[16]时间序列的自相关检验 - 论文选读：序列相关性的ACF与PACF验证

1Figure 1: Time series plot Athens University of Economics & BusinessV olatility Forecasting for Option Trading:Evaluating Estimators of Changes on Implied V olatilitiesGeorge Lilianov, George PapadakisGraduate Program in Decision Sciences, Department of Management and Technolo gy,Athens University of Economics & Business, 47A Evelpidon & 33 Lefkados - Athens 113 62 - Greece1. IntroductionIn this paper our goal is to find a reliable estimator of changes (first order differences) on implied volatilities on short maturity calls. We are using data from the Spanish Financial Futures Exchange (MEFF), containing historical intraday information of options on Index Futures on Ibex35, with sampling frequency at 60 minutes intervals and prediction interval one step ahead.. We examine three estimators of implied volatilities – a univariate time series (Box-Jenkins) model, a multivariate linear regression model and a multivariate backpropagation neural network model.Regarding the univariate model estimation, we firstly identify the significant lags upon which the model is specified as a second step. Then the model is estimated through regression and tested for adequacy with the Durbin-Watson statistic. As an additional step, we also test for ARCH-GA RCH effects, that is, for autocorrelation of squared residuals, and we specify and estimate an adequate model of volatility.In order to estimate an adequate multivariate model, we firstly regress the implied volatility against all 16 potentially significant factors (12 factors + 4 lags) and then we stepwise exclude the insignificant ones and repeat the process until the model is specified. Again the model is tested for existence of systematic error using the DW-stat.The third estimator is built upon a neural network, which uses the first 495 observations as an estimation dataset, the next 100 as validation dataset, and tests its forecast on the final 55 observations. Various combinations of hidden layers and hidden units are examined so that the best trade off between generalization error and training error is made.After comparing the three estimators on the basis of their ability to best explain changes of implied volatility, we examine the economic implications of having an adequate model for predicting implied volatility. We assume that the forecasts are used as the criterion to buy or sell a call option following the rule th at Δoption_price= ½ Δimplied_volatility.Finally, hourly and cumulative profit results of each estimator are calculated and compared using graphical representation, variance, standard deviation and the Sharpe ratio.2. Univariate Time Series EstimatorAs we plot the first 595 observations of the data we can see that all the series seems stationary, i.e. there is no noticeable trend and the mean seems to be constant (Figure 1).We need to examine the autocorrelation in the time series in order to determine the strength of association between the current and the lagged values of the price changes in implied volatility. As we can see on Figure 2 the autocorrelations are not persistently large and so we can concludeFigure 2: Autocorrelation function Figure 3: Partial autocorrelation function that the series has mean stationarity.From the partial autocorrelation function plot it seems that there is autocorrelation for the first, second, third, and fourth lags, and possibly the 15th lag (Figure 3).1. Model IdentificationAs we can see from the plots above, ACF spikes at the 1st lag and then cuts off, while PA CF tails off. This suggests that the process that best fits the analysis is the MA(1). Since from the PACF plot we can see that first 4 lags are significant, we can assume that MA(1) process is approximated with an AR(4) process.2. Model SpecificationAR(4) : Y t = α + β1Y t-1+ β2Y t-2 + β3Y t-3+ β4Y t-4 +εt3. Model EstimationWhen we run the regression we found that the constant and lags beyond 4th are insignificant. Thus the best estimation output is:Table 1: Auto-regression four lags estimationRegression StatisticsCoefficientst StatP-valueMultiple R 0.590562399 Intercept 0 #N/A #N/AR Square0.348763947 X Variable 1 -0.717960318 -17.63521603 2.94584E-56 Adjusted R Square 0.343774575 X Variable 2 -0.510407408 -10.38578741 2.51661E-23 Standard Error 21.28635869X Variable 3 -0.2581589 -5.254859507 2.07046E-07 Observations 596X Variable 4 -0.138370612 -3.401922481 0.000714268ANOVAF79.26014517 Significance F 7.97541E-54 DW2.011680944Therefore, the estimated model is:Y t = - 0.718Y t-1 - 0.510Y t-2 - 0.258Y t-3 - 0.138Y t-4 +εtIf we look at the t-Stat values and P-values of each coefficient we notice that all coefficients are significantly different from zero for a confidence level of 95%.An Adjusted R Square equal to 0.3437 indicates that 34.37% of the changes in last price change in the implied volatility are explained through this model. As the F value is greater than 4 we are more than 95% confident that our model explains 34.37% of the changes. As the Significance F is close to zero the probability of mistake is very small.If we calculate the DW statistic we get 2.01, which again indicates absence of autocorrelation in the residuals, thus our model again does not make systematic errors and is adequate.Figure 4: Correlogram of residuals squaredFigure 5: Correlogram of standardized residuals squared4. Volatility equationIf we test the above estimator for ARCH effects by looking at the autocorrelation function plots of the squared residuals below, we can see that the error term can be described as an ARCH process (Figure 4).The estimation output for the ARCH/GA RCH model is: L1_IMPLIED_V -0.132231 0.053047 -2.492691 0.0127 L2_IMPLIED_V 0.014605 0.051714 0.282426 0.7776 ARCH(1) 0.546330 0.092623 5.898443 0.0000 ARCH(2) -0.291951 0.089686 -3.255239 0.0011 Adjusted R-squared -0.026364 S.D. dependent var 26.34354 S.E. of regression 26.68853 Akaike info criterion 9.147399 Sum squared resid 418107.1 Schwarz criterion 9.206405 If we test the above model using the ACF of squared residuals below we can now see on Figure 5 that there is no correlation, thus the model has been able to capture all the A RCH effects.3. Multivariate Linear Regression EstimatorWe perform the regression analysis using the first 595 observations where the Implied Volatility is the dependent variable and the independent variables are: Day effect, Maturity, Maturity effect, Change spot, Moneyness, Time effect, Velocity, Average spread, Interest, Volume, Historic Volatility, 1st lag of Implied Volatility, 2nd lag of Implied Volatility, 3rd lag of Implied Volatility, 4th lag of Implied Volatility.After excluding the insignificant independent variables stepwise, we resulted with a regression of the Implied Volatility on Maturity effect, Change spot, Moneyness, and all the lags. The output was the following:Table 3: 3.Multivariate linear regression estimationRegression Statistics Coefficients t Stat P-value Multiple R 0.648269254 Intercept 0 #N/A #N/AR Square 0.420253026 Maturity_Eff 9.859003106 3.900671557 0.000106904 Adjusted R Square 0.41270078 Change_Spot -2.884722196 -5.466039204 6.78624E-08 Standard Error 20.08778398 Moneyness 175.5727989 5.066066937 5.43183E-07 Observations 600 L1_Implied_V -0.716523686 -18.58421945 4.28923E-61 ANOVA L2_Implied_V -0.487979582 -10.48799845 1.00687E-23 F61.40857399 L3_Implied_V -0.256806587 -5.531530048 4.76643E-08 Significance F 3.59169E-66 L4_Implied_V -0.128614495 -3.343912114 0.000878157Therefore, the estimated regression equation is:Implied_Vol = 9.859003106*Maturity_Effect - 2.884722196*Change_Spot ++ 175.5727989*Moneyness - 0.716523686*L1_Implied_Vol -- 0.487979582*L2_Implied_Vol - 0.256806587*L3_Implied_Vol -- 0.128614495*L4_Implied_VolWe can notice that a unit-change in Moneyness has the greatest impact on explaining the Implied Volatility changes, while unit-changes in Maturity Effect and Change Spot have much lower impact, b ut still greaterthan the unit-changes in past values of Implied Volatility.If we look at the t-Stat values and P-values of each coefficient we notice that all coefficients are significantly different from zero for a confidence level of 5%.An Adjusted R Square equal to 0.3437 indicates that 41.27% of the changes in last price change in the implied volatility are explained through this model. As the F value is greater than 4 we are more than 95% confident that our model explains 41.27% of the changes. As the Significance F is close to zero the probability of mistake is very small.If we calculate the DW statistic we get 1.95, which again indicates absence of autocorrelation in the residuals, thus our model again does not make systematic errors and is adequate.4.Neural Network EstimatorIn building the neural network estimator we use the first 495 observations as an estimation dataset, the next100 as a validation dataset, and the last 55 as the test dataset on which we will examine the forecasts fromthe model estimators.After examining various combinations of a number of hidden layers and hidden units in each of them, we deduced that the network with two hidden layers with 7 hidden units each, gives the least errors (the generalization error and training error trade-off is at lowest level). The resulting error graph after 1000 iterations is shown in Figure 6:We can observe that approximately after the 50th iteration the generalization error starts to increase, whilethe training error decreases at a lower pace. Given the trade-off between the generalization and training errors, we believe that at the global minimum of the generalization error graph the network will give thebest estimations.Thus, we rerun the neural network model with the same parameters, but for only 50 iterations (Figure 7)We can notice that the most significant impact on Implied Volatility changes is the unit-change in Maturity effect, followed by that in the 1st lag of Implied Volatility. Unit-changes in the Change spot and Moneynesshave the least impact.Figure 6: NN error graph (1000 iterations) Figure 7: NN error graph (50 iterations)Figure 8: Model forecast (line plot) Figure 9: Model forecast (scatter plot) 5. Estimators Cross-validationTo compare the three estimator models forecasts with the last 55 actual observations of Implied Volatility, firstly we plot the forecasted and actual series (Figures 8 and 9):It seems hard to conclude which of the three forecasts best fits the actual data, but it seems that all have similar mean values and lower deviation from the mean compared to the actual data.Next, we regress the actual observations of Implied Volatility on each forecas t from the three estimation models:Table 4: Univariate ModelVariable Coefficient Std. Error t-Statistic Prob.UNIVARIATE_ESTIM0.681629 0.180215 3.782315 0.0004 Adjusted R-squared 0.197693 S.D. dependent var 12.41328 S.E. of regression 11.11877 Akaike info criterion 7.690833 Sum squared resid 6552.238 Schwarz criterion 7.763827 Log likelihood-209.4979 F-statistic14.30591Table 5: Multivariate ModelC 2.227706 1.423447 1.565008 0.1235 R-squared 0.344662 Mean dependent var 0.145455 Adjusted R-squared 0.332298 S.D. dependent var 12.41328 S.E. of regression 10.14327 Akaike info criterion 7.507184 Sum squared resid 5452.957 Schwarz criterion 7.580178 Log likelihood -204.4476 F-statistic 27.87435Table 6: Neural Network ModelVariable Coefficient Std. Error t-Statistic Prob.NEURAL_NETWORK_E0.818160 0.250689 3.263648 0.0019Adjusted R-squared 0.151629 S.D. dependent var 12.41328 S.E. of regression 11.43351 Akaike info criterion 7.746660 Sum squared resid 6928.431 Schwarz criterion 7.819654 Log likelihood -211.0331 F-statistic 10.65140We can see that the R-square value of the multivariate estimator is the highest – 34.47%, i.e. 34.47% of the actual observations are forecasted correctly by this model.Another useful test of the models is the Array correlation coefficient, which shows therelationship between the actual observationsand the forecasts. This test confirms theobserved R-square values, i.e. that themultivariate estimator forecast has the closest correlation with the actual implied volatility values.If we look also at the predictive power of the estimator models in relation to the correct direction of the market rather than the levels, we notice that the multivariate estimator correctly predicts the direction 67% of the time, while the univariate estimator and the neural network estimator only 64% and 56% respectively.6.Economic implicationConsidering that the call option price and the volatility are positively related, we can assume that when next hour implied volatility is expected to increase (the next hour forecast of the first difference of implied volatility is positive) the price of the option contract will also increase, and vice versa.Thus, assuming that we can close our position and open a new one every hour, our trading rule is that when our model predicts the next hour first difference of implied volatility (Implied_Vol t+1) to be positive, we assume the option price will go up in period t+1, and our strategy is to buy the contract now (at period t) and sell it in the next hour (at period t+1). On the contrary, when the implied volatility is expected to go down, we short sell now (at period t) and buy the contract in the next hour (at period t+1).Since we assume that there are no transaction costs, we close our current position and open a new one every hour even if our forecast for the market direction is not changing.Thus, we calculate the hour-to-hour profit by the difference in the actual option price from the current andprevious hour given our trade rule.Figure 10: Cumulative profits If we bought a contract at hour t-1 and we sell it at hour t : Profit = Actual Option Price t - Actual Option Price t-1If we short sold a contract at hour t-1 and we buy it at hour t : Profit = Actual Option Price t-1 - Actual Option Price tThus, if our model correctly predicts the direction of the market, we realize profit, otherwise we have a loss.In order to calculate the actual call option prices, we assume that initially the price is 50 pesetas, and next hour price is increased/decreased by half of the actual first difference of implied volatility increase/decrease:Actual Option Price t = Actual Option Price t-1 + 0.5*(Implied_Vol t ) 7. Estimators in practiceIf we calculate the cumulative profit of each model using the trade rule and profit calculation described above, we can see in the graph (Figure 10) that the univariate estimator has the highest profits until the 39th forecast period, while the multivariate estimator forecasts results in the highest total profits.When we examine profits resulting from each of the estimators we can see that the multivariate estimator not only gives the highest total profit at the end of the week, but also has the lowest variance, as shown in the table below:Also the Sharpe ratio, measuring the average return relative to the total risk,of the multivariate estimator is thehighest, indicating that this model willresult in the best risk-adjusted performance.8. ConclusionsConcluding this paper we should summarize the results of this attempt to find the best possible model forimplied volatility and describe its economic implication:∙ The univariate model explains 34,37% of the changes in volatility using 4 lags as independentvariables.∙ The multivariate model suggests that apart from the 4 lags, the Maturity effect, Change spot andMoneyness also have explanatory power. Thus, R 2 rises to 41,27%∙ The neural network that was built with relatively strict constraints (2 hidden layers each with 7hidden units, 0.001 learning rate, validation every 5 iterations) failed to outperform the multivariate model, having an R 2 equal to 16,73%.∙ Applying this models so as to make profit by trading on a call option, produced correspondingresults: the multivariate model came up with the greatest accumulated profit (144 pes.), followed by the univariate model (113 pes.) and then by the neural network (86 pes.).∙ Adjusting thi s performance for risk (Sharpe ratio) ranked the multivariate model first (0,4642), theunivariate second (0,3512) and the neural network third (0,2605)。