intra-day quadratic variation, and not the squared inter-day returns. Following this, the concept of integrated volatility or it’s estimate has proven to have numerous applications, especially in derivatives. Inspired by Malliavin & Mancino (2002), who derive a estimator of integrated volatility by Fourier analysis, we derive an equivalent estimator using Wavelet methods. A wavelet approach might be preferable to the Fourier method for the several reasons: • Localization: Fourier basis functions are localized in frequency but not in time. Small frequency changes in the Fourier transform will produce changes everywhere in the time domain. Wavelets are local in both frequency/scale (via dilations) and in time (via translations). • Sparse coding: Function with discontinuities and functions with sharp spikes usually take substantially fewer wavelet basis functions than sine-cosine basis function to achieve a comparable approximation. • Low computational cost: Number of operations for DWT is proportional to the sample size n (i.e. O(n)), which is best possible. In contrast FFT requires O(n log2 n). We demonstrate that the performance of our wavelet estimator is very good compared to the Fourier method. This result emerges in a series of simulations of a stochastic volatility commonly used in financial economics. Later we also apply our estimator to a sample of stock prices from the Dow Jones Industrial Index. The rest of the paper is organized as follows. In the next section we describe the general theoretical setting. In section 3 we review the state of the art approaches to estimating integrated volatility. Section 4 contains our contribution to this research agenda, a wavelet estimator of integrated volatility. In section 5 we deliver a simulation study, where we compare our estimator to the Fourier estimators. Finally in section 8 we give some concluding remarks. 2
Høg & Lunde: Wavelet Estimation of Integrated Volatility
2. A General Framework for Assets Price Modelling As most of modern financial theory is based on semimartingales, we will embed our exposition in this setting (this is also the framework of e.g. Foster & Nelson (1996), Andersen, Bollerslev, Diebold & Labys (2000a), Andersen, Bollerslev, Diebold & Ebens (2000), Andersen, Bollerslev, Diebold & Labys (2001), Andreou & Ghysels (2002) and Barndorff-Nielsen & Shephard (2002)). Hence, we are considering a k-dimensional logarithmic asset price vector process, p = {p(t)}t∈[0,∞) . This is a semimartingale1 if it permits the following unique canonical decomposition: p(t) = p(0) + A(t) + M(t), where A(t) is a drift term having locally bounded variation on any finite subinterval of [0, ∞), M(t) is a local martingale. The intuition is that A(t) has locally deterministic drift delivering the instantaneous mean return, whereas M(t) represents the unpredictable innovation. One popular model within this frame is the stochastic volatility model, which we write in a generic form. The (univariate) log-price is assumed to follow a diffusion process of the following kind: dp(t) = µ(t)dt + σ (t)dw(t).
(1)
Where µ(t) and σ (t) are time-varying random functions and w(t) is a standard Brownian motion. It was suggested in Andersen & Bollerslev (1998a), that in this setting the relevant notion of volatility over a particular time span, normalized to [0, 1], is 1 σ 2 (u)du. 0
This is called integrated volatility or integrated variance (by e.g. Barndorff-Nielsen & Shephard (2002)), and is the focal point in the pricing of derivative securities under stochastic volatility (see e.g. Hull & White (87))
3. Estimators for Integrated Volatility One very simple proxy for the volatility over a given day is the squared daily returns. It is not surprising that squared daily returns produce a rather noisy measure of integrated volatility/variance. Better choices are measures that incorporate the additional information that intra-day returns have t to offer about σ 2∗ (t) = t−1 σ 2 (u)du. The simplest extension is the range-based proxy for σ 2∗ (t), which is based on the “open”, “low”, “high”, and “close” prices for day t, see, e.g., Parkinson (1983), Garman & Klass (1983), Beckers (1983), Ball & Torous (1984), Rogers & Satchell 1 For more on semimartingales see e.g. , Back (1991), Protter (1992, chapter 3), , Rogers & Williams (1988) or
Shiryaev (1999, pp 294-313).
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Høg & Lunde: Wavelet Estimation of Integrated Volatility
(1991), Wiggins (1991), Kunitomo (1992), Gallant, Hsu & Tauchen (1999), Yang & Zhang (2000) and McLeish (2002). A better, but also computational expensive, measure of daily volatility, is the realized volatility, suggested in Andersen & Bollerslev (1998a). Realized volatility is constructed by taking the sum of squared intra-day returns, which produces an unbiased measure of the conditional variance, σ 2∗ (t), under suitable regularity conditions. Adopting a notation similar to that of Andersen & Bollerslev (1998a), in our description of intra-day data. We define the (intra-day) returns, rt, ≡ pt − pt− , for > 0. So rt, is the return over the time interval with length , that preceded time t, and for integers of t we have rt,1 = rt , where rt is the daily return that we defined previously. Intra-day returns can be used to obtain a precise estimate of integrated volatility/variance. In particular if the intra-day return are serially uncorrelated, and σ 2 (t) has continuous sample paths, then it follows from the theory of quadratic variation, that
p lim
m→∞
1
m−1
σ 2 (u)du −
0
2 =0 rt− j 1 ,
j=0
m m
Hence, under these assumptions 2∗ (t) σˆ (m)
≡
m−1
2 rt− j 1 , ,
j=0
(2)
m m
is an unbiased estimator of integrated volatility/variance. We refer to (2) as the m-frequency of realized daily volatility. Market microstructure effects such as the bid-ask bounds and the irregular spacing of price quotes, could potentially distort the estimates. For example, they are unadjusted estimates based on tick-by-tick data are likely to be biased. Andersen & Bollerslev (1997, 1998a, 1998b) and Andersen, Bollerslev, Diebold & Ebens (2000) circumvent this obstacle by estimating the volatility from artificially constructed five-minutes returns. Another approach is to allow for a New-West type autocorrelation correction. This gives the following estimator 2∗ (t) σˆ NW(m)
≡
m−1 j=0
2 rt− j 1 m ,m
g 1− +2 h=1
h g+1
m−1
rt− j , 1 rt− j−h , 1 , m m
m
m
(3)
j=h+1
with g equal to the integer part of 4(n/100)2/9 . Both these estimator are potentially very dependent on how the intraday returns are constructed. Regularly spaced intraday prices are virtually never available, hence such prices must be constructed from tick-by-tick data. We now turn to this matter.
4
Høg & Lunde: Wavelet Estimation of Integrated Volatility
3.1. Generating Homogeneous Time Series Following the nomenclature of Dacorogna, Gencay, Muller, ¨ Olsen & Pictet (2001) we classify a time series according to its spacing of data points in time. If the series is regularly spaced then its is called homogeneous, whereas a irregularly spaced series is called inhomogeneous. In order to use the estimators (2) or (3) we need a homogenous time series of prices. Such a series is usual not at hand, hence it has to be constructed from the raw data. Let assume that the raw data is an inhomogeneous series, sampled at times t j with values z j = z(t j ). By interpolation we can construct homogenous time series with values at times t0 + i, starting at t0 . Now any of the desired homogenous time points t0 + i, i = 1, 2, . . . are surrounded by two times t j and t j +1 of the raw series: t j ≤ t0 + i ≤ t j +1 ,
with j = arg max t j ≤ t0 + i. j
Based on z j and z j +1 we ”estimate” z(t0 + i) using interpolation. The two most used methods are linear interpolation, given by z(t0 + i) = z j +
t0 + i − t j +1 (z j +1 − z j ), t j +1 − t j
and previous-tick interpolation z(t0 + i) = z j . A discussion of these two interpolation schemes are found in Dacorogna et al. (2001, sec. 3.2.1)
3.2. The Fourier Method This method was suggested by Malliavin & Mancino (2002), and has been applied by Barucci & Reno (2002b) and Barucci & Reno (2002a). It’s use only requires that the quadratic variation of the price process is bounded. Hence, the price follows a diffusion process like (1). Then the Fourier method deliver the integrated volatility estimated by 2π σ 2 (t)dt = 2πa0 (σ 2 ).
(4)
0
Hence, a time window of [t1 , t N ] with N observations at times t1 < t2 < . . . < t N , must be normalized into [0, 2π]. We denote the normalized time point by τi = 2π(ti − t1 )/(t N − t1 ) for i = 1, . . . , N . Now, we compute (4) by n π 2 a0 (σ ) = lim as (dp) + bs2 (dp) n→∞ 2n s=1 2
where 1 ak (dp) = π
2π
cos(kt)dp(t)
and
0
5
1 bk (dp) = π
0
(5)
2π
sin(kτ )dp(τ ).
Høg & Lunde: Wavelet Estimation of Integrated Volatility
The justification of this is given in Malliavin & Mancino (2002), and is – among other things – based on the Parseval identity for the drift term:
2π
µ2 (t)dt =
0
∞
2 ak (µ) + bk2 (µ) , k=0
which is a consequence of the fact that the trigonometric system is an orthonormal basis for L 2 ([0, 2π]). Now, following Barucci & Reno (2002b), we compute the fourier coefficients ak (dp) and bk (dp) using the following approximations N −1 p(τ N ) − p(τ1 ) 1 aˆ k (dp) = p(τi ) [cos(kτi ) − cos(kτi+1 )] + π π i=1
1 bˆk (dp) = − π
N −1
p(τi )(sin(kτi ) − sin(kτi+1 ))
i=1
which we plug into (4) to get the wavelet integrated volatility estimate:
2∗ σˆ fourier(n) (t) =
n
π2 2 aˆ s (dp) + bˆs2 (dp) n s=1
The granulity of the estimator is controlled by n.
4. The Wavelet Method The basic idea behind wavelet analysis is to decompose a time function into a number of components, each one of which can be associated with a particular scale at a particular time. In short, a wavelet ψ ∈ L 2 (R) is a function whose binary dilations and dyadic translations generate a Riesz basis of L 2 (R). Any f ∈ L 2 (R) can be expanded into a wavelet series, f (t) =
∞ ∞
w jk ψ jk (t),
j=−∞ k=−∞
where ψ j,k ∈ L 2 (R) denotes the dilated and translated wavelet, defined by ψ jk (t) = 2 j/2 ψ(2 j t − k). j ∈ Z is the scale of the wavelet, corresponding to a dilation by 2 j , and k ∈ Z is the position (translation). In the present setting we use this expansion to represent a diffusion d X (t) in a Wavelet basis. The wavelets are derived from a single function, ψ(t), called the mother wavelet. They are derived by translations and dilations. The coefficient w jk characterizes features of the wavelets 6
Høg & Lunde: Wavelet Estimation of Integrated Volatility
at scale j and position k. In this sense the wavelet transform as defined below offers a mixed position-frequency representation of the function. Daubechies (1988) constructed the so-named Daubechies wavelets of orders p = 1, 2, 3, · · · with number of non-zero coefficients 2 p = 2, 4, 6, · · · , respectively. In practice they are very important, as they have compact support: supp{ψ} = [−( p − 1), p], they are orthonormal, and they have p vanishing moments: x r ψ(x)d x = 0, r = 0, 1, · · · , p − 1. For a diffusion process d X the Continuous Wavelet Transform at time-position 2− j k and dilation (scale) 2− j is (Wψ f )(2− j , 2− j k) = d X, ψ(2− j ,2− j k) = 2 j/2 ψ(2 j t − k)d X (t). Here ·, · denotes the inner-product, and ψ(a,b) (t) = a
−1/2
ψ
t −b . a
4.1. The Wavelet method to estimate integrated volatility The Continuous Wavelet Transform is similar to the windowed Fourier Transform in that we integrate against a function with localized oscillations. Though we use the notion of time in the wavelet analysis, the frequency is replaced by the notion of scale. All functions used in computing the wavelet transform come from the same function (the mother wavelet). As before we consider the diffusion process dp(t) = µ(t)dt + σ (t)dw,
(6)
where dw is a Brownian motion, µ(t) and σ (t) are time dependent functions. Consider an orthogonal wavelet with mother function ψ. For any scale a > 0 and dilation2 b the continuous Wavelet transform of dp is t −b −1/2 ψ( (Wψ dp)(a, b) = a )dp(t) a t −b t −b −1/2 −1/2 )σ (t)dw(t) + a )µ(t)dt =a ψ( ψ( a a Lemma 1 The second order moment of the Wavelet transform in (7) is given by 2 t −b 2 −1 E (Wψ dp)(a, b) = a )σ (t)dt + [(Wψ µ)(a, b)]2 ψ 2( a 2 For example a = 2− j and b = 2− j k
7
(7)
Høg & Lunde: Wavelet Estimation of Integrated Volatility
Proof. This is easily shown by using the standard rules of Ito calculus. In the sequel we use the simple Haar wavelet ψ(t) = φ(2t)−φ(2t −1), where φ(t) = 1[0,1) (t) is the indicator function on the unit interval. Then if 0 ≤ t < 12 1 ψ(t) = −1 if 12 ≤ t < 1 0 otherwise It follows immediately that 2 E (Wψ dp)(a, b) = a −1
a+b
σ 2 (t)dt + [(Wψ µ)(a, b)]2 .
(8)
b
Without loss of generality we normalize the time window [0, T ] to [0, 1]. Let M denote the integer part of the base 2 logarithm of n. Using the scales a = 2− j and dilations b = 2− j k, for j = 0, · · · , M, and k = 0, · · · , 2 j − 1 we obtain a disjoint partition of the interval [0, 1] and the following so-called wavelet coefficients of dp(t) on the interval [0, 1], w jk (dp) = (Wψ dp)(2− j , 2− j k). From (8) it follows that the second order moments of the wavelet coefficients of dp are E[w 2jk (dp)]
=2
j
2− j (k+1)
2− j k
σ 2 (t)dt + [(Wψ µ)(2− j , 2− j k)]2 .
(9)
The following result shows how the integrated volatility can be estimated by using wavelets Theorem 2 Given a sample of n observations of dp(t) from (6), and defining jn = int(log2 (n)−1) we have the following result
1
− jn
σ (t) = lim 2 2
n→∞
0
jn −1 2
w2jn k (dp)
(10)
k=0
Remark: jn is the nearest power of two below (or equal to) n. Proof. From (8) it follows that for some given j E[
j −1 2
k=0
w2jk (d p)]
=2
1
j
σ (t)dt + 2
0
j −1 2
k=0
where w jk (µ) = (Wψ µ)(2− j , 2− j k).
8
w2jk (µ),
(11)
Høg & Lunde: Wavelet Estimation of Integrated Volatility
Since the chosen ψ(2 j t − k) is an orthogonal family of wavelets it satisfies the “Parseval Identity” for wavelets:
jn 2 −1
j
w2jk (µ)
1
=
µ2 (t)dt.
0
j=0 k=0
Hence the term involving the drift vanishes when we take averages in (11). The statement of the theorem follows.
4.2. Computation of Wavelet coefficients of dp(t) Given a series of n not necessarily evenly sampled observations (ti , p(ti )), i = 1, · · · , n we have using the Haar wavelets
1
w jk (dp) = 2 j/2
ψ(2 j t − k)dp(t)
0
= 2 j/2
2− j (k+1/2)
2− j k
dp(t) −
2− j (k+1)
2− j (k+1/2)
dp(t)
= 2 j/2 2 p(2− j (k + 1/2)) − p(2− j k) − p(2− j (k + 1)) which we plug into (10) to get the wavelet integrated volatility estimate: 2∗ σˆ wave( jn ) (t)
=
jn −1 2
2 2 p(2− jn (k + 1/2)) − p(2− jn k) − p(2− jn (k + 1))
k=0
5. Simulation study Let p(t) be the log of a generic asset price. Following Barucci & Reno (2002b) we assume that p(t) follows the continuous time GARCH model: d p(t) = σ (t)dW1 (t), dσ 2 (t) = θ(ω − σ 2 (t))dt +
√
2λθσ 2 (t)dW2 (t),
where W1 and W2 are independent Brownian motions. To illustrate the Wavelet approach together with the Fourier method, we simulate the diffusion by an Euler discretization scheme. We use the parameters θ = 0.035, ω = 0.636, λ = 0.296 estimated in Andersen & Bollerslev (1998a) on the daily return time series of the DEM-USD exchange rate. The Euler discretization scheme (Kloeden & Platen (1992)) is implemented as follows. First, dσ 2 (t) is generated according to the scheme. Then dσ (t) is used to generate d p(t), also by the
9
Høg & Lunde: Wavelet Estimation of Integrated Volatility
Euler scheme3 . We simulate 24 hours of trading with one observation at each of the 86.400 seconds. Then we sample from this ”true” distribution by extracting observations such that the time-difference between successive observations is exponentially distributed. We chose the mean of this exponential distribution to 35 seconds, which corresponds to approximately 86.400/35 ≈ 2469 observations per day. We simulate 10.000 days in this way. The performances of the Wavelet and Fourier estimates are evaluated by the statistics 86400 ξ and sξ
where
ξt =
2∗ ( p(i) − p(i − 1))2 − σˆ (•) (t) 86400 2 i=2 ( p(i) − p(i − 1))
i=2
, t
The results of the simulations are presented as histograms ξt in Figure 1 and Table 1. The conclusion is that the wavelet based results are indistinguishable from the alternative Fourier based method4 . However, the method developed here is superior to the Fourier method in the sense that the expression of the estimator of the integrated volatility is simpler for the wavelet based method. Also the wavelet based method is much faster: In the simulations used here the computational time used to estimate by the Fourier based method is over 340 times larger than the corresponding computational time used to estimate by the wavelet based method.
6. The Data The data are IBM transaction prices from the Trade and Quote (TAQ) database. The TAQ database is a collection of all trades and quotes in the New York Stock Exchange (NYSE), American Stock Exchange (AMEX), and National Association of Securities Dealers Automated Quotation (Nasdaq) securities. In our estimation of intra-day volatility we include transactions from the period running from January 3, 1995, through February 28, 2002, spanning a total of 1802 trading days.. From the trade files we used transactions in time spand from 9:30 am to 4 pm. We cleaned the trade files in the following way: 1. Remove prices equal to zero. 2. Remove prices stamped outside 9:30 am to 4 pm. 3. For a running window of 100 prices, remove all prices that are more that λ% different from the mean in their window. √
2 3 As in Andersen & Bollerslev (1998a) the simulated process is given by p t+ = pt + σt w p,t and σt+ = √ θ ω + σt2 (1 − θ + 2λθ )wσ,t , where w p,t and wσ,t denote independent standard normal variables. 4 This conclusion is also supported by several two sample tests of no distributional differences.
10
Høg & Lunde: Wavelet Estimation of Integrated Volatility
4. If a prices went up/down with more than λ% and then back down/up again, then remove all the peaky quotes. 5. If some prices all peaks more than λ% for a short while and then goes back, then remove all the peaky prices. Table 2 present some summary statistics of this filter.
7. Results and Evaluation The fact that the true volatility is unobservable makes it difficult to identify the best volatility model. It is, however, possible to construct benchmark criteria to assess the performance of different estimators. A collection of such benchmarks are complied from Andersen, Bollerslev, Diebold & Labys (2000b), Bollen & Inder (2002) and Barndorff-Nielsen & Shephard (2002). Criterion 1 Bias test (1) – If interday returns are uncorrelated, then the mean of the estimated interday volatility 2∗ σˆ Simple (t)
N 1 2 ≡ r , N t=1 t
is an unbiased estimator of the true daily volatilities σ 2∗ (t). Hence, an alternative estimator which give a mean significantly different from the mean of the simple estimator will be biased. Criterion 2 Iid-ness or Normality of Standardized returns – If the model for the log-price is a stochastic volatility like the one in Barndorff-Nielsen & Shephard (2002), t σ (s)dw(s), p(t) = µ(t) + 0
with w being a standard Brownian motion. Then we have r (t)|µ∗ (t), σ 2∗ (t) ∼ N id(µ∗ (t), σ 2∗ (t)) where r (t) = p(t) − p(t − 1) µ∗ (t) = µ(t) − µ(t − 1), σ (t) =
t
2∗
t
σ (u)du = 2
t−1
0
11
σ (u)du − 2
0
t−1
σ 2 (u)du.
Høg & Lunde: Wavelet Estimation of Integrated Volatility
Hence, it will be the case that H02 : z(t) =
r (t) − µ∗ (t) ∼ N id(0, 1). σ 2∗ (t)
and if σ 2∗ (t) is properly estimated then we should not reject this null. If we only know that w has independent increments, then we can only test
H02 : z(t) =
r (t) − µ∗ (t) ∼ iid(0, 1). σ 2∗ (t)
This involves testing for a unit variance and uncorrelated-ness of these residuals. The results of such tests are presented in Figures 2-7.
8. Conclusion In this paper we suggest a new method for estimating the integrated variance of a semimartingale, such as the classical stochastic volatility model. The method is based on wavelets, and is computationally much faster than a method based on Fourier analysis, see Malliavin & Mancino (2002). It is unbiased as is the alternative Fourier based method. The proposed method is also as efficient as the Fourier based method.
References Andersen, T. G. & Bollerslev, T. (1997), ‘Intraday periodicity and volatility persistence in financial markets’, Journal of Empirical Finance 4, 115–158. Andersen, T. G. & Bollerslev, T. (1998a), ‘Answering the skeptics: Yes, standard volatility models do provide accurate forecasts’, International Economic Review 39(4), 885–905. Andersen, T. G. & Bollerslev, T. (1998b), ‘Deutsche mark-dollar volatility: Intraday activity patterns, macroeconomic announcements, and longer run dependencies’, Journal of Finance 53(1), 219–265. Andersen, T. G., Bollerslev, T., Diebold, F. X. & Ebens, H. (2000), ‘The distribution of realized stock return volatility’, Journal of Financial Economics 61(1), 43–76. Andersen, T. G., Bollerslev, T., Diebold, F. X. & Labys, P. (2000a), ‘The distribution of exchange rate volatility’, Journal of the American Statistical Association 96(453), 42–55. Andersen, T. G., Bollerslev, T., Diebold, F. X. & Labys, P. (2000b), ‘Exchange rate return standardized by realized volatility are (nearly) Gaussian’, Multinational Finance Journal 4(3&4), 159–179. 12
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Andersen, T. G., Bollerslev, T., Diebold, F. X. & Labys, P. (2001), ‘Modeling and forcasting realized volatility’, Unpublished manuscript . Andreou, E. & Ghysels, E. (2002), ‘Rolling-sample volatility estimators: Some new theoretical, simulation, and empirical results’, Journal of Business & Economic Statistics 20. Back, K. (1991), ‘Asset prices for general processes’, Journal of Mathematical Economics 5(3), 387–409. Ball, C. A. & Torous, W. N. (1984), ‘The maximum likelihood estimation of security price volatility: Theory evidence, and application to option pricing’, Journal of Business 57, 97–112. Barndorff-Nielsen, O. E. & Shephard, N. (2002), ‘Estimating quadratic variation using realised volatility’, Journal of Applied Econometrics 17, 457–477. Barucci, E. & Reno, R. (2002a), ‘On measuring volatility and GARCH forecasting performance’, Journal of International Financial Markets 12, 183–200. Barucci, E. & Reno, R. (2002b), ‘On measuring volatility of diffusion processes with high frequency data’, Economics Letters 74, 371–378. Beckers, S. (1983), ‘Variances of security price returns based on high, low, and closing prices’, Journal of Business 56(1), 97–112. Bollen, B. & Inder, B. (2002), ‘Estimating daliy volatility in financial markets utilizing intraday data’, Journal of Empirical Finance 9, 551–562. Dacorogna, M. M., Gencay, R., Muller, ¨ U., Olsen, R. B. & Pictet, O. V. (2001), An Introduction to High-Frequency Finance. Daubechies, I. (1988), ‘Orthonormal bases of compactly supported wavelets’, Comm. Pure and Appl. Math. 41, 909–996. Engle, R. F. (1982), ‘Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation’, Econometrica 45, 987–1007. Foster, D. P. & Nelson, D. B. (1996), ‘Continuous record asymptotics for rolling sample variance estimators, ecobometrica, vol 64, no 1, p. 139-174’, Econometrica 64(1), 139–174. Gallant, A. R., Hsu, C.-T. & Tauchen, G. E. (1999), ‘Using daily range data to calibrate volatility diffusions and extract the forward integrated variance’, Review of Economics & Statistics 81(4), 617–631. Garman, M. B. & Klass, M. J. (1983), ‘On the estimation of security volatilities from historical data’, Journal of Business 53(1), 67–78. Hull, J. & White, A. (87), ‘The pricing of options on asssets with stochastic volatility’, Journal of Finance 42(2), 281–300. Kloeden, P. E. & Platen, E. (1992), Numerical Solution of Stochastic Differential Equations, Springer-Verlag. Kunitomo, N. (1992), ‘Improving the Parkinson method of estimating security price volatilities’, Journal of Business 64(2), 295–302. 13
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Malliavin, P. & Mancino, M. E. (2002), ‘Fourier series method for measurement of multivariate volatilities’, Finance and Stochastics 6(1), 49–61. McLeish, D. L. (2002), ‘Highs and lows: Some properties of the extremes of a diffusion and applications to finance’, Canadian Journal of Statistics 30(2), 243–267. Parkinson, M. (1983), ‘The extreme value method for estimating the variance of the rate of return’, Journal of Business 53(1), 61–65. Protter, P. (1992), Stochastic Integration and Differential Equations, New York: Springer-Verlag. Rogers, L. C. G. & Satchell, S. E. (1991), ‘Estimating variances from high, low, and closing prices’, Annals of Applied Probability 1(4), 504–512. Rogers, L. C. G. & Williams, D. (1988), Diffusions, Markov Processes and Martingales, Vol. Volume 2: Ito Calculus, John Wiliey & Sons. Shiryaev, A. N. (1999), Essentials of Stochastic Finance: Facts, Models, and Theory, World Scientific: Singapore. Wiggins, J. B. (1991), ‘Empirical tests of the baus and efficiency of the extreme-value estimator for common stocks’, Journal of Business 64(3), 417–432. Yang, D. & Zhang, Q. (2000), ‘Drift-independent volatility estimation based on high, low, open and close prices’, Journal of Business 73(3), 477–491.
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Høg & Lunde: Wavelet Estimation of Integrated Volatility
Table 1: Summary Statistics for Simulations Errors. Panel A: Wavelet Estimator M
Mean
tno bias
Std. dev.
Skewness
Kurtosis
JB
8 9 10 11 12
0.002089 0.001598 0.001420 0.001183 0.001287
0.0166 0.0179 0.0216 0.0237 0.0292
0.1259 0.0893 0.0657 0.0500 0.0441
-0.2736 -0.1698 -0.1617 -0.0911 -0.1169
3.187 2.972 2.979 2.989 2.985
139.3 48.4 43.7 13.9 22.9
Panel B: Fourier Estimator n 64 128 256 512 1024
Mean
tno bias
Std. dev.
Skewness
Kurtosis
JB
0.001704 0.000749 0.001351 0.001213 0.000969
0.0136 0.0084 0.0207 0.0238 0.0219
0.1254 0.0891 0.0653 0.0510 0.0442
-0.2105 -0.1920 -0.1879 -0.1540 -0.1346
3.040 3.051 3.098 3.023 2.970
74.5 62.5 62.9 39.8 30.6
Panel B: Two sample tests for equality, p-values M/n
Mean
Variance
Skewness
Kurtosis
8/64 9/128 10/256 11/512 12/1024
0.828 0.501 0.941 0.967 0.611
0.697 0.815 0.547 0.032 0.688
0.287 0.690 0.641 0.259 0.743
0.392 0.586 0.425 0.820 0.917
This table present summary statistics for the simulations.
15
Høg & Lunde: Wavelet Estimation of Integrated Volatility
Table 2: Some Summary Statistics of the Data Cleaning Filter. Panel A: Number of observations left after each data cleaning step
N.-obs.
Raw data
Zeros
Off hours
Clean 1
Clean 2
Clean 3
6544831
6544829
6499504
6499322
6493928
6472029
Panel B: Number of observations deleted in each data cleaning step
N.-obs.
Zeros
Off hours
Clean 1
Clean 2
Clean 3
Total
2
45325
182
5394
21899
72802
Panel C: Percentage of observations deleted in each data cleaning step
%-obs.
Zeros
Off hours
Clean 1
Clean 2
Clean 3
Total
0.000
0.693
0.003
0.083
0.337
1.112
Panel D: Number of observations deleted on each exchange after each data cleaning step Exchange Boston Cincinnati Chicago NYSE Pacific NASD Philadelphia
Zeros
Off hours
Clean 1
Clean 2
Clean 3
Total
2 0 0 0 0 0 0
2191 1061 2468 10025 5702 23520 358
27 14 22 36 3 77 3
464 457 393 684 307 2941 148
2378 1475 2033 5204 1432 8736 641
5062 3007 4916 15949 7444 35274 1150
Panel E: Percentage of observations deleted on each exchange after each data cleaning step Exchange
Zeros
Off hours
Clean 1
Clean 2
Clean 3
Total
Boston Cincinnati Chicago NYSE Pacific NASD Philadelphia
0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.301 0.300 0.845 0.305 2.568 1.491 0.410
0.004 0.004 0.008 0.001 0.001 0.005 0.003
0.064 0.129 0.135 0.021 0.138 0.186 0.169
0.327 0.417 0.696 0.158 0.645 0.554 0.734
0.695 0.850 1.684 0.486 3.352 2.237 1.316
In this table present some summary numbers for the data cleaning filter.
16
4 3 0
0
1
2
Fourier n=128
3 2 1
Wavelet M=8
4
Høg & Lunde: Wavelet Estimation of Integrated Volatility
-0.4
-0.2
0.0
0.2
0.4
-0.4
-0.2
0.2
0.4
0.2
0.4
0.2
0.4
0.2
0.4
6 4 3 0
1
2
Fourier n=256
5
5 4 3
Wavelet M=9
2 1 0
-0.4
-0.2
0.0
0.2
0.4
-0.4
-0.2
0.0
Error
4 0
0
2
2
4
Fourier n=512
6
6
8
8
Error
Wavelet M=10
0.0
Error
6
Error
-0.4
-0.2
0.0
0.2
0.4
-0.4
-0.2
0.0
Error
6 2
4
Fourier n=1024
6 4
0
2 0
Wavelet M=11
8
8
Error
-0.4
-0.2
0.0
0.2
0.4
-0.4
-0.2
Error
0.0
Error
Figure 1: Histograms of relative errors.
17
Høg & Lunde: Wavelet Estimation of Integrated Volatility
Sampling Frequency in n
Average Integrated Volatility - IBM 2 4 6 8 10 12
2048
1024
512
256
128
64
32
16
8
4
2
Fourier Wavelet no. Vols 2∗ σˆ Simple
1802
629
0.25
1.0
3.0
7.0
15.0
35.0
97.5
Sampling Frequency in Minutes
Figure 2: Volatility signature plots for Fourier and Wavelet estimators.
Sampling Frequency in n 512
256
128
64
32
16
8
4
2
0.8
1.0
1.2
Fourier Wavelet
0.6
Standard deviation - IBM
1.4
1024
0.25
0.50
1.0
2.0
4.0
8.0
20.0
40.0
97.5
Sampling Frequency in Minutes
Figure 3: Standard deviation of open to close return divided by square root of Fourier and Wavelet estimators, for several frequencies.
18
Høg & Lunde: Wavelet Estimation of Integrated Volatility
Sampling Frequency in n 512
256
128
64
32
16
8
4
2
0.0 -0.1
Fourier Wavelet
-0.3
-0.2
Skewness - IBM
0.1
1024
0.25
0.50
1.0
2.0
4.0
8.0
20.0
40.0
97.5
Sampling Frequency in Minutes
Figure 4: Skewness of open to close return divided by square root of Fourier and Wavelet estimators, for several frequencies.
Sampling Frequency in n 512
256
128
64
32
16
8
4
2
12
1024
8 6 2
4
Kurtosis - IBM
10
Fourier Wavelet
0.25
0.50
1.0
2.0
4.0
8.0
20.0
40.0
97.5
Sampling Frequency in Minutes
Figure 5: Kurtosis of open to close return divided by square root of Fourier and Wavelet estimators, for several frequencies.
19
Høg & Lunde: Wavelet Estimation of Integrated Volatility
Sampling Frequency in n 512
256
128
64
32
16
8
4
2
Average Integrated Volatility - IBM 0.0 0.2 0.4 0.6 0.8 1.0
1024
0.25
0.50
1.0
2.0
4.0
8.0
20.0
40.0
97.5
Sampling Frequency in Minutes
Figure 6: P-values for Jarque-Bera test of normality on open to close return divided by square root of Fourier and Wavelet estimators, for several frequencies.
Sampling Frequency in n 512
256
128
64
32
16
8
4
2
0.6 0.4 0.2 0.0
Ljung-Box(15) - IBM
0.8
1.0
1024
0.25
0.50
1.0
2.0
4.0
8.0
20.0
40.0
97.5
Sampling Frequency in Minutes
Figure 7: P-values for Ljung-Box(15) test of no serial correlation on squared open to close return divided by square root of Fourier and Wavelet estimators, for several frequencies.
20
