Fitting a Linear Information Rate Model

This section explores the application of a fitting procedure to the linear informa- tion rate BHM-model, Eq. (4.11), which is based on

where σˆ is the positive constant information rate parameter that will be fitted and all other notation is the same as in Sec. 4.2. The fitting of ˆσ is applied to

the winning last price matched signals (XT =1) which are described in the Table

2.1 and Sec. 2.2. The winning last price matched signals are filtered and sorted by the course distances, splitting the data into sub-samples. There tends to be a period of lower fluctuation in the last price matched winning signal at the start of the race compared to when the volatility cluster is observed, see Fig. 2.4 middle left. A volatility cluster is defined as tendency of large price movements to clump together, this is illustrated in Fig. 4.19 using the return series from the S&P500.

2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 Date −0.100 −0.075 −0.050 −0.025 0.000 0.025 0.050 0.075 0.100 S&P5 00 d ai ly pe rce nt ag e re tu rn s

Figure 4.19: This is the percentage return series for the S&P500 starting from 01/01/06 and ending 31/12/16. We see there are parts of the signal that oscil- late with large swings, as compared to the majority of the series, and these are clumped together. This is known as a volatility cluster and the most prevalent one is found round the period 2008-2009.

The volatility cluster found with the gambling data is assumed to be the point at which the information process is initiated, such that the time coordinate in the model is defined to start at this point. To find the time when the volatility

cluster starts to emerge we use an exponential weighted moving average model (EWMA) [18,81] which is calibrated on the increments of the last price matched: this running volatility measure is then compared to the signal’s standard devia- tion. If the EWMA measure of volatility exceeds the standard deviation then this is the point at which the volatility cluster starts and the point t=0 in the cali- bration of our BHM model fit. The sub-samples determined by the race distance are represented by 100 randomly picked winning signals which are averaged by binning and averaging the data points in those bins; these bins are equally spaced in the interval t∈ [0,1]. These averaged signals are then shifted up vertically so that the final value of price is 1 and then rotated about the end data point, which is the point (t =1, Last Price Match =1), ensuring that all the signals start at the same a priori probability (denoted as p1 in the BHM model Eq. (4.51)). The

fit is performed on each of the average price signals that are designated by the course distance. We define the information rate parameter space as the interval ˆ

σ = [0.5,2] which is split into evenly spaced values of step size 0.01. We define there to be Nσ=151 values of the ˆσparameter in the search space. We index the interval σˆ =0.5,0.51,0.52, . . . ,2 as σˆl =0.5+l(0.01) where l =0,1,2, . . .150. For each value of σˆl one generates1.0×104 information processes and thus 1.0×104 price signals using the BHM pricing kernel in Eq. (4.19). These price processes are denoted as {S(t,σˆl)}u=1,2,...,1×104 for each l =0,1,2, . . .150 and then they are

averaged over the 1.0×104 samples, denoted_S¯(l)

t = ⟨S(t,σˆl)⟩. The average empir- ical price is J(j)

t = ⟨LP M

(j)

t ⟩ where the index j =1,2, . . . Nd corresponds to the sub-samples filtered on course distance and⟨.⟩is the binned average over the 100

signals. The mean square error (MSE) on the set of parameters σˆl is defined as

{M SE(σˆl)}_j=1,2,...,Nd= 1 T T ∑ τ=1 (J(j) τ −S¯ (l) τ ) 2 (4.133) wherel=0,1, . . . ,150,τ=1,2, . . . , T is indexed timetwithin the averaged binned

price J(j)

t and the average BHM price is S¯

(l)

t . Using the MSE as the fitting measure, one can find the σˆl for each race distancej =1,2, . . . , Nd that gives the smallest value of MSE. This procedure is defined mathematically as the minimum

mean square error (MMSE) ˆ σj = {arg min σl (M SE(σl))} j=1,2,...,Nd (4.134) hence we estimate the σˆj for each race distance j. This process is repeated 100 times and σˆj for each j =1,2, . . . , Nd is shown in Fig. (4.20). On a log-log scale one observes a linear relationship between log(σˆj) and the logarithm of race course distance. A linear relationship on a log-log scale is a power-law where the gradient of the linear line is the exponent of the power law which is estimated as

ˆ

m =0.3993±0.0413. This positive value of mˆ indicates that races of a smaller distance are dominated more by noise than the longer races.

-1 -0.5 0 0.5 1 1.5 log(Distance) -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 lo g ( σ )

Figure 4.20: A log-log plot where the blue circles are the estimated values of σˆj for each of the distances j = 1,2, . . . , Nd. The red line is a linear model with intercept, the gradient is estimated to be mˆ =0.3993±0.0413. This estimation of mˆ has a p-value of 5.18×10−7 and _R2=_0.8164both indicating that this linear

An issue when calculating the variance process Eq. (4.59) with the fitted linear information rate ˆσ is demonstrated in Fig. 4.21. One observes from Fig.

4.21 that the model underestimates the variance after the time t ≈ 0.2. The volatility profiles in Fig. 4.21 of the model (blue line) and the data (red line) do have a similar structure but we can improve the fit by employing a non-linear information process in Sec. 4.7.2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time,t 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 hVti

Figure 4.21: The solid blue lines are the average variance calculated with the real price data. The average variance is calculated by Eq. (4.59) and averaged over 10 samples. The dashed red line is the average variance for the simulated BHM model with σˆj given by the blue circles in Fig. 4.20. Each of the lines in the two stack (red and blue) from top to bottom are increasing in race distance, and are estimates for the variance.