Simulated data

In reality, we can never know for sure what the true volatility was, but when we simulate the data, then we know the true volatility exactly. Simulation therefore provides a convinient tool to study different volatility models. On the other hand, the issue with simulation is always how close are simulated data to the real world. In order to convince the reader that the simulated data we chose are close to reality, we do not decide on the simulation by ourselves. We borrow the credibility of Alizadeh, Brand and Diebold (2002) and simulate the data in the following way. First we simulate the volatility process

lnσ_t= lnσ+ρ_H(lnσt−1−lnσ) +µεt−1 (3.28) with parameters ln (σ) =−2.5,ρH = 0.985 andµ= 0.75/√

257 = 0.048. Af-terwards we simulate for every dayt= 1,2, ...,100000 a Brownian motion¹³ with zero drift term and diffusion term equal toσ_t. We save the highest, the lowest and the final value of this Brownian motion. Both equation (3.28) and the parameter values are taken from Alizadeh, Brand and Diebold (2002), who found that the volatility dynamics (3.28) is broadly consitent with lit-erature on stochastic volatility. Note that there are no opening jumps in this these simulated data.

Note that the volatility process (3.28) does not favour directly either of the competing models (3.5) and (3.10). Volatility gradually evolves over the time, and neither past returns nor past high or low prices influence the future volatility in any way.

13We use 100000 discrete steps for the approximation of the continuous Brownian mo-tion.

In-sample analysis

Table 3.9 presents estimated coefficients for the standard GARCH model (3.5) and the RGARCH model (3.10) together with the values of Akaike Information Criterion (AIC). As expected, the RGARCH model performs better than the standard GARCH model.

Table 3.9: Estimated coefficients of GARCH(1,1) model σ_t² =ω+αr_t−1² + βσ_t−1² and the RGARCH(1,1) model σ_t² = ω +ασ\_P,t−1² +βσ²_t−1, reported together with the values of Akaike Information Criterion (AIC) of the re-spective equations for the simulated data.

GARCH(1,1) RGARCH(1,1)

ω α β AIC ω α β AIC

1.73E-04 0.044 0.933 -2.112 1.61E-04 0.122 0.857 -2.133

Coefficients in the RGARCH are changed in exactly the same way as in the previous section - coefficient α is increased and coefficient β is de-creased. Moreover, since there are no jumps in the simulated data, the Parkinson volatility estimator is an unbiased estimator of a daily volatility and therefore all the coefficientsω,α and β can be interpreted in the same way as in standard GARCH models. Note thatα+β is smaller than one for both models (implying stationarity) and α+β is the same (0.98) for both models. This means that both models imply the same volatility persistance, which is very natural, since both are estimated on the same data set.

Now we estimate the combined GARCH model (3.22)

Table 3.10: Estimated coefficients and p-values for the combined GARCH(1,1) model σ_t² =ω+α1r_t−1² +α2σ\_P,t−1² +βσ²_t−1 for the simulated data.

combined GARCH(1,1)

ω p-value α₁ p-value β p-value α₂ p-value

1.53E-04 0 -0.057 0 0.834 0 0.204 0

As we can see, the results are generally consistent with those in Table 96

3.2. The main difference is that the negative coefficient α1 is now clearly significant. As Garman and Klass (1980) showed, the optimal volatility forecast based on open, high, low and close price is (3.9). It is a weighted average of the Parkinson volatility estimator (3.8) and squared open-to-close returns, where squared returns have negative weight. This is the reason why coefficient α₁ is negative. Note that the ration between the coefficientsα₁ and α2 is very close to the ratio predicted from the Garman-Klass formula.

As previously mentioned, we use the Parkinson volatility estimator (3.8) instead of Garman and Klass (3.9) volatility estimator because of the data concerns (open prices are sometimes not available). Another reason is that for the purpose of volatility modelling, the Garman and Klass volatility esti-mator brings only a small improvement over the Parkinson estiesti-mator even in the ideal case. This can be seen from the coefficientβ, which decreases from 0.933 (for the standard GARCH) to 0.857 (for RGARCH), but afterwards only a little bit to 0.834 (for the combined GARCH, which is basically the same as GARCH based on the Garman and Klass volatility estimator).

Out–of-sample forecasting performance

Now we compare the forecasting performance of the RGARCH model and the standard GARCH model on the simulated data. Results are shown in Table 3.11.

These results are the main reason why we used simulated data too. Now we know exactly what the true volatility is and we can use it as a benchmark.

Additionally, simulation allows us to have much larger data sample (100000 observations of the simulated data vs. 4423 observations of the real data), what in turns mean that we can draw conclusions with certainty.

First note that the results obtained from the simulated data are

consis-Table 3.11: Comparison of the forecasting performance of the GARCH(1,1) modelσ_t² =ω+αr²_t−1+βσ_t−1² and it modified version RGARCH(1,1)σ²_t = ω+ασ\²_P,t−1+βσ_t−1² . Numbers in this table are 1000×RMSE of the one-day-ahead rolling window forecasts reported for different window sizes w and different benchmarks squared returns (r²) and the true volatility (σ²_true) for the simulated data. The differences in MSE are significant at any significance level.

bnch GARCH(1,1) RGARCH(1,1) σ_true²

w=300 400 500 600 300 400 500 600

r² 11.76 11.72 11.69 11.66 11.64 11.60 11.56 11.54 11.34 σ_true² 2.95 2.88 2.80 2.72 2.47 2.30 2.20 2.12 0

tent with results in Table 3.3. Table 3.3 shows that the RGARCH model outperforms the standard GARCH model most of the time. Since the sim-ulated data are much larger, we basically got rid of the noise and now we can see exactly how much better the RGARCH performs. The improvement seems to be small, just around 1% decrease in RMSE, when we use squared returns as a benchmark. However, use of the true volatility as a benchmark shows that the real improvement of the RGARCH in comparison to the standard GARCH model is much larger, around 20%.

In fact, the mean squared error (MSE) between the forecasted volatility (cσ²) and squared returns (r²) can be rewritten in the following way:

where r² is squared return and σ_true² is the true volatility. When squared returns are used as a benchmark, then the second term typically dominates and it is therefore difficult to choose between competing volatility models based on the MSE (RMSE).

98