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REALIZED VOLATILITY ESTIMATION
Master thesis
Miquel Masoliver, Guillem Roig, Shikhar Singla
25.06.2014
Advisors: Christian...
Acknowledgements
We would like to thank our advisors Christian Brownlees and Eulalia Nualart for
the guidance and support ...
Contents
1 Introduction 3
2 Theoretical background 4
2.1 Realized Volatility Estimator . . . . . . . . . . . . . . . . . ....
1 Introduction
In finance it is of utmost importance to understand volatility and its dynamics,
since it is the main driver...
The main purpose of this study is to try to find the optimal volatility esti-
mator in a non-parametric framework. In parti...
the following diffusion process,
yt =
t
0
µ(s)ds +
t
0
σ(s)dWs. (1)
The object of interest is the amount of accumulated var...
theory shows that
plim
n−1
i
(yi+1 − yi)2
=
T =1
0
σ2
s ds (4)
However, in real life, market microstructure noise appears ...
microstructure noise [Yu09].
One of these consistent estimators, named Two Scaled Realized Variance (TSRV),
was first propo...
where γh is matrix of autocovariances given by
γh =



n
j=h+2
rjrT
j−h, h ≥ 0
γh = γT
−h, h < 0
(10)
and k(x) is P...
i.e., it has to vanish slower than the continuity of Brownian motion in order to
have convergence in probability (consiste...
the Heston model. The price process is specified by
dyt = µytdt + σtytdWS
t (17)
where µ represents the return of the asset...
Figure 1: Frobenius and Infinite norm for matrices A as defined in (20) to test
the performance of the estimators on the ben...
The preferred choice of bandwidth for RK is H∗
= c∗
ξ4/5
n3/5
where c∗
= 3.5314
for Parzen function and ξ2
= ω2
/
√
IQ.
We...
In reality, however, simple Poisson processes are of little interest since they
only consider one possible jump size and a...
Figure 2: Frobenius and Infinite norm for matrices A as defined in (20)to test
the performance of the estimators in the pres...
Figure 3: Impact of noise using Frobenius norm
4 Data implementation
In the previous section we described how the estimato...
Data cleaning
Since the opening of the exchange, assets can be traded at will if two parties
agree to do so. This means th...
Figure 4: Performance of RV and Kernel
methods to assess which estimator is better have been hampered by the multi-
tude o...
frequency of 5 seconds outperforms RV 5 seconds due to the large number of
datapoints. However, as the number of data poin...
Figure 5: Performance of RV and Threshold RV
Estimator Variance
RCOV from daily data 5.370
TSRV from daily data 218.7957
R...
post quadratic variation of asset prices, giving rise to the Realized Volatility
estimator.
The next step in adding comple...
References
[AS05] Lan Zhang, Per A. Mykland, Yacine Ait-Sahalia. A tale of two time
scales: Determining integrated volatil...
[Ren10] Fulvio Corsi, Davide Pirino, Roberto Reno. Journal of Econometrics,
159, 2010.
[She08] Ole Barndorff-Nielsen, Peter...
TICKER COMPANY NAME GICS SECTOR NAME STATUS
MMM 3M CO Industrials
ABT ABBOTT LABORATORIES Health Care
ACN ACCENTURE PLC-CL...
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Realized Volatility Estimation (Paper)

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Barcelona GSE Master Project by Miquel Masoliver, Guillem Roig, Shikhar Singla

Master Program: Finance

About Barcelona GSE master programs: http://j.mp/MastersBarcelonaGSE

Published in: Economy & Finance
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Realized Volatility Estimation (Paper)

  1. 1. REALIZED VOLATILITY ESTIMATION Master thesis Miquel Masoliver, Guillem Roig, Shikhar Singla 25.06.2014 Advisors: Christian Brownless, Eulalia Nualart
  2. 2. Acknowledgements We would like to thank our advisors Christian Brownlees and Eulalia Nualart for the guidance and support they provided during the development of this thesis. 1
  3. 3. Contents 1 Introduction 3 2 Theoretical background 4 2.1 Realized Volatility Estimator . . . . . . . . . . . . . . . . . . . . 4 2.2 Subsampling to remedy for noise . . . . . . . . . . . . . . . . . . 5 2.3 Realized Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 Threshold Realized Variance . . . . . . . . . . . . . . . . . . . . . 8 3 Simulation 9 3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Finding optimal parameters for TSRV & RK . . . . . . . . . . . 11 3.3 Modeling price jumps . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Implementing market microstructure noise . . . . . . . . . . . . . 14 4 Data implementation 15 4.1 Data and methodology . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 5 Conclusions 19 2
  4. 4. 1 Introduction In finance it is of utmost importance to understand volatility and its dynamics, since it is the main driver of portfolio construction, in hedging and pricing of options and in the determination of a firm’s exposure to risk. It also plays a critical role in discovering trading and investment opportunities that provide an attractive risk-return trade-off [Yu09]. Therefore, it is not surprising that many estimators have been proposed to measure volatility from a discrete price sample: Parkinson [Par80], Rogers et al. [Yoo94] and Yang and Zhang [Zha00] among others, were the first proponents of new methodologies to estimate re- alized volatility using low-frequency daily data. Parametric models such as the seminal ARCH model first proposed by Engle [Eng82] and posterior contribu- tions from Bollerslev and Mikkelsen [Mik96] among many others, have provided new ways to estimate and forecast volatility. However, the validity of such volatility measures rely upon specific distributional assumptions, immediately calling to question the robustness of previous findings. Andersen et al. proposed a non-parametric approach based on summing squares and cross-products of intraday high-frequency returns to construct estimates of realized daily volatility. The underlying idea is to use the quadratic variation as an ex-post variation of asset prices [Ebe01]. This approach, however, has the weakness that it can be sensitive to market frictions when applied to returns obtained over shorter time intervals, since it estimates the magnitude of the noise term rather than volatility. To overcome these drawbacks Zhang et al., on the one hand, proposed the so-called Two-Scale Realized Variance (TSRV), an unbiased estimator that incorporates noise into the model and separates the sample into multiple ”grids” [AS05], and Barndorff-Nielsen et al., on the other hand, introduce the family of realized kernel estimators to carry out efficient feasible inference on the ex-post variation of underlying equity prices in the presence of market frictions [She08]. All of these estimators, however, turn out to be inconsistent in the presence of jumps. Corsi et al. introduce the family of threshold realized variance estima- tors (TRV), which detect discontinuities in prices and do not include those price points in volatility calculations [Ren10]. 3
  5. 5. The main purpose of this study is to try to find the optimal volatility esti- mator in a non-parametric framework. In particular, this study focuses on the estimation of the daily integrated variance-covariance matrix of stock returns us- ing simulated and high-frequency data in the presence of market microstructure noise, jumps, and non-synchronous trading. This work is structured in three building blocks:(i) price processes are simulated in the presence of jumps and market microstructure noise. This allows us to obtain some insight about the estimators’ performance. (ii) The aforementioned realized volatility estimators are applied to high-frequency data of the S&P 100 stocks of October 27th 2010 using 5-second, 10-second, 30-second, 1-minute and 2-minute time intervals. (iii) We use the estimated covariance matrices to construct the global minimum variance portfolio for each sampling frequency. These global minimum variance portfolios are used to build 30 day ex-post portfolio’s returns and we use the variance of these returns to compare between the performance of the estimators. 2 Theoretical background In this section we provide a review of the main theory on different non-parametric volatility estimators that take part in the study, as well as the rationale behind using high-frequency data. 2.1 Realized Volatility Estimator Volatility estimation and inference has attracted much attention in the financial econometric and statistical literature. Even in a discrete time framework one will often start with the sum of squared log-returns, as not only the simplest and most natural estimator, but also as the one with the most desirable properties as shown in [Yu09]. One of the recent achievements in financial econometrics is the introduction of the concept of realized volatility [AS05] , which allows to consistently estimate the accumulated price variation over some time interval by summing over the high-frequency squared returns. Let us consider that the logarithmic prices for a given asset are governed by 4
  6. 6. the following diffusion process, yt = t 0 µ(s)ds + t 0 σ(s)dWs. (1) The object of interest is the amount of accumulated variation for the asset price over a time interval τ = [0, 1] (representing a trading day), called integrated variance. Mathematically it is given by the following expression, IV = 1 0 σ2 (s)ds. (2) Since intraday prices are not continuous, it is not feasible to compute the previ- ous integral; therefore, one needs to come up with an estimator for the integrated variance process. One feasible solution is to discretize the time interval τ = [0, 1] into a grid of subintervals t0 = 0 ≤ · · · ≤ tn = 1 of length ti − ti−1 = ∆ = 1/n, and then set the prices yi = yti . Under such circumstances, the realized volatil- ity estimator,defined as the sum of the squared returns, i.e. RV 2 = n−1 i=1 (yi+1 − yi)2 , (3) turns out to be a natural estimator for the integrated variance. The asymptotic properties of this estimator are especially striking when sampling occurs at an increasing frequency that is, small δ which, when assets trade every few seconds, is a realistic approximation to what we observe using the now commonly available transaction or quote-level sources of financial data. In particular, fully observing the sample path of an asset will perfectly reveal the volatility of that path. 2.2 Subsampling to remedy for noise As pointed out in the previous section, the realized variance estimator will converge to the quadratic variation of the process, i.e., the approximation of integrated volatility as the RV estimator seems natural since stochastic processes 5
  7. 7. theory shows that plim n−1 i (yi+1 − yi)2 = T =1 0 σ2 s ds (4) However, in real life, market microstructure noise appears when dealing with high-frequency data. Such noise captures a variety of frictions inherent in the trading process: bid-ask bounces, discreteness of price changes, differences in trade sizes, etc. Assume a portfolio of N assets with M price points for each asset. Assume that the log prices are contaminated by market microstructure noise ui , i.e. yi = y∗ i + ui. (5) In this case, the observed return is given by: ri = r∗ i + i, (6) with the noise intraday increment it = uit −uit−1 . Therefore, the RV estimator can be decomposed as: RV = RV ∗ + 2 M i=1 r∗ i i + M j=1 2 j . (7) The last term on the right can be interpreted as the (unobservable) realized variance of the noise process, while the second term is induced by potential dependence between the efficient price and the noise. Based on this decompo- sition, RV is a biased estimator for IV [Lun06]. Two-Scaled Realized Variance Estimator One approach to overcome such a drawback consists in explicitly incorporating microstructure noise into the analysis, and some estimators have been devel- oped such that they make use of all the data, no matter how high the frequency and how noisy it is. These methods decompose the total observed variance into a component attributable to the fundamental price and another to the market 6
  8. 8. microstructure noise [Yu09]. One of these consistent estimators, named Two Scaled Realized Variance (TSRV), was first proposed by Zhang et al. [AS05]. The underlying idea is to sample sparsely at some lower frequency and to evaluate the quadratic variation at the two frequencies. Averaging the results over the entire sampling, and taking a suitable linear combination, one obtains a consistent and asymptotically unbi- ased estimator of IV. More precisely, suppose T = {t1, . . . , tn} is a vector containing the times of the observed log prices in a certain trading day. Then T is partitioned into K non-overlapping sub-grids with equal number of observations. The kth (k = 1, 2,..., K) sub-grid extracts the observations from the whole intraday data with following times attached: Tk = {tk−1, tk−1+K, ..., tk−1+nkK}, where nk is the largest integer so that the (tk−1+nkK)th observation is included in Tk. The TSRV is calculated as follows: TSRV 2 = 1 K K k=1 ti,ti+1∈Tk (yti+1 − yti )2 − n n tj ,tj+1∈T (ytj+1 − ytj )2 (8) where yj is log price process, n is the total observations in an intraday dataset, and n = n−K+1 K . 2.3 Realized Kernel Realized kernel estimators introduced by Bandorff-Nielsen et al. 2011 can be used to estimate the quadratic variation of an underlying price process from high frequency noisy data guaranteeing consistency and positive semi- definiteness[LS09]. The multivariate realized kernel is defined as K(X) = H h=−H k( h H + 1 )γh, (9) 7
  9. 9. where γh is matrix of autocovariances given by γh =    n j=h+2 rjrT j−h, h ≥ 0 γh = γT −h, h < 0 (10) and k(x) is Parzen function and is given by k(x) =    1 − 6x2 + 6x3 , 0 ≤ x ≤ 1 2 2(1 − x)3 , 1 2 ≤ x ≤ 1 0, x ≥ 1 (11) Here rj is the high frequency return. We focus on the Parzen function be- cause it satisfies the smoothness conditions and is guaranteed to produce a non-negative estimate. The preferred choice of bandwidth is H∗ = c∗ ξ4/5 n3/5 where c∗ = 3.5314 for Parzen function and ξ2 = ω2 / √ IQ denotes the noise- to-signal ratio, ω2 is a measure of microstructure noise variance and IQ is the integrated quarticity[LS09]. 2.4 Threshold Realized Variance The three estimators described above are inconsistent when there are jumps in the prices. For an estimator to work well in presence of jumps, it has to detect the jumps and not include those price points in volatility calculation where there is a jump. Corsi et al.(2010) introduce the family of threshold realized variance estimators which are consistent in presence of jumps [Ren10]. If the difference in log prices (returns) is above a certain threshold, it is not included in calculation of variance. Threshold realized variance is defined as follows: TRVδ(y)t = [T/δ] j=1 (yj − yj−1)2 1{(yj −yj−1)2≤Θ(δ)} (12) The threshold function has to satisfy lim δ→0 Θ(δ) = 0 and lim δ→0 δlog(1 δ ) Θ(δ) = 0 (13) 8
  10. 10. i.e., it has to vanish slower than the continuity of Brownian motion in order to have convergence in probability (consistency). The two threshold functions we work with are as follows: TRV1 = [T/δ] j=1 (yj − yj−1)2 1{(yj −yj−1)2≤log( 1 δ ) √ δ} (14) TRV2 = [T/δ] j=1 (yj − yj−1)2 1{(yj −yj−1)2≤ √ δ} (15) where T denotes the length of the interval (1 in our case) and δ = 1 number of observations (16) 3 Simulation The aim of this section is to study the behavior of the different estimators in a controlled environment so we can analyze its performance under various scenarios. The main advantage of simulating is that it provides us with the ”true” variance-covariance matrix, since under the Heston model volatility is also simulated, which will allow us to compare the performance of the estimators. The section starts by exposing the foundations of the framework under which we run the simulations as well as the main technical aspects one has to take into account. Later, we offer some observations and comments on the results to highlight the main properties of each estimator. 3.1 Setup The first step to simulate the covariance matrix is to impose some structure to the underlying asset’s price and volatility. In a discrete time framework GARCH family processes are widely used for its appeal. The analog in a continuous time setting corresponds to a Geometric Brownian Motion price process and a Heston structure for the volatility. Therefore, the market model will be implemented by first simulating a price process and then generating its implied volatility given 9
  11. 11. the Heston model. The price process is specified by dyt = µytdt + σtytdWS t (17) where µ represents the return of the asset and σt the instantaneous volatility. The instantaneous variance is determined by a CIR process of the form dσ2 t = κ(θ − σt)dt + ξσtdWν t (18) where dWS t and dWν t are Brownian Motion processes with covariance ρ. In the variance equation, θ is the long variance, or long run average price vari- ance; as t tends to infinity the expected value of σt tends to θ. κ is the rate at which the volatility reverts to θ and ξ is the volatility of the volatility, which determines the variance of σt[Hes93]. Simulating prices using the Heston model allows us to compute the integrated variance-covariance (i.e. the true volatility) by adding the variance-covariance matrix simulated at each step of the time interval. To compare the performance of the estimators we construct a matrix A defined as the subtraction of the variance-covariance matrix of each estimator minus the integrated variance-covariance, ARV = RV − IV (19) AT SRV = TSRV − IV ARK = RK − IV. RV(TSRV,RK) corresponds to the variance-covariance matrix associated with the RV estimator (TSRV estimator, RK estimator). Afterwards, these matrices are used to compute the Frobenius and the Infinite norm that are defined as, Frobenius Norm ||A||f = m i=1 m j=1 |aij|2 (20) Infinite Norm ||A||i = max|aij|. (21) 10
  12. 12. Figure 1: Frobenius and Infinite norm for matrices A as defined in (20) to test the performance of the estimators on the benchmark scenario i.e. no jumps nor noise. The intuition behind these norms is that they are used as a metric to compare the performance of the estimators by computing how much their estimated val- ues differ from the ”true” values. This distance is captured by the elements of the matrices ARV , AT SRV and ARK. The performance is assessed either by means of the Frobenius norm, i.e. by computing the sum of all of the elements of these matrices, either by means of the Infinite Norm, i.e. by comparing be- tween the elements with the highest value. Figure 1 shows the benchmark scenario i.e. simulation of prices without adding jumps nor noise. We can observe that RV is consistent and its perfor- mance is increasing with sampling frequency. This behavior also holds for all other estimators. It is important to point out that the TSRV outperforms the RV for low frequencies up to a threshold. Beyond that point the RV is the best among our estimators. 3.2 Finding optimal parameters for TSRV & RK The optimal value of K for TSRV is cn2/3 where c = 12ω2 IQ 1/3 and n is the number of data points. 11
  13. 13. The preferred choice of bandwidth for RK is H∗ = c∗ ξ4/5 n3/5 where c∗ = 3.5314 for Parzen function and ξ2 = ω2 / √ IQ. We estimate ω2 = RV/2n and IQ = RV 2 sparse as explained in [LS09]. We find subsampled realized variance based on returns from every 50th price point. More precisely, we compute a total of 50 realized variances by shifting the first observation. RVsparse is simply the average of these estimators. We get K and H for every stock from this exercise, global K and H is simply the average of these K’s and H’s. 3.3 Modeling price jumps In a liquid and efficient financial market prices are set so that they reflect all available information regarding all traded products in the exchange. Following this rationale, when new information is released or generated prices will change accordingly. This shift in the price can be either smooth or very sharp. We will focus on the latter, since a sharp increase or decrease in the price can be understood as a jump. In finance, the building block of a jump model is the Poisson process. Let us consider a sequence {τi}i≥1 of independent exponential random variables with parameter λ, that is, with cumulative distribution function defined as F(y) = P(τi ≥ y) = e−λy . (22) Let Tn = n i=1 τi, then the process N(t) = n≥1 1t≤Tn , (23) is called a Poisson process with rate λ. In our case, τ corresponds to the waiting times between jumps, and N(t) the number of jumps occurred up to time t. Poisson processes have turned out to be the paradigm for jump models in dif- fusion process, since it shares with the Brownian motion the property of in- crements being independent and stationary, i.e., for every t > s the increment Nt − Ns is independent of the history of the process up to time s. 12
  14. 14. In reality, however, simple Poisson processes are of little interest since they only consider one possible jump size and assume that they occur strictly one after the other. To relax such assumptions we considered a Compound Pois- son process where waiting times between jumps are exponentially distributed, whereas jump sizes can have an arbitrary distribution. More precisely, letting B1, B2, ... denote the i.i.d. sequential amplitude for jumps, the total amplitude of the jump at time t, A(t) is given by A(t) = N(t) n=1 Bn, (24) where N(t) are the number of jumps occurring at time t. In practice, we first generate the waiting times {Xn = tn − tn−1}, i.e., time intervals between events by considering that they are exponentially distributed thus following the same distribution as described in 22. From such a distribution on can explicitly compute the set of times {tn} at which the events take place, tn = tn−1 − 1 λ ln(U), (25) where U is a Uniform [0, 1] function. Therefore, by simulating a random [0, 1] vector and defining a λ such that only one jump takes place per day we obtained the set of times at which jumps occur. The next step consists on modeling the amplitude of those jumps. Efficient pricing theory dictates that a jump in the price can either be a consequence of supply and demand adjustments or due to new information released. In both cases this jump can move in either direction. Hence, the best way to model the direction and size of the shift in price is by assigning to each jump a draw from a (standard) normal distribution corresponding to its amplitude. Figure 2 plots the performance of our estimators when we add jumps to the price process. In this case, sampling frequency provides better estimates as well but the first three estimators are inconsistent, thus unreliable. On the other hand, the Threshold Realized Variance estimators are consistent under jumps 13
  15. 15. Figure 2: Frobenius and Infinite norm for matrices A as defined in (20)to test the performance of the estimators in the presence of jumps. and both of them outperform the inconsistent estimators and both behave very similarly. 3.4 Implementing market microstructure noise To study the effect of market microstructure on the behavior of estimators we need to add noise in the price process. A simple yet effective approach is to add a random term to the price. This term will consist of random draws form a normal distribution. Our main concern regarding noise is how will it affect the behavior of the estimators depending on its size. Figure 3 shows the impact of noise on the estimation of volatility depending on the size of this distortion measured using the Frobenius norm. On the horizontal axis we measure the sampling frequency in a way so that values far from the origin represent estimators using very few data points while points close to the origin are true high-frequency estimators. We can see that for small interferences the effect is insignificant. When we increase the size from σ2 noise = 0.025 onwards, only consistent estimators such as the TSRV and Kernel remain reliable, whereas the RV is clearly outperformed due to its asymptotic inconsistency in the presence of interferences. 14
  16. 16. Figure 3: Impact of noise using Frobenius norm 4 Data implementation In the previous section we described how the estimators behave under differ- ent ways of simulating prices. The next steps are assessing whether estimates from intraday data outperform estimates using daily data, determining which estimator perform the best and addressing the question whether intraday prices have jumps or not. 4.1 Data and methodology All the estimators are obtained by using intraday tick-by-tick prices on the trades executed on October 27th, 2010 in the S&P 100 index. This dataset comprises the prices of the 94 stocks traded on that day from 09:30 to 16:00, the opening and closing times of the exchange. The integrated volatility is also estimated using daily data comprising the closing prices for the same stocks during the previous 60 days (relative to October 27). 15
  17. 17. Data cleaning Since the opening of the exchange, assets can be traded at will if two parties agree to do so. This means that trades between agents can materialize at any time. Therefore, the first problem to overcome in tick-by-tick data is its lack of synchronization. Non-synchronous trading delivers fresh (trade or quote) prices at irregularly spaced times which differ across stocks. Raw data on trades is saved with actual times, so for example a trade would be saved with a timestamp 090345 if it were executed at 09:03:45. In order to standardize the timestamps we normalize the trading day consisting of 6.5 hours into a [0, 1] interval split into seconds (23400 seconds). As such, since the exchange is open for 6.5 hours, this particular trade would have a normalized timestamp of 0.00833. The second issue that needs to be tackled is the presence of simultaneous trades. In that case, the approach consists in keeping only the last trade recorded. Since they are simultaneous, there is no price distortion by arbitrarily choosing one of them, since we apply the same policy for all cases. Last but not least, we may have a problem of low liquidity in some assets in the sense that they are traded very few times during a day. This has a tremendous impact on the estimators since at some point we need to invert the variance-covariance matrix. If those assets are not traded frequently or the price does not change significantly the variance of those assets will be extremely small, resulting in the impossibility to compute its inverse. We set a threshold on the minimum daily variance at 0.001. An asset with a variance lower than the threshold is dropped out of the sample since it makes the estimation unfeasible. After running these cleaning procedures we obtain a clean database containing 91 assets. The list of stocks is available in the 2. 4.2 Analysis At the core of our work lies the idea that some estimators provide better es- timates than others, as discussed in previous sections. Asymptotic properties were discussed to assess the desirability to use certain candidates in the pres- ence of specific distortions in the observed prices such as noise and jumps. This section tries to determine if there is evidence of a systematic outperformance of any estimator over all other studied candidates. Explicit comparisons between 16
  18. 18. Figure 4: Performance of RV and Kernel methods to assess which estimator is better have been hampered by the multi- tude of metrics to use in forming the comparisons. The distance between two covariance matrices is not well defined, and it is certainly not obvious that all elements of this difference should be treated as equally important. To overcome this drawback, an asset allocation perspective is introduced to measure the value of covariance information. As shown by Engle in et al., realized volatility is the smallest for the correctly specified covariance matrix for any vector of expected returns [Col06]. Making use of this property, we will construct the Global Minimum Variance Portfolio using data on the S&P 100 index. By means of the previous theoretical result, the best estimator should yield a GMVP with the least variance for the same expected return. To assess the performance of each estimator, we compute the variance of returns of a Buy-and-Hold strategy on these portfolios over the following 30 trading days (October 27th to December 9th 2010) as a function of intraday sampling fre- quency. As explained by Lunde et al., estimators may yield implausible results when working on real data [SS11]. TSRV happens to exhibit this misbehvior with our data. Nonehteless, it is worth mentioning that TSRV computed using sampling 17
  19. 19. frequency of 5 seconds outperforms RV 5 seconds due to the large number of datapoints. However, as the number of data points decreases, the variance- covariance matrix obtained from TSRV is not invertible even after regulariza- tion. Figure 4 shows the variance of returns for the Kernel estimator against the Re- alized Volatility. We also include an equally weighted portfolio as a benchmark. We can clearly observe that the RV estimator systematically outperforms the Kernel with the exception of the highest sampling frequencies where market microstructure noise becomes relevant and RV is inconsistent. Additionally, Figure 5 plots the variance of the minimum variance portfolio as a function of sampling frequency for the Realized Volatility against the Threshold RV. In- terestingly, both specifications of the TRV and RV behave very similarly. This fact may lead us to think that there is no evidence of jumps in the data or size and/or frequency of jumps must be small as the three values are very close. To understand why the two specifications for TRV perform differently we need to go back to its definitions in section 2.4. The difference between the two arises when defining how we measure a jump, i.e., the threshold we set on the amplitude of the price change required to be considered as a jump. The first specification has a broader range, so only ex- treme events will be classified as jumps. In contrast, the second case is less strict and small variations can go beyond the threshold. It is important to note that the RV performs better at the maximum frequency as well as relatively low frequencies but the lowest variance is achieved by all estimators at a 30 seconds frequency. This is consistent with empirical studies such as [Pat11] in which beating a high-frequency (15 to 120 seconds) RV constitutes a serious (but as seen not impossible) challenge. To conclude this section we offer a final remark. One may wonder whether it makes sense to implement this sort of estimators to the conventional daily data approach in case they could improve volatility estimates. Table 1 below shows the volatility estimates of the Buy-and-Hold minimum variance portfolio as described previously using daily data instead of intraday data. The covariances were computed using data on the previous 60 trading days to October 27. Not 18
  20. 20. Figure 5: Performance of RV and Threshold RV Estimator Variance RCOV from daily data 5.370 TSRV from daily data 218.7957 RK from daily data 16.9148 Uniformly weighted portfolio 0.8372 Table 1: Performance of estimators using daily data surprisingly, the outcome of this strategy is extremely poor especially in the TSRV, the one that relies more on data abundance to perform the sub-sampling. 5 Conclusions In this study we analyzed the performance of multiple methods to estimate volatility using high-frequency data. We saw that in contrast with stochastic processes theory, using the highest available frequency does not necessarily re- sult in the best approximation to a continuous sample path because financial markets’ microstructure plays a distortionary role that leads to (in some cases) inconsistent estimators. This study chooses a series of non-parametric volatility estimators to mea- sure realized volatility. The starting point was to measure volatility as an ex- 19
  21. 21. post quadratic variation of asset prices, giving rise to the Realized Volatility estimator. The next step in adding complexity to the process of measuring volatility is correcting for jumps and noise. To do that, we first decided to compare the performance of the Realized variance estimator that is inconsistent under noise with the performance of consistent estimators under noise, i.e., TSRV and Real- ized Kernel estimators. As expected, we found that when noise is present TSRV and Realized Kernel outperform Realized Variance, specially for small sampling frequencies. The same procedure was applied in the presence of jumps. Therefore, we com- pared Realized Variance (inconsistent in the presence of jumps) with Threshold Realized variance estimators, that are consistent under jumps. In this case and consistent with the literature, we found that they outperform the other estima- tors. Our results support the hypothesis of Corsi et al.. that price jumps do have an impact on future volatility [Ren10]. Lastly, we tested again the estimators using data from S&P 100. Excluding TSRV, that fails to give plausible outcomes, the results are consistent with the ones obtained using simulated prices. Realized Kernel performs better than Realized Variance estimator for sampling frequencies smaller than 5 seconds. Therefore, we conclude that market microstructure noise emerges for sampling frequencies smaller than 5 seconds. Minimum values for the ex-post portfolio’s variance are found for sampling frequencies equal to 30 seconds for RV, TRV1 and TRV2 estimators. Although TRV1 outperforms RV we cannot conclude that jumps do exist in intraday data, since both specifications behave very similarly. To get more insights into the nature of jumps, threshold versions of TSRV and Realized Kernel should be implemented to assess their performance. 20
  22. 22. References [AS05] Lan Zhang, Per A. Mykland, Yacine Ait-Sahalia. A tale of two time scales: Determining integrated volatility with noisy high-frequency data. Journal of the American Statistical Association, 101 (472):1394– 1411, 2005. [Col06] Robert Engle,Ricardo Colaccito. Testing and valuing dynamic correla- tions for asset allocation. Journal of Business and Economic Statistics, 24(2), 2006. [Ebe01] Torben G. Andersen, Tim Bollerslev, Francis X. Diebold, Heiko Ebens. The distribution of realized stock return volatility. Joural of Financial Econometrics, 61:43–76, 2001. [Eng82] Robert Engle. Autoregressive conditional heteroskedasticity with es- timates of the variance of u.k. inflation. Econometrica, 50:987–1007, 1982. [Hes93] Steven L. Heston. A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6(2):327–343, 1993. [LS09] O.E. Barndoff-Nielsen,P. Reinhard Hansen, A. Lunde and N. Shephard. Realized kernels in practice: traded and quotes. The Econometrics Journal, 2009. [Lun06] Peter R. Hansen, Asger Lunde. Realized variance and market mi- crostructure noise. Journal of Business and Economic Statistics, 24(2):127–161, 2006. [Mik96] Tim Bollerslev, Hans O. Mikkelsen. Modeling and pricing long memory in stock market volatility. Journal of Econometrics, 73:151–184, 1996. [Par80] M. Parkinson. The extreme value method for estimating the variance of the rate of return. Journal of Business, 53:67–78, 1980. [Pat11] Andrew J. Patton. Data-based ranking of realised volatility estimators. Journal of Econometrics, 3:284–303, 2011. 21
  23. 23. [Ren10] Fulvio Corsi, Davide Pirino, Roberto Reno. Journal of Econometrics, 159, 2010. [She08] Ole Barndorff-Nielsen, Peter R. Hansen, Asger Lunde, Neil Shephard. Realized kernels to measure ex post variation of equity prices in the presence of noise. Econometrica, 76(6):1481–1536, 2008. [SS11] Asger Lunde, Neil Shephard and Kevin Sheppard. Econometric analysis of vast covariance matrices using composite realized kernels. Working paper, 2011. [Yoo94] L.C.G Rogers, S. Satchell, Y. Yoon. Estimating the volatility of stock prices: a comparison of methods that use high and low prices. Applied Financial Economics, 4:241–247, 1994. [Yu09] Yacine Ait-Sahalia, Jianlin Yu. High frequency market microstructure noise estimates and liquidity measures. Journal of Mathematical Statis- tics, 161:422–457, 2009. [Zha00] Dennis Yang, Qiang Zhag. Drift-independent volatility estimation based on high, low, open, and close prices. Journal of Business, 73(3):477–491, 2000. 22
  24. 24. TICKER COMPANY NAME GICS SECTOR NAME STATUS MMM 3M CO Industrials ABT ABBOTT LABORATORIES Health Care ACN ACCENTURE PLC-CL A Information Technology ALL ALLSTATE CORP Financials MO ALTRIA GROUP INC Consumer Staples AMZN AMAZON.COM INC Consumer Discretionary AXP AMERICAN EXPRESS CO Financials AIG AMERICAN INTERNATIONAL GROUP Financials AMGN AMGEN INC Health Care APC ANADARKO PETROLEUM CORP Energy APA APACHE CORP Energy AAPL APPLE INC Information Technology T AT&T INC Telecommunication Services BAC BANK OF AMERICA CORP Financials BK BANK OF NEW YORK MELLON CORP Financials BAX BAXTER INTERNATIONAL INC Health Care BIIB BIOGEN IDEC INC Health Care BA BOEING CO/THE Industrials BMY BRISTOL-MYERS SQUIBB CO Health Care COF CAPITAL ONE FINANCIAL CORP Financials CAT CATERPILLAR INC Industrials CVX CHEVRON CORP Energy CSCO CISCO SYSTEMS INC Information Technology C CITIGROUP INC Financials dropped KO COCA-COLA CO/THE Consumer Staples CL COLGATE-PALMOLIVE CO Consumer Staples CMCSA COMCAST CORP-CLASS A Consumer Discretionary dropped COP CONOCOPHILLIPS Energy COST COSTCO WHOLESALE CORP Consumer Staples CVS CVS CAREMARK CORP Consumer Staples DVN DEVON ENERGY CORPORATION Energy DOW DOW CHEMICAL CO/THE Materials DD DU PONT (E.I.) DE NEMOURS Materials EBAY EBAY INC Information Technology EMC EMC CORP/MA Information Technology EMR EMERSON ELECTRIC CO Industrials EXC EXELON CORP Utilities XOM EXXON MOBIL CORP Energy FDX FEDEX CORP Industrials F FORD MOTOR CO Consumer Discretionary FCX FREEPORT-MCMORAN COPPER Materials GD GENERAL DYNAMICS CORP Industrials GE GENERAL ELECTRIC CO Industrials dropped GILD GILEAD SCIENCES INC Health Care GS GOLDMAN SACHS GROUP INC Financials GOOG GOOGLE INC-CL C Information Technology HAL HALLIBURTON CO Energy HPQ HEWLETT-PACKARD CO Information Technology HD HOME DEPOT INC Consumer Discretionary HON HONEYWELL INTERNATIONAL INC Industrials INTC INTEL CORP Information Technology IBM INTL BUSINESS MACHINES CORP Information Technology JNJ JOHNSON & JOHNSON Health Care JPM JPMORGAN CHASE & CO Financials LLY ELI LILLY & CO Health Care LMT LOCKHEED MARTIN CORP Industrials LOW LOWE’S COS INC Consumer Discretionary MA MASTERCARD INC-CLASS A Information Technology MCD MCDONALD’S CORP Consumer Discretionary MDT MEDTRONIC INC Health Care MRK MERCK & CO. INC. Health Care MET METLIFE INC Financials MSFT MICROSOFT CORP Information Technology MON MONSANTO CO Materials MS MORGAN STANLEY Financials NOV NATIONAL OILWELL VARCO INC Energy NKE NIKE INC -CL B Consumer Discretionary NSC NORFOLK SOUTHERN CORP Industrials OXY OCCIDENTAL PETROLEUM CORP Energy ORCL ORACLE CORP Information Technology PEP PEPSICO INC Consumer Staples PFE PFIZER INC Health Care PM PHILIP MORRIS INTERNATIONAL Consumer Staples PG PROCTER & GAMBLE CO/THE Consumer Staples QCOM QUALCOMM INC Information Technology RTN RAYTHEON COMPANY Industrials SLB SCHLUMBERGER LTD Energy SPG SIMON PROPERTY GROUP INC Financials SO SOUTHERN CO/THE Utilities SBUX STARBUCKS CORP Consumer Discretionary TGT TARGET CORP Consumer Discretionary TXN TEXAS INSTRUMENTS INC Information Technology TWX TIME WARNER INC Consumer Discretionary USB US BANCORP Financials UNP UNION PACIFIC CORP Industrials UNH UNITEDHEALTH GROUP INC Health Care UPS UNITED PARCEL SERVICE-CL B Industrials UTX UNITED TECHNOLOGIES CORP Industrials VZ VERIZON COMMUNICATIONS INC Telecommunication Services V VISA INC-CLASS A SHARES Information Technology WMT WAL-MART STORES INC Consumer Staples WAG WALGREEN CO Consumer Staples DIS WALT DISNEY CO/THE Consumer Discretionary WFC WELLS FARGO & CO Financials Table 2: S&P 100 index constituents for October 27th 2010 23

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