Disentangle the components of high-frequency financial data

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high-frequency financial data

Stanislaw Laniewski

Master’s Thesis to obtain the degree in Actuarial Science and Mathematical Finance University of Amsterdam

Faculty of Economics and Business Amsterdam School of Economics Author: Stanislaw Laniewski Student nr: 11086769

Email: stanislaw.a.laniewski@gmail.com Date: July 14, 2016

Supervisor: Mr. Merrick Zhen Li Second reader: Prof. Roger J.A. Laeven

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ii Stanislaw Laniewski — _{The components of price movement}

Statement of Originality

This document is written by Student Stanislaw Laniewski who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in

creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

This thesis explains components of price movements of financial as-sets: Brownian motion, microstructure noise and unexpected jumps, for each I provide economical and mathematical background. Combin-ing them in pairs and all three together, I create models to disentangle components, which is successful in every case. I find maximum likeli-hood estimators which are precise with a relative small variance.

Keywords Brownian motion, high-frequency data, market microstructure noise, Monte Carlo simulations, maximum likelihood estimation, jump diffusion

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Introduction

The technological advances in the last couple of years made it possible to create huge databases and analyse them in no time. One of its application in financial market is the possibility to sample price data at a very high-frequency for a plethora of variables such as asset returns, currencies or interest rates. Thanks to that investors and researchers can use the realised volatility (which is the sum of squares of log-returns calculated every short interval, for example a few minutes) as an estimator for variance of returns. Price of financial assets is changing constantly. People all over the world trade them for plenty reasons. This seemingly chaotic movement is generally modelled by Brownian motion, which often can be seen in nature.

However, apart from continuous price movement, there are other components which determine the factual price. One part is discontinuous jump process reflecting shocks -that is unexpected arrival of new and crucial information. Researchers usually assume that jumps behave as Poisson process, but nowadays more complex types of processes are considered like Levy processes (compare early Merton (1976) with Markov chain approach in Eraker et al. (2003)).

Another essential component of price movement and unique to frequent sampling is the microstructure noise. There are many sources of that noise: usually the highest impact is due to transaction costs, bid-ask spread and information asymmetry. The more often I sample data, the more contamination in the results. It might seems as there is a pay-off between noise and a general principle that people want more data. However (A¨ıt-Sahalia et al.,2005) shows that by accounting for normal and independent noise it is optimal to sample as frequent as possible (use all the data). Furthermore, in the same paper authors prove that the answer does not change if noise terms are misspecified.

Decomposing each part is vital. When it is possible, hedging is more accurate, risk calculation and management can be improved and behaviour of financial market is easier to understand. In (A¨ıt-Sahalia,2004) author proves that through frequent sampling and considering only Brownian motion with jumps maximum likelihood (ML) can disentan-gle components perfectly. On the other hand, (Xiu,2010) and (A¨ıt-Sahalia et al.,2005) consider Brownian motion with noise. The former shows that estimators of integrated volatility and variance of noise are consistent, efficient and robust using quasi-maximum likelihood (QML), while the latter use ML (under assumption that volatility is constant) to study behaviour of asymptotic variance. Moreover, there are many extensions to this approach, like letting volatility to be stochastic ((Zhang et al., 2005)) or using kernels ((Barndorff-Nielsen et al.,2008)).

Nevertheless, in my research I focus on yet different approach. Jumps and noise are two distinctive components which have different sources, behaviour and implications. Therefore my model consider them simultaneously with Brownian motion. Firstly, I briefly introduce the theory behind each component separately and together. Then I set up Monte Carlo simulations of price movement. I analyse them, disentangle and find ML estimators. My results show that ML can be used to successfully distinguish each

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part of price movement not only in pairs, but also in full model. Furthermore, I show that it is surprisingly simple for daily data.

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Chapter 2

The components

I study a price movement process consisting of three components: Brownian motion, jumps and noise. In this chapter I introduce each part and underline its economical meaning.

2.1 Brownian motion

For a long time now after popularization of Black-Scholes formula Brownian motion is a cornerstone of financial market modelling. In Mathematics it is more often used under the name of Wiener process (Wt), which is defined as continuous-time stochastic

process that starts from 0, has independent Gaussian increments and continuous paths. One of its extensions especially useful in economics is Wiener process with a drift, which introduces movement with deterministic and stochastic part (more on it later).

Wiener process is often seen in nature, for example when you pour a drop of fat into milk or see movement of pollens in water. Robert Brown noticed them and studied it, since we call them Brownian motion. It was later at the beginning of 20th century when mathematicians Marian Smoluchowski and Albert Einstein proved that this movement can be described by Wiener processes.

Brownian motion was introduced to financial markets by (Merton,1969) and pop-ularised by (Black and Scholes, 1973) and (Merton, 1973). It assumes that the in-stantaneous log-return is a geometric Brownian motion. There were many additional assumptions like constant volatility, constant interest rate and so on, however, further models allowed to remove these conditions.

While Black-Scholes formula calculates no-arbitrage price for options, the assump-tion that the movement of log-returns (or log-price) on financial assets bases on Brow-nian motion is omnipresent. In my research it is no different - the main part of price movement, the major contributor, is Brownian motion.

2.1.1 Mathematics behind it

Now I introduce mathematics behind Brownian motion. This stochastic process can be written as

Xt= µt + σWt,

where X₀ = 0, µ is the drift coefficient or function, t is as usual time, σ is the volatility (thus σ2 is diffusion coefficient) and Wt is a standard Wiener process.

Moreover, I use high-frequency approach, thus the diffusive part is dominant. In-deed, the drift is of order dt, while stochastic part is of order (dt)1/2. That means by sampling very often the deterministic part can be omitted. This is in accord with empir-ical results (see for example (Merton, 1980)). Furthermore, using high-frequency data the estimation of the deterministic parameter has relatively huge error. As shown in

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(A¨ıt-Sahalia et al.,2005), adding drift does not change final results. Recapitulating all this, I set for simplicity µ = 0.

Finally, it is vital to properly define σ. Volatility on financial markets changes for some assets quite often. Especially during periods of higher uncertainty after negative shocks, the majority of financial instruments sees an increase in volatility. Thus it rea-sonable to consider stochastic volatility as opposed to constant. Nevertheless, crucial to this approach are results from (Xiu,2010): by intentionally taking volatility as constant (therefore misspecifying the model) and ”performing maximum likelihood estimation results are robust to stochastic volatility”. Bearing this is mind, I assume volatility to be constant.

To sum it up, from now on I focus on

Xt= σWt,

where σ is constant.

2.2 Jumps

In this section I explain what are jumps and how they differ from Brownian motion. I show why any Levy process suits modelling jumps and why I only work with Poisson, which is a special case with finite number of increments. Finally, I introduce Poisson process which I use as my jump component.

2.2.1 Unexpected significant changes

In a similar way as Brownian motion, Poisson process can be found in many applications from mathematics, physics, biology to financial economics. For my study especially the latter is important. This approach was introduced by Merton in (Merton,1976).

In my study I use Levy process to explain jumps in my model. The idea is that while Brownian motion is changing price continuously, jumps happen occasionally but they are usually more impactful. A price jump would be unanticipated and relatively drastic change of a price which cannot be explained just by (contaminated) Brownian motion. If we consider short time period and no jumps, the price movement would be bounded to small changes. By introducing jump component the change can be enormous (even in dt). This is further confirmed by empirical results and by intuition - unexpected yet vital information can drastically increase or decrease price.

The ’usual’ information which investors learn everyday causes marginal changes and therefore is a part of Brownian motion. Only extraordinary news move price by a great deal, but that happens rarely. Thus jumps can be considered as discrete. Researchers for many years modelled them using Poisson process.

Despite that, in modern research Poisson process is often replaced by more gener-alised Levy process. The main difference is that the latter can have infinite number of jumps in any time period. However, due to decomposition of Levy process to three independent simple Levy processes (linear transformation of Brownian motion, com-pound Poisson and pure jump martingale) this approach is not more complicated. In (A¨ıt-Sahalia,2004) it is shown that such decomposition has an economic interpretation. Each part represent continuous changes, huge jumps (which are rarely and therefore discrete) and small jumps (main difference between Poisson and Levy since there can be infinitely many), depending on how crucial and influential is the news. Yet author shows that maximum likelihood estimation is just as good as with Poisson process.

Different approach to explain jumps can be found in (Joulin et al., 2008) where authors find as the main source of jump lack of liquidity. It can be sudden or persistent, even inefficient ways to improve liquidity can lead to jump. Despite that it does not

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6 Stanislaw Laniewski — _{The components of price movement}

change my main goal - whatever the reasons for jump, modelling them is the same in my focus.

Recapitulating, I choose to model unexpected significant changes with Poisson pro-cess. Now I present mathematical background for it.

2.2.2 Poisson process

I consider time between jumps Ti. As news which create a shock happen unexpectedly,

the distribution of those events should be independent and identically distributed. Fur-thermore, just because there is a jump on one day it does not mean that tomorrow is relatively less likely to produce a new one. Therefore Ti should be memoryless, which

means ∀_s,t0 P (Ti > s + t | Ti > s) = P (Ti > t).

I use exponential distribution to describe time between occurrence of jumps (T_i ∼ Exp(λ). The insensitivity parameter λ characterises how often new jump appears.

I call N_t counting process, because it counts how many jumps where between 0 and t. I put N0 = 0 and Nt = sup{n : T1+ T2+ ... + Tn ¬ t}. As Ti are i.i.d. exponential

random variables with parameter λ I can write Nt2 − Nt1 ∼ P oisson(λ(t2 − t1)) for t2 > t1. Therefore process Nt has independent and stationary increments.

To sum it up, I call a stochastic process N_t Poisson process. It starts from N₀ = 0 and has independent and stationary increments with Poisson distribution.

Since jump size is unknown and differs one from another, the size of the jump itself is a random variable. I focus on log-size J_i. As I do not focus on modelling the jump, any known distribution suits me. I choose independent normal variables with mean j and variance β2. Furthermore, I assume that the jump size is independent from both Poisson process and Brownian motion. Dependency on those does not change model significantly, but it makes calculations more complicated and give little economic value.

Here I present basic properties of the aggregated jump size S(t) = PNt

i=1Ji. They

can be found in (Rolski et al.,2009).

1. Process {S(t)} has independent and stationary increments.

2. Moment generating function for S(t) is m_S(t)(s) = eλt(mY(s)−1)_{= e}λt(exp(jt+1₂β2t2)−1)_, where mJ is MGF of Yi.

3. Expected value is equal ES(t) = λtE(J) = λtj. 4. Variance is equal V arS(t) = λtE(J2) = λt(β2− j2_).

2.3 Market microstructure noise

Inseparable element of high-frequency financial data is microstructure noise. On liquid stocks there are thousands of operations every second, the price is in constant motion. However when studying each data without accounting for noise realised volatility is far from expected result. This is due to ’contamination’ of the data. Thus, instead of observing the real price movement, researchers see price and random noise term simultaneously.

One of the main sources of noise is bid-ask spread. It is the difference between the price for which sellers are ready to sell and for which buyers are ready to buy. It means there is no market efficient price but rather interval where true price lies. Usually less liquid assets have higher spread.

Other source is information asymmetry. When trading companies and traders have better knowledge, this superior information can lead to profit in bid-ask spread - even in risk-neutral world. Investors assume that since some people are buying for prices offered by trader then they have some additional information. In order to make up for losses caused by lack of information, trader can for example increase the spread even

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wider. More about information asymmetry in noise can be read inGlosten and Milgrom

(1985).

There are numerous other components of noise, such as but not limited to the discreteness of prices on some assets, the impact of huge size brokered operation or transaction costs.

As with many noises, I assume that microstructure noise is symmetrically distributed around 0. Moreover, I assume that it is independent of the log-price process and time. The latter assumption is often relaxed and studied - for long periods inA¨ıt-Sahalia et al.

(2005) or short periods (tick time) inLaeven et al. (2016).

As I mentioned before, it might seem as there is a pay-off between noise and how frequent to sample data. Yet (A¨ıt-Sahalia et al., 2005) proves that by accounting for normal and independent noise it is optimal to sample all the time (use all the data). What’s more, the answer does not change if noise terms are misspecified. That means in my study I model noise using normal random variables U_t with mean 0.

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Chapter 3

Disentangling the components

Now that I introduced each component of price movement, I run Monte Carlo simula-tions in Matlab. I start with a model as simple as it can be

Yi= ˜Xti− ˜Xti−1, where Y_i is the log-return, ˜Xti is the observed log-price.

Each following one is more complex until I reach full model with three components, that is

˜

Xti = Xti+ Uti+ Jti Yi = ˜Xti− ˜Xti−1

First I set starting values, some of them like volatility are set to be similar to real one on the market. Nevertheless, some parameters like λ (jump parameter which reflects how often jump appears) are purposely increased to make disentangling even more difficult.

3.1 The cornerstone: Brownian motion only

I start with the simplest case: price movement is just a Brownian motion. It is easier to understand the mathematics behind it, yet it provides good intuition on what happens when I add more components.

I assume that sampling intervals are deterministic and equal, that Brownian motion has constant volatility σ and the drift (deterministic) component µ equals 0. The impact of all assumption on both complexity of model and its ability to reflect real world market were discussed in previous chapter.

Therefore, log-price is equal to observed price ˜Xti = Xti (no contamination). Then since the drift (deterministic) component µ is set to 0 I have

Yi = ˜Xti− ˜Xti−1 = σ(Wti − Wti−1).

Thus Yiare normally distributed i.i.d. random variables with expected value equal 0

and variance σ2∆, where ∆ = ti− ti−1. I focus on a finite time period T which consists

of N intervals ∆, so ∆ = T N.

I calculate log-likelihood function for N observations

l(σ2) = −N ln(2πσ2∆)/2 − (2σ2∆)−1ΣN_j=1Yj.

Taking derivative with respect to σ2 yields −N 2 4π∆ 2πσ2_∆+ 4∆ 4σ4_∆2Σ N j=1Yj2= = −N σ2 + ΣN_j=1Yj σ4_∆ 8

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and by solving I get the maximum likelihood estimate ˆ σ2 = 1 N ∆ N X j=1 Y_j2 = 1 T N X j=1 Y_j2.

I calculate its moments

E[ˆσ2] = 1 N ∆ N X j=1 E[Y_j2] = N N ∆σ 2_{∆ = σ}2 var[ˆσ2] = 1 T2var   N X j=1 Y_j2  = 1 T2   N X j=1 var[Y_j2]  = N T22σ 4_∆2 ₌ 2σ4∆ T . Finally I find asymptotic variance to be equal

avar(ˆσ2) = ∆Eh−¨l(σ2)i−1= 2σ4∆ and therefore asymptotic distribution

T1/2(ˆσ2− σ2_{) −→}T →∞_{N (0, 2σ}4_∆).

The final two results are especially important - it is not the only time asymptotic variance is 2σ4_{∆, as I show in next sections.}

3.2 What changes when I make some noise

Firstly, I focus on the most popular approach - modelling price movement by Brownian motion and microstructure noise. Its principle is that investors do not observe real, fair price but it is contaminated by random error. Therefore my model is

Yi= ˜Xti− ˜Xti−1 = σ(Wti− Wti−1) + Uti− Uti−1, where Uti is the noise.

As I mentioned before, I assume that microstructure noise is symmetrically dis-tributed around 0. Furthermore, I assume that it is independent of the log-price process and time. I model Ut as i.i.d. normal random variables with variance a, because ( A¨ıt-Sahalia et al.,2005) and (Xiu,2010) show that misspecified noise distribution is not a problem (maximum likelihood estimators are robust to non-Gaussian contamination).

Before I calculate log-likelihood function, I want to stop here and point two interest-ing and vital differences between this model and the previous one with just Brownian motion.

Firstly, I observe that by introducing i.i.d. random variables with mean 0 and vari-ance α2, I can rewrite Y_i as moving-average (MA(1)) process. Indeed,

Yi= ˜Xti− ˜Xti−1 = σ(Wti− Wti−1) + Uti− Uti−1 = i+ θi−1, where

var[Yi] = α2(1 + θ2) = σ2∆ + 2a2

cov(Yi, Yi−1) = α2θ = −a2

∀k2 cov(Yi, Yi−k) = 0.

Therefore variance in presence of noise increases. Without microstructure contam-ination var[Yi] = σ2∆, but now noise component adds 2a2 to that. I expected this

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result, as negative effect of noise in estimating models led to the term ’microstructure contamination’. It is also confirmed by theoretical models which try to explain the noise. Secondly, it is interesting to note that from cov(Yi, Yi−1) = α2θ = −a2 I get θ < 0.

This result means that (some part of) empirical autocorrelation of log-returns is due to microstructure noise, since adding noise component made them negatively autocorre-lated. Among others, it was studied by (French and Roll,1986) who suggested changing variance estimators, however, it does not touch my focus.

There are two approaches for log-likelihood function - either use quasi-maximum likelihood (QML) method on misspecified model or change the model to incorporate the noise.

The former has simpler mathematics, however, in that case I should only sample in fixed time periods as shown by (A¨ıt-Sahalia et al.,2005), thus neglecting the majority of data. Nevertheless, I start with it to give background for the more complex latter model. Then I focus on incorporating the noise and altering the model to get (from statistical point of view) desired situation in which I use all the data.

By maximising l(σ2) from previous section, thus not accounting for noise in the model itself, I get following moments (calculations of variance can be found for example in (A¨ıt-Sahalia et al.,2005)). E[ˆσ2] = 1 N ∆ N X j=1 E[Y_j2] = N N ∆(σ 2_{∆ + 2a}2_{) = σ}2₊2a2 ∆ var[ˆσ2] = 1 T2var   N X j=1 Y_j2  = 1 T2   N X j=1 var[Y_j2]  = 1 T2   N X j,k=1 cov[Y_j2, Y_k2]   but

cov[Y_j2, Y_k2] = 2cov[Yj, Yk]2+ cum(Yj, Yj, Yk, Yk),

where cum stands for cumulants. To derive final result I use their properties. I write down general derivation (when U has any symmetrical distribution around 0) which uses forth cumulant cum4(U ) = E[U4] − 3(E[U2])2. However, this definition means that

in case of normal random variables, cum₄(U ) = 0.

var[ˆσ2] = 2(σ4∆2+ 4σ2∆a2+ 6a4+ 2cum₄(U ))/(T ∗ ∆) − 2(2a4+ cum₄(U ))/T2

U ∼ N(0,a) simplifies result to

var[ˆσ2] = 2(σ4∆2+ 4σ2∆a2+ 6a4)/(T ∗ ∆) − 4a4/T2.

Let me underline that this result suggests taking as large ∆ as possible. Indeed, when T → ∞ then var[ ˆσ2_{] → 0, thus root mean squared error depends mostly on}

bias 2a

2

∆ . But when noise fade (a → 0) the situation is typical again - optimal ∆ goes to 0. Nevertheless by taking such small ∆ when noise with variance a is present, ˆσ2 is basically estimator of a - useful approach when volatility of log-price movement is stochastic.

The above calculations are the base for the first Monte Carlo simulations. Before I present them, let me first show what happens when I do model noise.

Log-likelihood function is then

l(σ2, a2) = −N ln(2π)/2 −1

2ln det(Ω) − 1 2Y

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where Y0 = (Y₁, ..., Yn) and Ω is           2a2+ ∆σ2 −a2 ₀ _{. . .} ₀ −a2 _2a2_{+ ∆σ}2 _−a2 . .. .._. 0 −a2 _2a2_{+ ∆σ}2 . .. ₀ .. . . .. . .. . .. −a2 0 . . . 0 −a2 _2a2_{+ ∆σ}2          

For my Monte Carlo simulations I am particularly interested in asymptotic variance. As mentioned before, log-price movement process can be considered as MA(1). Therefore I can use known algorithms such as delta method to get the following result:

avar(ˆσ2, ˆa2) = 4(σ6∆(4a2+ σ2∆))12 + 2σ4∆ −σ2∆h −σ2_∆h _∆/2(2a2_{+ σ}2_{∆) ∗ h} ! (3.1) where h ≡ h(∆, σ2, a2) = 2a2+ σ2∆ + (σ2∆(4a2+ σ2∆))12.

Full derivation can be checked in for example (A¨ıt-Sahalia et al., 2005), where it is also noted that sampling as often as possible is optimal in that case (asymptotic variance increases when ∆ does).

Let me now briefly discuss about relaxing two assumptions - the drift of price and normality of noise. The former, as mentioned before, is set to 0 since it does not change anything from economical point of view. What is more, even if the drift component would not be 0, the formula for asymptotic variance stays the same, so it would not change my simulations.

Furthermore, it is noted in (A¨ıt-Sahalia and Yu,2009) that if noise is non-Gaussian, asymptotic variance does not change much, namely

avarnon−Gaussian(ˆσ2, ˆa2) = avarGaussian(ˆσ2, ˆa2) + cum4[U ]

0 0 0 ∆

! .

This result derives from the fact that when noise distribution is non-Gaussian, the second order moment structure of log-prices does not differ. Therefore the moment functions are not biased.

What is more, it is better to misspecify noise and model it as Gaussian rather than omit it at all. By factoring out a I show that

avar(ˆσ2) = 4(σ6∆(4a2+ σ2∆))12 + 2σ4∆ = 8σ3a∆12 + 2σ2∆ + o(∆)

Therefore if there was no noise modelled, I would lose ∆12 efficiency. To sum it up, my assumption of normally distributed noise let me calculate better estimators without having to worry about misspecification of its distribution.

This asymptotic variance let me do the following simulations.

3.2.1 Values for Monte Carlo simulations

First, I set parameters that I use throughout my simulations. Even though I do not model jumps yet, I want to maintain the same values of constants for each case, thus I bear this in mind when setting numbers.

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I choose time period of one year. It is long enough to witness some jumps and it is easy to find real data on yearly basis about many statistics (like yearly volatility).

While using different sampling times sounds interesting in case of model without noise (which exists nonetheless), the majority of estimators are be better off by sampling as often as possible. Bearing in mind that a financial year is 252 days long and a financial day is 6.5 hours long (that is the time when stock exchange is open) I set N = 252 ∗ 6.5 = 1638. However I often change this parameter to show different results. Therefore, my interval ∆ is T /N = 1 hour.

I assume I model the price of an asset with yearly volatility of 30%. It is a realistic value for indexes or some equities. As for variance of noise, I follow other researchers and set it at a = 0.0015, which means that price is subject to noise with standard deviation 0.15% of the efficient price.

In each case, I simulate 2000 different scenarios - this is number of paths I choose for Monte Carlo.

This is the base for my simulations. Sometimes I alter N and ∆ to show different results - in each case I underline the change. To accentuate high-frequency approach, I also simulate and disentangle data from one trading day (T = 1 day) but I sample every 5 seconds (thus N = 4680).

To sum it up, my parameters are as follows: • T = 1 (financial) year • N = 1638 • ∆ = T /N = 1/1638 of financial year • σ = 0.3 • a = 0.0015 • Number of simulations: 2000.

3.2.2 Simulation of Brownian motion with noise not modelled

The simulation is simple - first I create Brownian motion, then add noise U and derive the estimators. I calculate their bias and variance for each path.

For those parameters I calculate also theoretical values which are

E[ˆσ2] = σ2+2a 2 ∆ = 0.097371 std[ˆσ2] = (var[ˆσ2])12 = h 2(σ4∆2+ 4σ2∆a2+ 6a4)/(T ∗ ∆) − 4a4/T2i 1 2 = 0.003407 From my simulation I get the following result: ˆσ2 = 0.097351 with standard deviation of 0.003415. Comparing them with theoretical values we see that the result is close. However, due to the fact I am not accounting for noise, both theoretical and empirical estimators are off from real value (0.0974 instead of 0.09). That means also that hourly sampling is a bit too often. Let me now show two more cases: sample twice a day (N = 504) and every 20 minutes (N = 4914). In the first case I get mean 0.092177 and standard deviation 0.005952, while in the second mean equals 0.112152 with standard deviation of 0.002270. Those results show that by sampling less often noise has lower impact on σ, which is closer to real value, but its standard deviation increases. On the other hand smaller interval rises mean as I sample too frequently - noise impact is greater.

Finally for daily sampling (N = 252), which is also used further in my study, ˆσ2 = 0.091548 and standard deviation of 0.007907. This result is really close to the real value,

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which suggest that if daily sampling is frequent enough, there is less need to model noise explicitly.

As I change ∆, there is a trade-off between variance of estimator and the impact of noise. Therefore A¨ıt-Sahalia et al. (2005) introduces optimal sampling interval as the one which minimise root mean square error

RM SE[ˆσ2] = E[ˆσ2] − σ22+ var[ˆσ2] 1/2 .

Below I plot how different ∆ changes RMSE. Optimal sampling intervals for T = 1 day is 22 minutes and for T = 1 month it is around 1 hour.

Figure 3.1: RMSE of estimator ˆσ2 - orange line is for T = 1 month, blue for T = 1 day, ∆ is in hours

3.2.3 Simulation of Brownian motion with noise modelled

With the same set of parameters as above, I do Monte Carlo simulation taking mi-crostructure noise into account. This changes sampling - I sample as often as I can now so that my sampling internal ∆ goes down. Thus I consider asymptotic behaviour of my estimator.

Variance of the estimator in the sample is quite close to asymptotic one in ∆ = 20 minutes intervals (equal 0.003368).

As expected, the result is even better if I take smaller ∆ = 4 minutes, when it goes down to 0.001764, as shown on the next page.

Those results are promising: high-frequency approach, usually contaminated by mi-crostructure noise, can be disentangled precisely. Moreover, I want to sample very often - even though intuitively it only increases the impact of noise, by modelling it I cope with that problem. This is vital for full model with three components. Below in a model with Brownian motion and jumps I show that in order to disentangle jumps I need to also sample quite often, which could create too much microstructure noise. Yet this result shows that if I model it, disentangling is still going to be quite precise.

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Figure 3.2: Plot shows unbiased estimator ˆσ2

Figure 3.3: Blue graph is calculated with ∆_blue= 20 minutes, yellow ∆_yellow= 4 minutes

Finally I use very high-frequent data, which means I sample every 5 seconds. Because Brownian motion component can be dominant over small changes (as its variance is σ2∆), sampling extremely often gives more space for contamination. Thus it may be more challenging to disentangle the parts of the price movement.

I do more simulations (7500) with a daily scope (T = 1 day) and sample every 5 seconds (N = 4680). My results still hold - disentangling is precise. The asymptotic behaviour of the maximum likelihood estimator is as in3.1.

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Figure 3.4: High-frequency data gives tighter interval for ˆσ2

3.3 Brownian flow disturbed by unexpected jumps

In this part I model Brownian motion with jumps. This approach is useful when noise is negligible, for example when researchers do not sample very often or when they deal with noise using different methods.

I want to point out that disentangling jumps from Brownian motion is not as simple as it may seem given that the former is continuous and the latter is discrete and possibly much greater in size. First of all, different parameters describing jump process can lead to the same variance - that is, there are many combinations of λ (how often jump appear) and β (variance of the jump size) that yields the same variance to the model.

Secondly, even when I see a huge change in price this still could be a large realization of the Brownian motion. Especially when the chances of jump appearing are relatively low, a huge tail of even 3 standard deviations is more likely to be because of Brownian motion than due to the jump. All the data is discrete, so it certainly does depend on ∆, which leads me also to my next point.

Lastly, when I sample rarely to avoid microstructure noise (for example once a week), then the sum of all Brownian motion that happened during this time has huge variance thus making it (intuitively) difficult to distinguish between jump and the usual behaviour.

However this difficult problem of disentangling those two components has been solved by (A¨ıt-Sahalia,2004). In his work he proved a theorem which is a base for my Monte Carlo simulations in this case: using high-frequency data I can get an estimator for σ2 which is the closer to the real value the more often I sample. All above remarks are not a problem thanks to discreteness of Poisson jumps - high-frequency helps to separate them. Furthermore, it is not very complicated - just a maximum likelihood estimation. The theorem states that asymptotic variance of ML estimator ˆσ2 does not change when contaminated by Poisson jumps. Therefore my previous result is seen again, namely

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This result is further extended for any Levy process. Decomposing it into three parts, two of them (Brownian motion and Poisson jumps) are already taken care of. The only new component has infinite jumps, however, there are distinguishable from Brownian moves (during set period t they are larger than Brownian).

Nonetheless, such high-frequency approach in real world would be contaminated by microstructure noise provided I do not deal with it using different methods. I see this as a motivation for my final, full model with 3 components. But first I show Monte Carlo simulations for Brownian motion with jumps.

3.3.1 Simulations

Let me remind my assumptions: Brownian motion has negligible drift part, jumps follow Poisson process, Brownian motion, jump size and their numbers are all independent. All the extensions to those assumptions are not complex, but require a lot of work which does not help interpret the economical implications.

My equation is as follows ˜ Xti = Xti+ Jti Yi = ˜Xti− ˜Xti−1 = σW∆+ M∆ X i=1 Ji,

where ∆ = t_i− t_i−1 (since I assume same length intervals, it is constant for any i) and M_∆ is number of jumps that happened in time period ∆.

As I do not focus on modelling the jump, any known distribution suits me. I model it using independent normal variables with mean j and variance β2_.

For simulation I use the same set of parameters as before extended by:

• I start with daily data, N = 252. Even though the more frequent I sample, the better the result, in real life sampling more often would contain a lot of noise. Since I do not model it, it is economically reasonable to not sample more often. As I show further, daily sampling is enough to give precise estimation.

• Mean jump size j = 0. While it can be interesting to model negative jumps as more volatile as seen in empirical data, this is not the focus of my study.

• Variance of jump size β2 _{= 0.49 so that it has standard deviation of β = 0.7.}

• While empirical data shows that jumps for indexes happen relatively rarely (once a couple of years), I set λ = 4 to make disentangling more difficult.

My log-likelihood function is l = N X n=0 ln " _M X m=0 e−λ∆(λ∆)m m! 1 p 2π(σ2_{∆ + mβ}2₎exp −(Yn)2 2(σ2_{∆ + mβ}2₎ !# .

I focus on estimating σ2. Because it is analytically virtually impossible to disentangle it, I use numerical methods, that is minimising log-likelihoods in Matlab.

It is worth pointing out that thanks to high-frequency sampling, choosing appropri-ately small ∆ (that is huge N ), M rarely exceeds 1.

I find maximum likelihood estimator equal ˆσ2 _{= 0.089750, which is close to real}

value 0.09. In the figure below I drew asymptotic distribution - it can be seen that it is a good approximation to results from Monte Carlo simulations. The standard deviation of estimator is just 0.008199.

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Figure 3.5: Plot shows unbiased estimator ˆσ2

Now I take λ as unknown parameter as well. To make sure estimator ˆλ stays positive I add control component to my log-likelihood (increasing it by enormous amount provided λ is negative, for example by (sign(λ) − 1) ∗ 10000))).

I get daily ˆλdaily= 0.015972 which translates to yearly ˆλ = 4.024944. My estimator

for variance is still satisfying - ˆσ2 = 0.089421 with standard deviation 0.008373.

Figure 3.6: Modelling both ˆλ and ˆσ2 does not harness ability to disentangle components

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interested in daily jumps now. They happen quite often, but due to time smoothing effect they are not seen as Brownian motion component (with variance σ2∆) can be dominant over such small jumps given ∆ is great enough.

Now I focus on small jumps that happens on average 15 times daily, which implies my yearly λ = 3780. I choose period of one financial day (T = 1 day) and sample every 5 seconds (N = 4680). As shown below, disentangling is good, but I find maximum likelihood estimator equal ˆσ2 = 0.091432 with standard deviation 0.002946. This result shows some bias.

Figure 3.7: High-frequency shows bias in estimator ˆσ2

3.4 Full model of price movement consisting of Brownian

motion, microstructure noise and unexpected jumps

Finally I merge all the three components of price movement - Brownian motion, mi-crostructure noise and unexpected jumps - into one model. This is the closest model to real life applications, as it copes with high-frequency related problems (contamination) and the shocks of different nature (for example Brexit).

On the one hand, the idea seems easy. Even if the movement is rather complex, as stated above, when taking two components, disentangling is possible and quite accurate. The idea is therefore simple - let me take Brownian motion and contaminate it with microstructure noise. Because when I model it explicitly I can distinguish between those components, adding independent jump component should not pose much difficulties (as it is possible to disentangle it from Brownian motion which is the core of my interest). Having said that, I want to emphasize that to decompose jump component I need high-frequency approach. Thanks to that jumps are easy to pick off, however, there is the need to deal with the noise. If I were to neglect its impact, my variance would be even twice too big. Thus I choose the model with noise explicitly modelled and add jump component to it.

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˜

Xti = Xti+ Uti+ Jti Yi = ˜Xti− ˜Xti−1.

Let me gather all parameters in one place. I change some of them to show different approaches and results.

• A time period of T = 1 (financial) year. Itself it does not change modelling - it is T /N = ∆ which is vital. Nonetheless yearly approach is popular in economic data, so by keeping everything yearly I make it simple to confront and compare this study with others.

• I start with daily data N = 252. This is not very high-frequency, however, I show that it is enough to distinguish σ2 precisely given T =1 year. Then I change it to hourly data N = 1658.

• ∆ = T /N of financial year, which changes respectively whenever I change N . • As in every case, I set σ = 0.3 - real life value amongst others for indexes. • Similarly volatility of noise is a = 0.0015, because empirical data suggests so. • Variance of jump size β2_{= 0.49 so that it has standard deviation of β = 0.7.}

Nev-ertheless this has almost no implications on disentangling σ2 - by high-frequency approach I pick off jumps with ease.

• While empirical data shows that jumps for indexes happen relatively rarely (once a couple of years), I set λ = 4 to make disentangling more difficult. Because I showed that estimating λ does not harness ability to disentangle components and my focus is on σ2, I estimate the latter only. The former could be taken from historical data.

• Number of simulations: 2000.

With daily I get ˆσ2 = 0.090687 with standard deviation 0.008446. Furthermore, I showed in3.2.2that daily frequency is rare enough that there is no need to model noise! Therefore if daily frequency suits the researchers, they can get precise data simply by taking the same steps as in 3.3.

I show now that this is not the case for hourly data (N = 1638). As expected from

3.2.2, even though disentangling jumps is painless, noise has powerful impact.

Therefore I need to model the noise explicitly again, as in 3.2.3. By combining the models I get ˆσ2 = 0.089812 with standard deviation of 0.003423, more satisfying result. This result is superb as it combines all the intuitions with a good approximation. I sample as often as I want (desired from statistic and econometric point of view), I use all components yet my maximum likelihood estimator ˆσ2 is really close to real value.

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Figure 3.8: Sampling daily give great estimator ˆσ2 - both with and without modelling noise

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Figure 3.10: High frequency yields ˆσ2 close to real value when noise is modelled (please notice tighter X axis)

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Chapter 4

Conclusion

Behaviour of price movement is quite complex analytically, yet it gets easier thanks to Monte Carlo simulations. The movement is composed of three parts and each has economical explanation. I show that disentangling those components is possible with a good precision. Using maximum likelihood I calculate estimators which are close ap-proximation to the real value and how precise they are depending on how frequent is sampling.

Furthermore, real life data with noise can be accurately disentangled without the need to model the noise with daily and more sporadic sampling. If there is a need for more frequent data, it can be still done, only this time noise has to be modelled.

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Yacine A¨ıt-Sahalia. Disentangling diffusion from jumps. Journal of Financial Eco-nomics, 74(3):487–528, 2004.

Yacine A¨ıt-Sahalia and Jialin Yu. High frequency market microstructure noise estimates and liquidity measures. Annals of Applied Statistics, 3(1):422457, 2009.

Yacine A¨ıt-Sahalia, Per A Mykland, and Lan Zhang. How often to sample a continuous-time process in the presence of market microstructure noise. Review of Financial studies, 18(2):351–416, 2005.

Ole E Barndorff-Nielsen, Peter Reinhard Hansen, Asger Lunde, and Neil Shephard. Designing realized kernels to measure the ex post variation of equity prices in the presence of noise. Econometrica, 76(6):1481–1536, 2008.

Fischer Black and Myron Scholes. The pricing of options and corporate liabilities. The Journal of Political Economy, pages 637–654, 1973.

Bjørn Eraker, Michael Johannes, and Nicholas Polson. The impact of jumps in volatility and returns. The Journal of Finance, 58(3):1269–1300, 2003.

Kenneth R French and Richard Roll. Stock return variances: The arrival of information and the reaction of traders. Journal of Financial Economics, 17(1):5–26, 1986. Lawrence R Glosten and Paul R Milgrom. Bid, ask and transaction prices in a specialist

market with heterogeneously informed traders. Journal of Financial Economics, 14 (1):71–100, 1985.

Armand Joulin, Augustin Lefevre, Daniel Grunberg, and Jean-Philippe Bouchaud. Stock price jumps: news and volume play a minor role. Wilmott Magazine, September-October(26), 2008.

Roger JA Laeven, Merrick Zhen Li, and Michel H Vellekoop. Dependent microstructure noise and integrated volatility estimation from high-frequency data. 2016.

Robert C Merton. Lifetime portfolio selection under uncertainty: The continuous-time case. The Review of Economics and Statistics, pages 247–257, 1969.

Robert C Merton. Theory of rational option pricing. The Bell Journal of Economics and Management Science, pages 141–183, 1973.

Robert C Merton. Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1-2):125–144, 1976.

Robert C Merton. On estimating the expected return on the market: An exploratory investigation. Journal of Financial Economics, 8(4):323–361, 1980.

Tomasz Rolski, Hanspeter Schmidli, Volker Schmidt, and Jozef Teugels. Stochastic Processes for Insurance and Finance, volume 505. John Wiley & Sons, 2009.

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Dacheng Xiu. Quasi-maximum likelihood estimation of volatility with high frequency data. Journal of Econometrics, 159(1):235–250, 2010.

Lan Zhang, Per A Mykland, and Yacine A¨ıt-Sahalia. A tale of two time scales. Journal of the American Statistical Association, (100):1394–1411, 2005.

Disentangle the components of high-frequency financial data

high-frequency financial data

Stanislaw Laniewski

Contents

Chapter 1

Introduction

Chapter 2

The components

2.1

Brownian motion

2.2

Jumps

2.3

Market microstructure noise

Chapter 3

Disentangling the components

3.1

The cornerstone: Brownian motion only

3.2

What changes when I make some noise

3.3

Brownian flow disturbed by unexpected jumps

3.4

Full model of price movement consisting of Brownian

motion, microstructure noise and unexpected jumps

Chapter 4

Conclusion