The first author would like to thank University College London (UCL), for awarding Overseas Research Scholarship (ORS) and Graduate Research Scholarship (GRS) funding.

FTSE 100 Returns and Volatility estimation using Higher Order Neural NetworksJanti Shawash, Dr David R. Selviah

Department of Electrical and Electronic Engineering

Conclusion

Simulations

Results

CHONN-EGARCH

CHONN-GARCH

Linear-EGARCH

CHONN

1.55 1.60

CHONN-EGARCH

CHONN-GARCH

Linear-EGARCH

CHONN

1.0 1.2 1.4

Data

Kurtosis = 1.7157Skewness = -0.0941

! = 882.5572

Year Frequency

FTSE

100

daily

pri

ce se

ries

3000

4000

5000

6000

7000

2002 2004 2006 2008 0 50 100 150 200 250

(a)

Kurtosis = 9.1589Skewness = -0.0976

! = 1.3487

Year Frequency

FTSE

100

daily

retu

rn se

ries

(%)

-10

-5

0

5

10

2002 2004 2006 2008 0 50 100 150 200 250

(b)

Auto

Corr

elat

ion

Coef

ficie

nts

-0.4

-0.2

0

0.2

0.4

Lag (n)0 5 10 15 20 25 30

Lag (n)0 5 10 15 20 25 30

(c) (d)

Daily returns Daily returns2

# inputs

# Hiddenneurons

#parameters

0

500

1000

0

100

11

NN CHONN HorizHONN CPHONN FXPHONN

Returns Volatility

Kurtosis = 7.9794Skewness = -0.3904

! = 1.335

Year Frequency

Residual

-10

-5

0

5

10

2002 2004 2006 2008 0 50 100 150 200

(a)

Auto

Cor

rela

tion

Coe

ffici

ents

-0.4

-0.2

0

0.2

0.4

Lag (n)0 5 10 15 20 25 30

Lag (n)0 5 10 15 20 25 30

(c) (d)

Residual Residual2

Year Frequency

Dai

ly v

olat

ility

0

1

2

3

4

5

2002 2004 2006 2008 0 20 40 60 80 100 120 140

(a)

Kurtosis = 3.2290Skewness = -0.2720

! = 1.0007

Year Frequency

Stan

dard

ized

Resi

dual

-10

-5

0

5

10

2002 2004 2006 2008 0 20 40 60 80 100

(b)

Auto

Cor

rela

tion

Coe

ffici

ents

-0.4

-0.2

0

0.2

0.4

Lag (n)0 5 10 15 20 25 30

Lag (n)0 5 10 15 20 25 30

(c) (d)

Standardized Residuals Standardized Residuals2

J =δLayer

δWLayer

ModelMSE = min

W(1

N

N�

t=1

�2t )

Wnew = Wold − [∇2J − µI]−1J�

Time

0

5

Epochs

510

HRo 50

52

HRi 55

60

RMSEo

1.41.61.8

RMSEi

1.01.2

AIC

1.8

2.0

2.2

NN CHONN HorizHONN CPHONN FXPHONN

MSE MAE

Acknowledgements

(a) FTSE 100 daily price series(b) FTSE 100 daily returns series(c) Autocorrelation coefficients or returns and returns-squared

A graph showing the simulations of networks with input size range of [1-11] input lags. Hidden neurons in the range of [0-10] in models with a single output neuron. The stripped pattern in this graph matches the (input-hidden) parameter count of the graph on the far left. We observe that the in-sample (RMSE, HR) performance increases with the number of parameters in the estimation model. Out-of-sample (RMSE, HR) performance is better in smaller models. The training time was also correlated to the number of parameters. However, the epochs required for convergence were independent of the number of parameter with a mean value of 12 epochs. Two minimisation criteria were used for training as indicated in the legend.

(a) FTSE 100 Volatility estimate. (c,d) Autocorrelation coefficients or returns and returns-squared. (b) FTSE 100 daily returns residual after volatility compensation.Results show that most of the volatility and returns information was extracted leaving a residual close to a normal-Gaussian distribution.

(a) Residual of FTSE 100 returns estimation using the Correlation Higher Order Neural Network.

Even after returns estimation Kurtosis and ! levels remained the same.

HorizHONN

CHONN

CPHONN

FPXHONN

Linear

NN

1.32 1.34 1.36

CHONN

HorizHONN

NN

CPHONN

FPXHONN

Linear

1.65 1.70

Linear

HorizHONN

CHONN

CPHONN

FPXHONN

NN

58 60

AIC RMSEo Hit Rate RMSEo

AIC

(c) Autocorrelation of Residuals(d) Autocorrelation residuals-squared.

These results show that the autocorrelation levels were reduced in the residuals domain, however; Residuals-squared stayed at the same level indicating no change in volatility.

AIC performance improvement* when using HONNs over standard linear models.

Out of sample Root Mean square error slightly improved as well.

Hit Rate performance was worse for Linear model andthe best was NN.* AIC variation was in the 3rd significant figure indicating consistent performance.

Both AIC and RMSEo indicators were improved significantly after extracting the volatility information from the residual of the FTSE-100 index daily returns.

The FTSE 100 series was converted into a returns series with properties closer to a stationary Gaussian distributed data set using the following equation.

W11

W21

W22

W23

W2M

W12

W1N

yt!n

yt!2

yt!1

yt

NeuralNetworkHigher

Order Function

!!

t

2

!1

!2

!q

!t"1

2

!t"2

2

!t"q

2

!t"1

2

!t"2

2

!t" p

2

!1

!2

!p

Feedback

GARCH

Number of parameters of a Neural Network and 4 Higher Order Neural Networks.

Higher Order Neural Network output y(t) as a function of Weights (W,b) and inputs Xi.

Optimisation criterion for network weight update estimation.

Error gradient estimation represented as the Jacobian matrix

Weight update using Levenberg-Marquardt algorithm.

Root Mean Squared Error, evaluation criteria used to benchmark performance of model estimation.

Hit Rate (HR), is an estimate of the rate of correct direction of the estimated output.

Generalised AutoRegressive Conditional Heteroskedasticity (GARCH)

Volatility estimation from previous error values along with the previous volatility estimate.

Log-Likely-hood minimisation criterion that incorporate changes in volatility along with the error.

Akaike information criterion, used to select the best models based on the error and the number of parameters used for estimation.

This work compared Higher Order Neural Networks (HONN) with Neural Networks, and linear regression for short term forecasting of stock market index daily returns.

Two new HONNs, the Correlation HONN (CHONN) and the Horizontal HONN (HorizHONN) outperform all other models tested in terms of the Akaike Information Criterion, out-of-sample root mean square error, of FTSE100 and NASDAQ giving out-of-sample Hit Rates of up to 60% with AIC improvement up to 6.2%. New hybrid models for volatility estimation are formed by combining CHONN with E/GARCH are compared with conventional EGARCH, providing up to 2.1% and 2.7% AIC improvement for FTSE100 and NASDAQ.