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There exist many ways to measure financial asset volatility. In this paper, we introduce a new joint model for the high-low range of assets prices and realized measure of volatility: Realized CARR. In fact, the high-low range and realized volatility, both are efficient estimators of volatility. Hence, this new joint model can be viewed as a model of volatility. The model is similar to the Realized GARCH model of Hansen et al. (2012), and it can be estimated by the quasi-maximum likelihood method. Out-of-sample volatility forecasting using Standard and Poors 500 stock index (S&P), Dow Jones Industrial Average index (DJI) and National Association of Securities Dealers Automated Quotation (NASDAQ) 100 equity index shows that the Realized CARR model does outperform the Realized GARCH model.

Modeling the volatility of financial asset returns is of fundamental importance to option pricing, assets portfolio and risk management. Many ways exist to model financial asset volatility, such as ARCH/GARCH family of models and stochastic volatility (SV) model. The strength of these models lies in their flexible adaptation of the dynamics of volatility. With the increasing availability of high frequency financial data, considerable literature on the use of intra-day as set price data to measure daily volatility has been expanded. The research has introduced many realized measures of volatility, such as realized variance [

In practice, the daily return is less subject to the market microstructure noise but contains less information of volatility, while the realized volatility is heavily contaminated by the noise but still includes much information. In this context, recently, numerous researchers have devoted to study the joint model for daily returns and realized measure of volatility. The joint model can be classified into two categories according to different points of view. We will describe the two kinds of joint model in details in the following paragraph.

The first kind of the joint model is MEM (Multiplicative Error Model) and HEAVY (High-Frequency-Based Volatility Model) models, which deal with multiple latent volatility processes. The first joint model is introduced by Engle and Gallo (2006) [

The high-low range of daily return is an alternative way of measuring volatility; Parkinson (1980) showed that the high-low range is a more efficient estimator of volatility than the daily return, because the formation of the range is from the entire price process [

In this paper, we will introduce a new joint model which combines a CARR model for range with realized volatility, named realized CARR model. Comparing with the first joint MEM model, this new joint realized CARR model gives up the shortcoming of it that deals with three latent volatility processes. Meanwhile, the new model retains the superiority of the realized GARCH model which contains only two latent volatility processes, while more informative than the latter. The model proposed by this paper can be used to calculate Value-at-Risk and Expected Shortfall which are helpful for financial risk managers and portfolio managers.

The structure of this article is as follows. In Section 2, we first give a review of the CARR model and Realized GARCH model, and then we propose the Realized CARR model. The estimation of the Realized CARR model is described in Section 3. The results of the simulation for Realized CARR model are show in Section 4. In Section 5, we apply our model to Standard and Poors 500 stock index, Dow Jones Industria Average index (DJI) and National Association of Securities Dealers Automated Quotation (NASDAQ) 100 equity index, and provide out-of-sample forecasting comparison between the Realized CARR and Realized GARCH model. The Section 6 concludes the paper.

In this section, we introduce the Realized CARR model. We start with a brief of the CARR and the Realized GARCH model which provide the motivation for our introduced joint model.

Let

where

The CARR (p, q) model is first introduced by Chou (2005), which is specified as:

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The Realized GARCH model proposed by Hansen et al. (2012) [

In this model,

The first two equations (4) and (5) are referred as the return equation and the GARCH equation, see Hansen et al. (2012) [

Motivated by the CARR model of Chou (2005) [

The CARR model is similar to the ACD model by Engle and Russell (1998) [

In this section, we will introduce the estimation of the Realized CARR model. By the results of Chou (2005) [

Although the estimation of the Realized CARR model is similar to the Realized GARCH model, it is still somewhat different. In the following paragraph, we will describe the structure of QMLE analysis for the Realized CARR model. The first and second derivatives of the log likelihood function are provided in this section.

The log-likelihood function is specified as:

According to Hansen et al. (2012), the joint conditional density can be factorized as:

In the model,

In the estimation of the joint model, we can ignore the constant term which does not affect the parameter estimation. Therefore, the likelihood function can be abbreviated to:

Before taking derivatives, we simplify the joint model by:

where

Then, we will provide the first and second derivatives of the log-likelihood function.

Lemma 1. Define

Proposition 2. 1) The first derivative of the log-likelihood function,

where

2) The second derivative,

where

The details of Lemma 3.1 and Theorem 3.2’s proof can be seen in Hansen et al. (2012) [

According to the corollary of Lee and Hansen (1994) [

We will show the simulation results of Realized CARR (1,1) model in this section. In order to understand the performance of the model, we use two parameter settings in this simulation and set the sample size as T = 1000, 1500 and 2000. Two parameter settings: Case 1,

The results of the simulation for Realized CARR (1,1) model indicate that the model performs well and isn’t affected by the initial values. Therefore, the estimation method of this model is very robust.

In this section, we introduce the empirical analysis of our proposed model using daily range data, returns and realized measures for Standard and Poors 500 stock index (S&P), Dow Jones Industria Average index (DJI) and National Association of Securities Dealers Automated Quotation (NASDAQ) 100 equity index. The in sample period is from January 3, 2005 to August 30, 2013 and out of sample period is from September 2, 2013 to December 31, 2013. These data are downloaded from Oxford-Man Institute of Quantitative Finance Realized Library (Library Version: 0.2 [

T | LL | |||||||
---|---|---|---|---|---|---|---|---|

Init | Case 1 | 0.18 | 0.4 | 0.37 | −0.2 | 0.9 | 0.2 | |

3*1000 | par | 0.1817 | 0.4028 | 0.02916 | −0.3053 | 1.1904 | 0.0388 | 3*−1144.012 |

std error | 0.0175 | 0.0499 | 0.0373 | 0.0538 | 0.1552 | 0.0017 | ||

p value | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||

3*1500 | par | 0.1848 | 0.3906 | 0.3940 | −0.1822 | 0.8448 | 0.0383 | 3*−1598.929 |

std error | 0.0159 | 0.0448 | 0.0244 | 0.0211 | 0.0554 | 0.0014 | ||

p value | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||

3*2000 | par | 0.1790 | 0.4152 | 0.3507 | −0.2025 | 0.8901 | 0.0407 | 3*−2133.815 |

std error | 0.0145 | 0.0400 | 0.0249 | 0.0250 | 0.0679 | 0.0013 | ||

p value | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

T | LL | |||||||
---|---|---|---|---|---|---|---|---|

Init | Case 1 | 0.15 | 0.5 | 0.3 | −0.3 | 1 | 0.1 | |

3*1000 | par | 0.1487 | 0.4913 | 0.2848 | −0.2859 | 0.9932 | 0.0094 | 3*−2055.816 |

std error | 0.0153 | 0.0494 | 0.0599 | 0.0538 | 0.0643 | 0.0018 | ||

p value | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||

3*1500 | par | 0.1445 | 0.5190 | 0.2817 | −0.3058 | 1.0249 | 0.0099 | 3*−3026.141 |

std error | 0.0159 | 0.0448 | 0.0244 | 0.0211 | 0.0554 | 0.0014 | ||

p value | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | ||

3*2000 | par | 0.1299 | 0.5709 | 0.2328 | −0.3355 | 1.1186 | 0.0097 | 3*−4108.952 |

std error | 0.0112 | 0.0355 | 0.0414 | 0.0612 | 0.2040 | 0.0003 | ||

p value | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

Before estimation, we give the description of the sample data. Figures 1-3 show the time plot of the daily range, returns, realized kernel, log realized kernel and Tables 3-5 present the descriptive statistics for these data. The skewness and kurtosis show that the realized kernel is not normal but its logarithm is nearly normal, so we use logarithmic realized kernel data rather than realized kernel in this paper. The Jarque-Bera (JB) statistic is to test the normality of the sample data and its critical value is 5.99 (5%), which indicates the non-normal distribution of the sample data. It might be better to assume the distribution of

It might be better to assume the distribution of ut non-normal, which we will leave for future study. The Ljung-Box (LB) test (see Diebold (1988) [

mean | stdev | skewness | kurtosis | max | min | JB | LB (10) | |
---|---|---|---|---|---|---|---|---|

Return | 0.0002 | 0.0127 | −0.2632 | 12.7100 | 0.1022 | −0.9351 | 8546.00 | 54.048 |

High-low range | 0.0138 | 0.0083 | 1.9215 | 9.8749 | 0.0847 | 0.0024 | 5551.983 | 4715.5 |

Realized kernel | 0.0001 | 0.0003 | 12.671 | 280.140 | 0.0093 | 0.0000 | 6,999,454 | 5860.0 |

Log-realized kernel | −9.7301 | 1.08118 | 0.8118 | 3.7880 | −4.6763 | −12.238 | 294.415 | 11055 |

mean | stdev | skewness | kurtosis | max | min | JB | LB (10) | |
---|---|---|---|---|---|---|---|---|

Return | 0.0002 | 0.0122 | 0.0617 | 13.362 | 0.1075 | −0.0840 | 9700.15 | 56.089 |

High-low range | 0.0137 | 0.0112 | 3.6175 | 23.1100 | 0.1201 | 0.0019 | 41,260.40 | 8260.6 |

Realized kernel | 0.0001 | 0.0003 | 12.438 | 263.740 | 0.0091 | 0.0000 | 61972 | 6090.4 |

Log-realized kernel | −9.7421 | 1.0684 | 0.8754 | 3.9475 | −4.6966 | −12.168 | 357.979 | 10862 |

mean | stdev | skewness | kurtosis | max | min | JB | LB (10) | |
---|---|---|---|---|---|---|---|---|

Return | 0.0001 | 0.0117 | −0.3192 | 8.2382 | 0.0611 | −0.0724 | 2517.87 | 11.818 |

High-low range | 0.0146 | 0.0105 | 3.1056 | 18.028 | 0.1085 | 0.0025 | 23909.42 | 7741.5 |

Realized kernel | 0.0001 | 0.0002 | 9.3750 | 143.108 | 0.0050 | 0.0000 | 18067 | 6667.8 |

Log-realized kernel | −9.7659 | 0.9630 | 0.8344 | 4.0213 | −5.2965 | −12.42 | 342.166 | 10909 |

As is shown in the time plots of all the sample data, the financial assets have high volatilities during the financial crisis. The sample should be divided into pre and post financial crisis periods which indicate a regime switching model is needed. This is out of scope for the purposes of this article and we will leave this for further research.

The details of model estimation results are presented in this section. According to the plot of partial autocorrelation (PACF) and autocorrelation function (ACF), we determine the order of the two models as GARCH (1,1) and CARR (1,1). And in a practical application, the model GARCH (1,1) and CARR (1,1) are sufficient for most of the asset returns (see Bollerslev et al. (1992) [

In order to compare the estimated models, we calculate the Akaike information criterion (AIC) and Schwarz information criterion (SC) according to the following formulas:

where

As is shown in

Realized CARR | Realized GARCH | |||||
---|---|---|---|---|---|---|

S&P | Dji | Nasdaq | S&P | Dji | Nasdaq | |

0.110(0.001) | 0.109(0.001) | 0.090(0.003) | 0.111(0.001) | 0.109(0.005) | 0.090(0.005) | |

0.801(0.000) | 0.760(0.000) | 0.817(0.000) | 0.782(0.000) | 0.760(0.000) | 0.800(0.000) | |

0.160(0.015) | 0.150(0.049) | 0.131(0.048) | 0.200(0.046) | 0.160(0.047) | 0.159(0.047) | |

−0.160(0.040) | −0.15(0.048) | −0.090(0.047) | −0.102(0.047) | −0.101(0.044) | −0.104(0.044) | |

1.210(0.021) | 1.030(0.016) | 1.020(0.049) | 1.100(0.007) | 0.980(0.025) | 0.990(0.025) | |

0.340(0.000) | 0.391(0.00) | 0.331(0.000) | 0.450(0.000) | 0.449(0.000) | 0.450(0.000) | |

−0.05(0.027) | −0.047(0.018) | −0.187(0.028) | ||||

0.030(0.047) | 0.028(0.047) | 0.030(0.047) | ||||

−1090.218 | −1253.339 | −1862.539 | −1804.343 | −2136.193 | −2140.188 | |

AIC | 2192.436 | 2518.678 | 3737.078 | 3624.686 | 4288.386 | 4296.376 |

SC | 2225.939 | 2552.181 | 3770.581 | 3669.356 | 4333.056 | 4341.046 |

shape of the empirical distribution diverges from the exponential density whose function is monotonically decreasing. It is consistent with the descriptive statistics as showed in

To assess the forecasting power of realized CARR model, we perform out-of-sample forecasts and make comparisons with realized GARCH model. We choose the forecast horizons to be from 1 day to 80 days which is from September 2, 2013 to December 31, 2013. Two ex post volatilities: daily return squared (DRSQ) and daily high-low range (DHLR) are used as measures in this paper. Then the root-squared (RMSE) and the mean-abso- lute-errors (MAE) are computed to compare the forecasting power of realized CARR model with realized GARCH model. RMSE and MAE are defined as:

where h means the forecast horizon, MV and FV denote the measure volatility and forecasted volatility, respectively.

Rolling samples of 2173 observations are used to modeling the two models and 100 data are made for out-of- sample forecast. The

In this paper, we introduce a new joint model for the high-low range of assets prices and realized measure of volatility: Realized CARR. The model is easy to be estimated by the quasi maximum likelihood method. The empirical results show the superiority of fitting volatility than the realized GARCH model and it yields more precise in forecast comparisons. The new joint model gives up the shortcoming of MEM, which deals with multiple latent volatility processes, but retains the superiority of the realized GARCH model which contains only two latent volatility processes, while more informative than the latter. The model proposed by this paper can be

S&P | Dji | Nasdaq | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

DRSQ | DHLR | DRSQ | DHLR | DRSQ | DHLR | |||||||

Horizon | RC | RG | RC | RG | RC | RG | RC | RG | RC | RG | RC | RG |

RMSE | ||||||||||||

1 | 3.281 | 3.312 | 2.013 | 2.243 | 4.181 | 4.312 | 3.117 | 3.219 | 3.194 | 3.292 | 2.16 | 2.242 |

5 | 3.332 | 3.391 | 2.371 | 2.604 | 4.352 | 4.382 | 3.334 | 3.59 | 3.402 | 3.493 | 2.351 | 2.53 |

10 | 3.436 | 4.048 | 2.712 | 2.979 | 4.476 | 4.527 | 3.569 | 3.891 | 3.51 | 3.848 | 2.674 | 2.881 |

MAE | ||||||||||||

1 | 1.673 | 1.695 | 0.987 | 1.104 | 2.101 | 2.183 | 1.894 | 1.912 | 1.591 | 1.635 | 1.125 | 1.143 |

5 | 1.707 | 1.803 | 1.105 | 1.167 | 2.23 | 2.302 | 1.906 | 1.935 | 1.611 | 1.663 | 1.196 | 1.209 |

10 | 1.841 | 1.97 | 1.194 | 1.23 | 2.411 | 2.517 | 1.951 | 2.09 | 1.73 | 1.809 | 1.214 | 1.271 |

F | |||||
---|---|---|---|---|---|

2*S&P | RC | −0.023 (0.028) | 0.078 (0.126) | 1.867 | 0.213 |

RG | 0.314 (0.043) | −0.316 (0.19) | 10.671 | 0.189 | |

2*Dji | RC | 0.003 (0.031) | 0.019 (0.005) | 0.986 | 0.143 |

RG | 0.03 (0.047) | −0.086 (0.008) | 13.569 | 0.037 | |

2*Nasdaq | RC | 0.005 (0.013) | −0.22 (0.178) | 2.373 | 0.429 |

RG | 0.245 (0.031) | 0.844 (0.427) | 17.370 | 0.127 |

used to calculate Value-at-Risk and Expected Shortfall which are helpful for financial risk managers and portfolio managers. In fact, we only give the most general form of the model which can be extended much more, such as includes leverage effect, exogenous variables, heavy tail, regime switching, etc., which we leave for further study.

This work is supported by the Guangxi China Science Foundation (2014GXNSFAA118015) and the Scientific Research Project of Guangxi Colleges and Universities (KY2015ZD054).

Yunqian Ma,Yuanying Jiang, (2016) Modeling and Forecasting Financial Volatilities Using a Joint Model for Range and Realized Volatility. Open Journal of Business and Management,04,206-218. doi: 10.4236/ojbm.2016.42022