Formación del profesorado

3.5.1 Model

Previously, when we described the Extended G-M model, the constant volatility is been applied for the evolution of true underlying stock price. However, in real financial market, the volatility varies from time to time, it is unlikely to be constant over the time and in the meantime, large price change tends to be followed by large price change and vise versa. Therefore, in order to better describe such phenomenon, autoregressive conditional heteroskedasticity (ARCH) model was often used by economists. Assume we have a error variance in a time series data follows an autoregressive (AR(q)) model. yt=a0+ q X i=1 aiyt−i+t (3.19)

Lett denote the error term, where it is the product of a stochastic piecezt(white noise) and a time dependent standard deviationσt, whereσt2 is the conditional volatility. The following defines an ARCH(q) model: t =σtzt (3.20) σ2_t =γ0+γ1t−2 1+· · ·+γq2t−q =γ0+ q X i=1 γ2t−i (3.21) γ0>0 and γi ≥0, ∀i >0

This model can be estimated by using ordinary least squares.

Under certain conditions, if the error variance has the assumption of autoregressive moving aver- age, then the model become the generalized autoregressive conditional heteroskedasticity (GARCH) model. The following defines a GARCH(p, q) model:

t =σtzt (3.22) σ2_t =ω+β12t−1+· · ·+βq2t−q+γ1σt−2 1+· · ·+γpσ2t−p (3.23) =ω+ q X i=1 βi2t−i+ p X i=1 γiσ2t−p

Herepis the order of the termσ2andq is the order of the term2To ensure the non-negativity and stationarity of the variance process, we need to have the following constraints:

ω >0, βi≥0, γi ≥0, p X i=1 βi+ q X i=1 γj <1 (3.24)

Now, by setting p = 1 and q = 1, we use the GARCH(1,1) volatility to replace the original constant volatility Extended G-M model. The true underly stock price will following evolution described as following: Vt =Vt−1+t (3.25) t =σtzt, zti.i.d∼ N(0,1) (3.26) σ_t2 =ω+βσ2t−1+γ2t−1 (3.27) =ω+βσ2_t−1+γσ_t−2 ₁z_t−2 ₁ ω >0, β >0, γ >0, β+γ <1

where the restrictions listed in the last line are for stationary purposes. And in general, both ω and γ have small values close to 0 and β is very close to 1.

3.5.2 Simulation Study

For the GARCH(1,1) volatility model, we also need to conduct the simulation study to see whether our Bayesian MCMC algorithm still works, now we have three additional new parameters ω, β and γ. Other than those prior already listed in Equation (3.12), those newly introduced parameters will have priors listed below.

ω ∼ U nif(0, uω)

β ∼ U nif(uβ,1) (3.28) γ ∼ U nif(0,1−uβ)

The likelihood functions keep the same as Equation (3.13). Detailed MCMC algorithms are provided in Appendix B.2. Below we presents the trace plot for all parameters, the posterior summary table for parameters and the Gelman-Rubin convergence diagnosis plots.

Table 3.4: Posterior Results Summary for Extended G-M Model with GARCH(1,1) Volatility

Par True Min Q1 Medium Mean Q3 Max Sd Mode

ω 0.08 0.0001 0.0673 0.0813 0.0769 0.0916 0.1000 0.0189 0.0958 β 0.98 0.9500 0.9619 0.9746 0.9748 0.9878 1.0000 0.0147 0.9578 γ 0.025 0.0000 0.0109 0.0231 0.0237 0.0362 0.0500 0.0145 0.0040 σ2_u 0.20 0.0015 0.0973 0.1694 0.1854 0.2540 0.7639 0.1124 0.1007 α 0.20 0.0000 0.0575 0.1325 0.1671 0.2539 0.4999 0.1312 0.0410 c 2.00 0.0076 1.0172 1.9494 2.2626 3.1551 13.5449 1.6034 0.8875 η 0.80 0.0000 0.5367 0.7603 0.6986 0.9089 1.0000 0.2446 0.9523 δ2 - 0.0000 0.3592 0.4377 0.4000 0.4756 0.5000 0.1042 0.4846 δ2_x - 0.0000 0.1337 0.2066 0.2318 0.3173 0.5000 0.1209 0.1272

3.5.3 Empirical Study

For the empirical study of Extended G-M model with GARCH(1,1) volatility, we use the same dataset and following the same data processing method. After which, both in sample and out of sample validation are conducted for the two stocks BAC and GOOG.

Figure 3.18: Absolute Difference for BAC in Sept, 2017

Figure 3.20: Absolute Difference for GOOG in Sept, 2017

CHAPTER 4

Modified Hasbrouck’s Model

4.1 Model

The original Glostein and Milgrom (1985) is a very theoretical paper, neither simulation nor empirical study is actually conducted. Our main focus here is still incorporating the trading volume when we are studying market micro-structure: the stock price evolution, the informed trade’s ratio and impact, etc. Hasbrouck (1991) used a very simple VAR model to introduce trading volume. Inspired by his simple microstructure model, we begin with building our own model.

Following the same idea of the true underly price of a stock is a random walk.

Vt=Vt−1+t, where t∼N(0, σ2) (4.1)

And the trading volume Xtwill have the following evoluation

Xt=c(Vt−1−Qt−1) +ut (4.2)

wherec >0,ut∼N(0, σ_u2).

The quote-midpoint price isQt= (At+Bt)/2, which have the following evolution

Qt=Vt+a(Qt−1−Vt−1) +bXt (4.3)

4.2 Simulation Study

For our simulation study, we still first need to set parameter value and true values in our simulation. We specify σ2 = 0.85, σ_u2 = 0.25, a = 0.4, b = 1, c = 0.3. Based on equation (4.1), (4.2), (4.3), correspondingly, We could compute the price and trading volume and plot their evolution.

Figure 4.1: Evolution of mid-quote and true underlying price

Here we are still using the Bayesian MCMC Algorithm, the detailed algorithm could be found in Section B.3. The priors we assigned here as:

σ2 ∼U nif(u1σ, u2σ) σ2_u∼U nif(u1σu, u2σu)

a∼Beta(αa, βa) b∼GAM M A(αb, βb) c∼GAM M A(αc, βc)

Similarly as the Extended G-M simulation study, we run 60,000 iterations and treat the first 20,000 iterations as burn-in runs. For the left 40,000 iterations, based on the posterior sample values, we can compute the summary table. Since both the posterior mean and median for all parameters are very close to the true parameter’s values. We are considering our algorithm results in a good estimation.

Table 4.1: Posterior Results Summary for Modified Hasbrouk’s Model with Constant Volatility

Par True Min Q1 Median Mean Q3 Max Sd Mode

σ 0.85 0.0001 0.8038 0.9191 0.8497 0.9687 1.0000 0.1796 0.9786 σu 0.25 0.0000 0.1254 0.2508 0.2503 0.3749 0.5000 0.1443 0.1298 a 0.40 0.0177 0.2343 0.3434 0.3662 0.4788 0.9484 0.1751 0.2654 b 1.00 0.0098 0.5649 0.8541 0.9300 1.1874 4.6238 0.5165 0.7345 c 0.30 0.0001 0.1016 0.2358 0.3262 0.4505 3.0266 0.3197 0.0533 δ - 0.0000 0.1254 0.2507 0.2503 0.3761 0.5000 0.1448 0.3749 δx - 0.0000 0.1238 0.2496 0.2500 0.3752 0.5000 0.1444 0.3726

The trace plots of each parameter are also presented below.

And the Gelman-Rubin plot to diagnosis convergence can be showed in following for convergence diagnosis.

4.3 Empirical Study

For the empirical study in Modified Hasbrouck’s Model, we still choose BAC and GOOD as our two study stock share, below are two tables summarizing the posterior values of parameters.

Table 4.2: Posterior Results Summary for BAC

Par Min Q1 Median Mean Q3 Max Sd Mode

σ 0.0001 0.8337 0.9305 0.8664 0.9735 1.0000 0.1710 0.9816 σu 0.0000 0.1249 0.2505 0.2503 0.3768 0.5000 0.1442 0.4039 a 0.0044 0.6085 0.7168 0.6866 0.7989 0.9841 0.1591 0.7449 b 0.0190 1.5952 2.1401 2.1809 2.7164 6.3188 0.8802 1.9437 c 0.0001 0.1125 0.2398 0.3576 0.4668 4.6705 0.3820 0.0659 δ 0.0000 0.0139 0.0352 0.0665 0.0839 0.5000 0.0841 0.0092 δx 0.0000 0.0144 0.0425 0.0969 0.1328 0.4999 0.1202 0.0117

Table 4.3: Posterior Results Summary for GOOG

Par Min Q1 Median Mean Q3 Max Sd Mode

σ 0.0002 0.7988 0.9144 0.8487 0.9691 1.0000 0.1768 0.9798 σu 0.0000 0.1251 0.2507 0.2505 0.3764 0.5000 0.1445 0.1077 a 0.0040 0.6135 0.7185 0.6891 0.7985 0.9726 0.1573 0.7626 b 0.0023 0.1289 0.2225 0.3065 0.3738 5.1513 0.3086 0.1389 c 0.0000 0.1170 0.2789 0.3984 0.5582 4.1461 0.3928 0.0724 δ 0.0000 0.0136 0.0366 0.0687 0.0842 0.4999 0.0869 0.0091 δx 0.0000 0.0136 0.0432 0.1062 0.1563 0.4999 0.1301 0.0122

Then, for each parameter we estimated, we use mean, median and mode to work as the point estimator and then regenerate 100 set of data. For each set of the data, at time pointt, we take the absolute value of difference between the true bid/ask price and simulated bin/ask price, also apply the same rule for the true aggregated trade volume and simulated xt, below shows the absolute difference for both in sample and out of sample check.

Figure 4.5: Absolute Difference for BAC in Sept, 2017

Figure 4.7: Absolute Difference for GOOG in Sept, 2017

4.4 GARCH Volatility Model

4.4.1 Simulation Study

Then, follow the same working flow of Extended G-M model, we changed the constant volatility to GARCH(1,1) as well. For our simulation study of GARCH(1,1) volatility in extended Hasbrouck’s model, we still first need to set parameter value and true values in our simulation. We specify ω = 0.08, β = 0.98, γ = 0.025, σ_u2 = 0.25, a = 0.7, b = 1.5, c = 0.3 here. Once we finished running the bayseian MCMC algorithm, the posterior results summary table, trace plots and Gelman Rubin diagnosis plots are showing below consecutively

Table 4.4: Posterior Results Summary for Modified Hasbrouk’s Model with GARCH Volatility

Par True Min Q1 Median Mean Q3 Max Sd Mode

ω 0.08 0.0000 0.0827 0.0914 0.0866 0.0964 0.1000 0.0151 0.0979 β 0.98 0.9500 0.9621 0.9745 0.9747 0.9871 1.0000 0.0144 0.9609 γ 0.025 0.0000 0.0096 0.0212 0.0225 0.0346 0.0500 0.0145 0.0037 σu 0.25 0.0000 0.1268 0.2522 0.2514 0.3763 0.5000 0.1442 0.3876 a 0.7 0.0197 0.6075 0.7168 0.6871 0.7958 0.9859 0.1573 0.7459 b 1.5 0.0073 1.1988 1.5727 1.5708 1.9207 4.6602 0.5381 1.6632 c 0.3 0.0003 0.1010 0.2252 0.3268 0.4425 4.4115 0.3394 0.0610 δ - 0.0000 0.0141 0.0360 0.0683 0.0871 0.4998 0.0860 0.0096 δx - 0.0000 0.0143 0.0373 0.0886 0.1091 0.5000 0.1156 0.0113

4.4.2 Empirical Study

For the empirical study in Modified Hasbrouck’s Model with GARCH(1,1) Volatility, we still choose BAC and GOOD as our two study stock shares, below are plot for the in sample and out of sample aggregated absolute difference.

Figure 4.11: Absolute Difference for BAC in Sept, 2017

Figure 4.13: Absolute Difference for GOOG in Sept, 2017

CHAPTER 5

Discussion