Volatility is an important parameter for financial risk management and it is applied in many issues such as option pricing, portfolio optimization, VaR methodology and hedging; thus the forecasting of volatility or variance can be regarded as a problem of financial modelling. The objective of this paper is to forecast FTSE 100 Stock Prices of top 100 companies listed on London Stock Exchange by using the Exponential Weighted Moving Average (EWMA) Model. The data for this model are directly obtained from the UK FTSE 100 Index. In this research paper, we have examined the daily returns of FTSE 100 Stock Prices of top 100 companies listed on London Stock Exchange from the thirtieth day of June 2009 to the first day of December 2014 and equally forecasted the daily returns from the first day of December 2014 to the fifth day of February 2015 with the Exponential Weighted Moving Average (EWMA) Model. We found that there is a very high possibility that the stock prices will start to fall as from 5th February 2015 downwards.
It is well established that to estimate the volatility of a stock price empirically, the stock price is usually observed at fixed intervals of time. One particular objective of EWMA is to track changes in the volatility. For small λ values, recent observations affect the estimate promptly. For λ values closer to one, the estimate changes slowly based on recent changes in the returns of the underlying variable.
The aim of this research paper is to study the daily returns of FTSE 100 Stock Prices of top 100 companies listed on London Stock Exchange from 30th June 2009 to 1st December 2014 and thereby forecast the daily returns from 1st December 2014 to 5th February 2015 with the Exponential Weighted Moving Average (EWMA) Model. The data for this model will be directly obtained from the UK FTSE 100 Index. The paper will make use of Monte Carlo Simulation by writing the codes on the Excel VBA and thereby predict the future stock prices of FTSE 100 Stock Prices.
The paper is structured into four sections. Section 1 will introduce the paper and gives the benefits of modelling in detail. Section 2 will review the literature briefly while Section 3 will be dealing with the data and its analysis. The last section will give the conclusion of the paper.
Benefits of ModellingA model is a representation of real world events. A model can be defined as “a simplified description of reality that is at least potentially useful in decision-making” [
Recently, modelling the time-varying nature of the volatility of emerging stock markets has attracted the focus of researchers. [
Aside from all these, [
In actual sense, stock market prediction is a process of determining the future value of a stock or other financial instrument traded on stockexchange market. Any successful prediction of a stock’s future price usually result in high profit because of the common problem associated with forecasting the stock prices which is “uncertainty”. According to theefficient-market hypothesis, stock price movements which is quite unpredictable are controlled by the random walk hypothesis, implying that the best forecasting on tomorrow’s price is today’s price value.Some researchers identified a large number of statistical models and financial variables that are useful to predict the future price of stock market.
The Exponentially Weighted Moving Average (EWMA) model was derived by JP Morgan in 1989 for their Risk Metrics framework [
where,
According to [
The Autoregressive Conditional Heteroskedasticity (ARCH) model was first introduced by Engle in 1982 [
[
Many researchers have worked on GARCH extensions which had led to development of EGARCH, IGARCH, etc. In essence, these models are the most popularly known for forecasting the financial volatility and returns. In 1993, [
These models have been extended many times but out of them, we decided to use EWMA because one of the main objectives of EWMA is to estimate the next?day (or period) volatility of a time series and closely track the volatility as it changes. That is, the volatility of a market variable on day n, as estimated at the end of day
It has been observed that one of the major advantages of EWMA is that it gives more weight to the recent returns while calculating the returns. In this paper, we will look at how volatility is calculated using EWMA.
Firstly, we would to calculate the log returns of the price series.
To determine the stock prices, we first calculate the daily lognormal returns, using the formula ln(Pi/Pi − 1), where P represents each day’s closing stock price. We need to use the natural log because we want the returns to be continuously compounded. We will now have daily returns for the entire price series.
Step 2: The variance rate is the square of volatility.
The next step is the take the square of long returns. This is actually the calculation of simple variance or volatility represented by the following formula:
Here, u represents the returns, and m represents the number of days.
Step 3: Assign weights.
In order to assign weights in a way that recent returns have higher weight and older returns have lesser weight, there is need a factor called Lambda (λ), which is a smoothing constant or the persistent parameter. The weights are assigned as (1 − λ)λ0. Lambda must be less than 1. The Risk Metrics database (produced by JP Morgan and made public available) uses the EWMA with 0.94 for updating daily volatility. The first weight will be (1 − 0.94) = 6%, the second weight will be 6% × 0.94 = 5.64% and so on. In EWMA all the weights sum to 1, however they are on reducing basis with a constant ratio of λ.
Step 4: We now multiply the Returns-squared with the weights and take the addition of R2*w, and thereby gives the final EWMA variance. The volatility will be the square root of variance.
Lambda | 0.94 | |||||||||||||
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Prices | ||||||||||||||
Date | Open | High | Low | close | AvgVol | Adj Close* | Log Return | Sq Log Return | Weights | EWMA | Rand() | Dates | Simulated prices | |
01-Dec-14 | 6722.60 | 6753.20 | 6144.70 | 6566.10 | 519,695,500 | 6566.10 | 0.098326 | 0.0002% | 01-Dec-14 | 6,566.10 | ||||
03-Nov-14 | 6546.50 | 6773.10 | 6444.90 | 6722.60 | 755,287,400 | 6722.60 | 0.001307 | 1.71E−06 | 0.06 | 1.03E−07 | −0.121857 | 02-Dec-14 | 6566.031167 | |
01-Oct-14 | 6622.72 | 6622.81 | 6072.70 | 6546.50 | 789,204,100 | 6546.50 | 0.001346 | 1.81E−06 | 0.0564 | 1.02E−07 | 0.8826343 | 03-Dec-14 | 6566.529776 | |
01-Sep-14 | 6819.75 | 6904.86 | 6601.62 | 6622.72 | 0 | 6622.72 | 0.001327 | 1.76E−06 | 0.053016 | 9.33E−08 | −0.55954 | 04-Dec-14 | 6566.213685 | |
01-Aug-14 | 6730.11 | 6831.19 | 6528.73 | 6819.75 | 0 | 6819.75 | 0.00129 | 1.66E−06 | 0.049835 | 8.29E−08 | −0.913908 | 05-Dec-14 | 6565.697439 | |
01-Jul-14 | 6743.94 | 6875.31 | 6643.62 | 6730.11 | 0 | 6730.11 | 0.001312 | 1.72E−06 | 0.046845 | 8.06E−08 | 1.3079467 | 06-Dec-14 | 6566.436287 | |
02-Jun-14 | 6844.51 | 6878.96 | 6701.59 | 6743.94 | 0 | 6743.94 | 0.001307 | 1.71E−06 | 0.044034 | 7.52E−08 | −0.194625 | 07-Dec-14 | 6566.326343 | |
01-May-14 | 6780.03 | 6894.88 | 6766.84 | 6844.51 | 0 | 6844.51 | 0.001288 | 1.66E−06 | 0.041392 | 6.87E−08 | −0.477825 | 08-Dec-14 | 6566.056422 | |
01-Apr-14 | 6598.37 | 6794.88 | 6,507.08 | 6780.03 | 0 | 6780.03 | 0.001303 | 1.7E−06 | 0.038909 | 6.6E−08 | −0.250323 | 09-Dec-14 | 6565.915021 | |
03-Mar-14 | 6809.70 | 6827.22 | 6492.62 | 6598.37 | 0 | 6598.37 | 0.001337 | 1.79E−06 | 0.036574 | 6.54E−08 | −0.301044 | 10-Dec-14 | 6565.744973 | |
03-Feb-14 | 6510.44 | 6866.35 | 6416.72 | 6809.70 | 0 | 6809.70 | 0.001291 | 1.67E−06 | 0.03438 | 5.73E−08 | −0.755814 | 11-Dec-14 | 6565.318059 | |
02-Jan-14 | 6749.09 | 6867.42 | 6421.26 | 6510.44 | 0 | 6510.44 | 0.001356 | 1.84E−06 | 0.032317 | 5.94E−08 | 0.7109219 | 12-Dec-14 | 6565.71962 | |
02-Dec-13 | 6650.57 | 6768.44 | 6422.23 | 6749.09 | 0 | 6749.09 | 0.001301 | 1.69E−06 | 0.030378 | 5.14E−08 | −1.980534 | 13-Dec-14 | 6564.600994 | |
01-Nov-13 | 6731.43 | 6780.11 | 6613.98 | 6650.57 | 0 | 6650.57 | 0.001326 | 1.76E−06 | 0.028555 | 5.02E−08 | 0.6550123 | 14-Dec-14 | 6564.970934 | |
01-Oct-13 | 6462.22 | 6819.86 | 6316.91 | 6731.43 | 0 | 6731.43 | 0.001308 | 1.71E−06 | 0.026842 | 4.59E−08 | 2.5368915 | 15-Dec-14 | 6566.403917 | |
02-Sep-13 | 6412.93 | 6659.12 | 6412.93 | 6462.22 | 0 | 6462.22 | 0.001364 | 1.86E−06 | 0.025231 | 4.69E−08 | −0.941926 | 16-Dec-14 | 6565.871829 | |
01-Aug-13 | 6621.06 | 6696.63 | 6386.72 | 6412.93 | 0 | 6412.93 | 0.001368 | 1.87E−06 | 0.023718 | 4.44E−08 | −0.423072 | 17-Dec-14 | 6565.632854 | |
01-Jul-13 | 6215.47 | 6662.19 | 6185.21 | 6621.06 | 0 | 6621.06 | 0.001324 | 1.75E−06 | 0.022294 | 3.91E−08 | −1.19889 | 18-Dec-14 | 6564.955696 | |
03-Jun-13 | 6583.09 | 6583.09 | 6023.44 | 6215.47 | 0 | 6215.47 | 0.001416 | 2E−06 | 0.020957 | 4.2E−08 | 0.7979789 | 19-Dec-14 | 6565.406407 | |
01-May-13 | 6430.12 | 6875.62 | 6409.81 | 6583.09 | 0 | 6583.09 | 0.001327 | 1.76E−06 | 0.019699 | 3.47E−08 | −0.284465 | 20-Dec-14 | 6565.245737 | |
02-Apr-13 | 6411.74 | 6501.78 | 6214.36 | 6430.12 | 0 | 6430.12 | 0.001367 | 1.87E−06 | 0.018517 | 3.46E−08 | −0.113474 | 21-Dec-14 | 6565.181647 | |
01-Mar-13 | 6360.81 | 6533.99 | 6308.56 | 6411.74 | 0 | 6411.74 | 0.001368 | 1.87E−06 | 0.017406 | 3.26E−08 | 0.5829359 | 22-Dec-14 | 6565.510908 | |
01-Feb-13 | 6276.88 | 6412.44 | 6216.72 | 6360.81 | 0 | 6360.81 | 0.001378 | 1.9E−06 | 0.016362 | 3.11E−08 | 0.0813039 | 23-Dec-14 | 6565.556834 | |
02-Jan-13 | 5897.81 | 6451.01 | 5897.81 | 6276.88 | 0 | 6276.88 | 0.001395 | 1.95E−06 | 0.01538 | 2.99E−08 | 0.0966209 | 24-Dec-14 | 6565.611413 | |
03-Dec-12 | 5866.82 | 5997.04 | 5852.88 | 5897.81 | 0 | 5897.81 | 0.001483 | 2.2E−06 | 0.014457 | 3.18E−08 | −1.423437 | 25-Dec-14 | 6564.807436 | |
01-Nov-12 | 5782.70 | 5921.78 | 5605.59 | 5866.82 | 0 | 5866.82 | 0.00148 | 2.19E−06 | 0.01359 | 2.98E−08 | 0.8790136 | 26-Dec-14 | 6565.303907 | |
01-Oct-12 | 5742.07 | 5928.27 | 5738.59 | 5782.70 | 0 | 5782.70 | 0.001501 | 2.25E−06 | 0.012775 | 2.88E−08 | 0.5888577 | 27-Dec-14 | 6565.636519 | |
03-Sep-12 | 5711.48 | 5932.62 | 5634.88 | 5742.07 | 0 | 5742.07 | 0.001509 | 2.28E−06 | 0.012008 | 2.73E−08 | 0.5780796 | 28-Dec-14 | 6565.963059 | |
01-Aug-12 | 5712.82 | 5876.22 | 5662.30 | 5711.48 | 0 | 5711.48 | 0.001515 | 2.3E−06 | 0.011288 | 2.59E−08 | 1.1960531 | 29-Dec-14 | 6566.638724 |
02-Jul-12 | 5640.64 | 5714.19 | 5498.32 | 5635.28 | 0 | 5635.28 | 0.001535 | 2.36E−06 | 0.01061 | 2.5E−08 | 0.4410876 | 30-Dec-14 | 6566.887918 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
01-Jun-12 | 5260.19 | 5622.29 | 5260.19 | 5571.15 | 0 | 5571.15 | 0.00155 | 2.4E−06 | 0.009974 | 2.4E−08 | −1.184174 | 31-Dec-14 | 6566.218943 | |
01-May-12 | 5812.23 | 5812.23 | 5266.41 | 5320.86 | 0 | 5320.86 | 0.001621 | 2.63E−06 | 0.009375 | 2.46E−08 | 2.2376251 | 01-Jan-15 | 6567.483108 | |
02-Apr-12 | 5874.89 | 5874.89 | 5595.55 | 5737.78 | 0 | 5737.78 | 0.001495 | 2.24E−06 | 0.008813 | 1.97E−08 | −1.521539 | 02-Jan-15 | 6566.62348 | |
01-Mar-12 | 5931.25 | 5965.58 | 5742.03 | 5,768.45 | 0 | 5768.45 | 0.0015 | 2.25E−06 | 0.008284 | 1.86E−08 | −0.222751 | 03-Jan-15 | 6566.497643 | |
01-Feb-12 | 5790.72 | 5945.25 | 5790.72 | 5,871.51 | 0 | 5871.51 | 0.001475 | 2.18E−06 | 0.007787 | 1.69E−08 | 1.0059187 | 04-Jan-15 | 6567.06594 | |
03-Jan-12 | 5699.91 | 5795.20 | 5612.26 | 5,681.61 | 0 | 5681.61 | 0.001527 | 2.33E−06 | 0.00732 | 1.71E−08 | −0.957089 | 05-Jan-15 | 6566.525233 | |
01-Dec-11 | 5489.34 | 5572.28 | 5364.99 | 5,572.28 | 0 | 5572.28 | 0.001551 | 2.41E−06 | 0.006881 | 1.66E−08 | −0.236699 | 06-Jan-15 | 6566.391518 | |
01-Nov-11 | 5421.57 | 5567.34 | 5127.57 | 5,505.42 | 0 | 5505.42 | 0.001567 | 2.45E−06 | 0.006468 | 1.59E−08 | −1.100002 | 07-Jan-15 | 6565.770139 | |
03-Oct-11 | 5075.50 | 5713.82 | 4944.44 | 5,544.22 | 0 | 5544.22 | 0.001554 | 2.41E−06 | 0.00608 | 1.47E−08 | −0.175304 | 08-Jan-15 | 6565.671119 | |
01-Sep-11 | 5418.65 | 5418.65 | 5041.61 | 5,128.48 | 0 | 5128.48 | 0.001681 | 2.83E−06 | 0.005715 | 1.61E−08 | 0.0666619 | 09-Jan-15 | 6565.708776 | |
01-Aug-11 | 5774.43 | 5774.43 | 5007.16 | 5,394.53 | 0 | 5394.53 | 0.001584 | 2.51E−06 | 0.005372 | 1.35E−08 | 0.9478725 | 10-Jan-15 | 6566.244214 | |
01-Jul-11 | 5989.76 | 6054.55 | 5752.81 | 5,815.19 | 0 | 5815.19 | 0.001478 | 2.18E−06 | 0.00505 | 1.1E−08 | 1.3323002 | 11-Jan-15 | 6566.996883 | |
01-Jun-11 | 5928.61 | 5945.71 | 5674.38 | 5,945.71 | 0 | 5945.71 | 0.001458 | 2.13E−06 | 0.004747 | 1.01E−08 | −0.395315 | 12-Jan-15 | 6566.773548 | |
03-May-11 | 6082.88 | 6082.88 | 5835.89 | 5,989.99 | 0 | 5989.99 | 0.001451 | 2.1E−06 | 0.004462 | 9.39E−09 | −1.050028 | 13-Jan-15 | 6566.180364 | |
01-Apr-11 | 6009.92 | 6069.90 | 5870.08 | 6,069.90 | 0 | 6069.90 | 0.001433 | 2.05E−06 | 0.004194 | 8.61E−09 | 0.4490809 | 14-Jan-15 | 6566.434057 | |
01-Mar-11 | 5935.76 | 6005.09 | 5598.23 | 5,908.76 | 0 | 5908.76 | 0.001474 | 2.17E−06 | 0.003943 | 8.57E−09 | −0.120398 | 15-Jan-15 | 6566.366044 | |
01-Feb-11 | 5957.82 | 6091.33 | 5919.98 | 5,994.01 | 0 | 5994.01 | 0.001449 | 2.1E−06 | 0.003706 | 7.78E−09 | −1.521385 | 16-Jan-15 | 6565.506649 | |
04-Jan-11 | 6013.87 | 6056.43 | 5862.94 | 5,862.94 | 0 | 5862.94 | 0.001484 | 2.2E−06 | 0.003484 | 7.67E−09 | −0.094939 | 17-Jan-15 | 6565.453026 | |
01-Dec-10 | 5642.50 | 6008.92 | 5642.50 | 5,899.94 | 0 | 5899.94 | 0.001471 | 2.16E−06 | 0.003275 | 7.08E−09 | 2.1301881 | 18-Jan-15 | 6566.656347 | |
01-Nov-10 | 5694.62 | 5875.35 | 5528.27 | 5,528.27 | 0 | 5528.27 | 0.001571 | 2.47E−06 | 0.003078 | 7.59E−09 | 0.5406405 | 19-Jan-15 | 6566.961786 | |
01-Oct-10 | 5592.90 | 5757.86 | 5555.97 | 5,675.16 | 0 | 5675.16 | 0.001518 | 2.31E−06 | 0.002893 | 6.67E−09 | 1.5339269 | 20-Jan-15 | 6567.828463 | |
01-Sep-10 | 5366.41 | 5602.54 | 5366.41 | 5,548.62 | 0 | 5548.62 | 0.001558 | 2.43E−06 | 0.00272 | 6.6E−09 | −2.32595 | 21-Jan-15 | 6566.514339 | |
02-Aug-10 | 5397.11 | 5410.52 | 5109.40 | 5,225.22 | 0 | 5225.22 | 0.00165 | 2.72E−06 | 0.002557 | 6.96E−09 | 0.1536171 | 22-Jan-15 | 6566.601125 | |
01-Jul-10 | 4805.75 | 5365.67 | 4805.75 | 5,258.02 | 0 | 5258.02 | 0.001628 | 2.65E−06 | 0.002403 | 6.37E−09 | 0.1557975 | 23-Jan-15 | 6566.689143 | |
01-Jun-10 | 5163.30 | 5299.11 | 4914.22 | 4,916.87 | 0 | 4916.87 | 0.001742 | 3.04E−06 | 0.002259 | 6.86E−09 | 0.4118873 | 24-Jan-15 | 6566.921843 | |
04-May-10 | 5411.11 | 5433.73 | 4940.68 | 5,188.43 | 0 | 5188.43 | 0.001638 | 2.68E−06 | 0.002124 | 5.7E−09 | 1.2097533 | 25-Jan-15 | 6567.605347 | |
01-Apr-10 | 5744.89 | 5825.01 | 5553.29 | 5,553.29 | 0 | 5553.29 | 0.00154 | 2.37E−06 | 0.001996 | 4.74E−09 | 0.4761172 | 26-Jan-15 | 6567.874371 | |
01-Mar-10 | 5405.94 | 5727.65 | 5405.94 | 5,679.64 | 0 | 5679.64 | 0.001518 | 2.3E−06 | 0.001876 | 4.32E−09 | −0.065574 | 27-Jan-15 | 6567.837321 | |
01-Feb-10 | 5247.41 | 5358.17 | 5060.92 | 5,354.52 | 0 | 5354.52 | 0.001614 | 2.61E−06 | 0.001764 | 4.6E−09 | −0.924597 | 28-Jan-15 | 6567.314908 | |
04-Jan-10 | 5500.34 | 5538.07 | 5145.74 | 5,188.52 | 0 | 5188.52 | 0.001655 | 2.74E−06 | 0.001658 | 4.54E−09 | −1.514631 | 29-Jan-15 | 6566.459204 | |
01-Dec-09 | 5312.17 | 5437.61 | 5196.81 | 5,412.88 | 0 | 5412.88 | 0.00158 | 2.5E−06 | 0.001558 | 3.89E−09 | −0.176388 | 30-Jan-15 | 6566.359562 | |
02-Nov-09 | 5104.50 | 5382.67 | 5037.21 | 5,190.68 | 0 | 5190.68 | 0.001656 | 2.74E−06 | 0.001465 | 4.02E−09 | −0.235421 | 31-Jan-15 | 6566.226573 | |
01-Oct-09 | 5047.81 | 5281.54 | 4988.70 | 5,044.55 | 0 | 5044.55 | 0.001696 | 2.88E−06 | 0.001377 | 3.96E−09 | −0.288304 | 01-Feb-15 | 6566.063713 | |
01-Sep-09 | 4819.70 | 5172.89 | 4796.75 | 5,133.90 | 0 | 5133.90 | 0.001661 | 2.76E−06 | 0.001294 | 3.57E−09 | −0.145283 | 02-Feb-15 | 6565.981647 | |
03-Aug-09 | 4682.46 | 4916.80 | 4645.01 | 4,908.90 | 0 | 4908.90 | 0.00174 | 3.03E−06 | 0.001217 | 3.69E−09 | −0.424027 | 03-Feb-15 | 6565.742129 | |
01-Jul-09 | 4340.71 | 4631.61 | 4127.17 | 4,608.36 | 0 | 4608.36 | 0.001844 | 3.4E−06 | 0.001144 | 3.89E−09 | 0.1548313 | 04-Feb-15 | 6565.82959 | |
30-Jun-09 | 4249.21 | 4249.21 | 4249.21 | 4,249.21 | 0 | 4249.21 | 0.001985 | 3.94E−06 | 0.001075 | 4.24E−09 | −0.52282 | 05-Feb-15 | 3.74372E+14 |
EWMA is basically a special form of an ARCH() model, with such characteristics which include the fact that the ARCH order is equal to the sample data size and the weights are exponentially declining at rate λ throughout time. The main reason for using EWMA is that it is particularly useful to estimate the next-day (or period) volatility of a time series and closely track the volatility as it changes.
In this research paper, we have examined the daily returns of FTSE 100 Stock Prices of top 100 companies listed on London Stock Exchange from the thirtieth day of June 2009 to the first day of December 2014 and equally forecasted the daily returns from the first day of December 2014 to the fifth day of February 2015 with the Exponential Weighted Moving Average (EWMA) Model. We found that there is a very high possibility that the stock prices will start to fall as from 5th February 2015 downwards (
Adejumo WahabAdewuyi, (2016) Modelling Stock Prices with Exponential Weighted Moving Average (EWMA). Journal of Mathematical Finance,06,99-104. doi: 10.4236/jmf.2016.61011