For many financial market applications, including option pricing and investment decisions, volatility forecasting is crucial. Therefore, the research of volatility forecasting has been an active area of study since the past years. In recent years, the emergence of many financial time series methods for volatility forecasting has proved the importance of understanding the nature of volatility in any financial instruments.
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Often, people will think â€˜priceâ€™ is used as an indicator of the stock market performance. Due to the non-stationary nature of price series of the stock market, most researchers actually transformed series of â€˜price change (return)â€™ or â€˜absolute price changes (absolute return)â€™ in their studies. There is a difference between the term â€˜returnâ€™ and the term â€˜volatilityâ€™. The term â€˜volatilityâ€™ is used as a crude measure of the total risk of financial assets. Actually, volatility is the standard deviation or the variance of returns whereas â€˜returnâ€™ is merely the changes of prices. An increasingly commonly adopted tool for the measurement of the risk exposure associated with a particular portfolio of assets known as â€˜Value at Riskâ€™ (VaR) involves calculation of the expected losses that might result from changes in the market prices of particular securities (Jorion, 2001; Bessis, 2002). Thus, the VaR of a particular portfolio is defined as the maximum loss on a portfolio occurring within a specified time and with a given (small) probability. Under this approach, the validity of a bankâ€™s internally modeled VaR is â€˜backtestedâ€™ by comparing actual daily trading gains or losses with the estimated VaR and noting the number of â€˜exceptionsâ€™ occurring, in the sense of days when the VaR estimate was insufficient to cover actual trading losses, with concerns naturally arising where such exceptions frequently occur, and that can result in a range of penalties for the financial institution concerned (Saunders & Cornett, 2003). A crucial parameter in the implementation of parametric VaR calculation methods is an estimate of the volatility parameter that describes the asset or portfolio, or more accurately a forecast of that volatility where the simplifying assumption of constancy is relaxed and time-varying volatility is acknowledged. While it has long been recognized that returns volatility exhibits â€˜clustering,â€™ such that large (small) returns follow large (small) returns of random sign (Mandelbrot, 1963; Fama, 1965), it is only following the introduction of the generalized autoregressive conditional heteroskedasticity (GARCH) model (Engle, 1982; Bollerslev, 1986) that financial economists have modeled and forecast these temporal dependencies using econometric techniques, and a variety of adaptations of the basic GARCH framework are now widely used in modeling time-varying volatility. In particular, the significance of asymmetric effects in stock index returns has been widely documented, such that equity return volatility increases by a greater amount following positive shocks, usually associated with the â€˜leverage effect,â€™ whereby a firmâ€™s debt-to-equity ratio increases when equity values decline, and holders of that equity perceive future income streams of the firm as being more risky (Black, 1976; Christie, 1982). Such variance asymmetry has been successfully modeled and forecast in a variety of market contexts (Henry, 1998) using the threshold-GARCH (TGARCH) model (Glosten et al., 1993), and the exponential-GARCH (EGARCH) model (Nelson, 1991) in particular.
While risk management practises in financial institutions often rely on simpler volatility forecasting approaches based on heuristics and moving average, smoothing or â€˜RiskMetricsâ€™ techniques, symmetric and asymmetric GARCH models have also recently begun to be considered in the VaR context. However, the standard GARCH model and variants within that class of model impose rapid exponential decay in the effect of shocks on conditional variance. In contrast, empirical evidence has suggested that volatility tends to change slowly and that shocks take a considerable time to decay (Ding et al., 1993). The fractionally integrated-GARCH (FIGARCH) model (Baillie et al., 1996; Chung, 1999) has provided a popular means of capturing and forecasting such non-integrated but highly persistent â€˜long memoryâ€™ dynamics in volatility in the recent empirical literature, as well as its exponential (FIEGARCH) variant (Bollerslev & Mikkelsen, 1996) which parallels the EGARCH extension of the basic GARCH form, and therefore provides a generalization capable of capturing both the volatility asymmetry and long memory in volatility which are potential characteristics of emerging equity markets.
This paper therefore seeks to extend previous research concerned with the evaluation of alternative volatility forecasting methods under VaR modeling in the context of the Basle Committee criterion for determining the adequacy of the resulting VaR estimates in two ways. First, by broadening the class of GARCH models under consideration to include more recently proposed models such as the FIGARCH and FIEGARCH representations described above, which are capable of accommodating potential fractional integration and the associated long memory characteristics of return volatility, as well as the more simple and computationally less intensive methods commonly used in financial institutions. Second, extending the scope of previous research through evaluative application of these methods to daily index data of nine stock market indexes.
The extensive research of volatility forecasting plays an important role for investment, financial risk management, security valuation, and also business decision-making process. Without a proper forecasting tools and research on this field, many financial decision making process will be difficult and risky to be implemented. The positive contribution of volatility forecasting in the field of finance is no doubt a fact as it given many practitioners a mean of guidelines to estimate their management risk such as option pricing, hedging and estimating investment risk. Therefore, it is crucial to study on the performance of different approaches and methods of forecast model to determine the best suitable practical application for different situation. The most common form of financial instrument is the stock market. The stock indices consist of a particular countryâ€Ÿs most prominent stocks. Thus, in this study our aim is to focus on forecasting the stock indices volatility of eight different stock indices that provide us the ability to test the forecast approaches. There are quite a number of forecast models since the recent years. However, the new concern is on the performance of these forecast model when incorporated with higher frequency data with the realized volatility method. There are still gap for researching the intra-day data effects on forecasting model which is comparative new as compared to daily data volatility forecasting. The significant role of this study also include whether intra-day data can really help at improving the performance of forecast model to estimate volatility for the stock index.
In this proposal, the report is mainly subdivided into three chapters. Chapter 1 is about the overview of this research which includes the background of the study, the research objective, problem statement, and the significance of this study. Chapter 2 presents the literature review of volatility forecasting, GARCH models, exponentially smoothing and realized volatility.
Volatility forecasts are produced by either market-based or time-series methods. Market-based forecasting involves the calculation of implied volatility from current option prices by solving the Black and Scholes option pricing model for the volatility that results in a price equal to the market price. In this paper, our focus is on the development of a new time series method. These methods provide estimates of the conditional variance, Ïƒ2t = var(rt | It-1), of the log return, rt, at time t conditional on It â€“ 1, the information set of all observed returns up to time t â€“ 1. This can be viewed as the variance of an error (or residual) term, Îµt, defined by Îµt = rt â€“ E(rt | It â€“ 1 ), where E(rt | It â€“ 1 ) is a conditional mean term, which is often assumed to be zero or a constant. Îµt is often referred to as the price â€œshockâ€? or â€œnewsâ€?.
GARCH models (Engle, 1982; Bollersle, 1986) are the most widely used statistical models for volatility. GARCH models express the conditional variance as a linear function of lagged squared error terms and lagged conditional variance terms. For example, the GARCH(1, 1) model is shown in the following expression: Ïƒ2t = Ï‰ + Î±Îµ2t â€“ 1 + Î²Ïƒ2t â€“ 1, where Ï‰, Î±, and Î² are parameters. The multiperiod variance forecast, , is calculated as the sum of the variance forecasts for each of the k periods making up the holding period: where is the one-step-ahead variance forecast. Empirical results for the GARCH(1, 1) model have shown that often Î² â‰ˆ (1 â€“ Î±). The model in which Î² = (1 â€“ Î±) is term integrated GARCH (IGARCH) (Nelson, 1990). Exponential smoothing has the same formulation as the IGARCH(1, 1) model with the additional restriction that Ï‰ = 0. The IGARCH(1, 1) multiperiod forecast is written as Stock return volatility is often found to be greater following a negative return than a positive return of equal size. This leverage effect has promted the development of a number of GARCH models that allow for asymmetry. The first asymmetric formulation was the exponential GARCH model of Nelson (1991). In this log formulation for volatility, the impact of lagged squared residuals is exponential, which may exaggerate the impact of large shocks. A simpler asymmetric model is the GJRGARCH model of Glosten et al. (1993). The GJRGARCH(1, 1) model is given by
where Ï‰, Î±, Î³, and Î² are parameters; and I[.] is the indicator function. Typically, it is found that Î± > Î³, which indicates the presence of the leverage effect. The assumption that the median of the distribution of Îµt is zero implies that the expectation of the indicator function is 0.5, which enables the derivation of the following multiperiod forecast expression: GARCH parameters are estimated by maximum likelihood, which requires the assumption that the standardized errors, Îµt / Ïƒt, are independent and identically distributed (i.i.d.). Although a Gaussian assumption is common, the distribution is often fat tailed, which has prompted the use of the Student-t distribution (Bollerslev, 1987) and the generalized error distribution (Nelson, 1991). Stochastic volatility models provide an alternative statistical volatility modelling approach (Ghysels et al., 1996). However, estimation of these models has proved difficult and, consequently, they are not as widely used as GARCH models. Andersen et al. (2003) show how daily exchange rate volatility can be forecasted by fitting long-memory, or fractionally integrated, autoregressive and vector autoregressive models to the log realized daily volatility constructed from half-hourly returns. Although results for this approach are impressive, such high frequency data are not available to many forecasters, so there is still great interest in methods applied to daily data. A useful review of the volatility forecasting literature is provided by Poon and Granger (2003).
Exponentially Weighted Moving Average (EWMA) is simple and well-known volatility forecast method. The method is based on the simple average of past squared residuals to estimate its variance forecasts. The EWMA allows the latest observations to have a stronger weighted impact on the volatility forecast of past data observations. The equation for the EWMA is shown and written as exponential smoothing in recursive form. The Î± parameter is the smoothing parameter. The equation: There is no proper guideline or statistic model for exponential smoothing. Generally, literature suggested using reduction in the sum of in-sample one-step-ahead estimation of errors (Taylor, 2004 cited from Gardner, 1985). In RiskMetrics (1996), volatility forecasting for exponential smoothing is recommended to use the following minimisation: In the above equation, Îµ2t is the in-sample squared error which acted as the proxy for actual variance whereby it is said to be not observable. By using Îµ2t as a proxy for variance, the actual squared residual, Îµ2t, is said to be biased and noisy. In Andersen et al. (1998), the research showed the evaluation of variance forecasts using realised volatility as a more accurate proxy. The next section would discuss more on the literature of realised volatility. The usage of high frequency data for realised volatility in forecast evaluation can be applied in parameter estimation for exponential smoothing with the following minimisation expression:
The recent researchâ€™s interest in using a comparative volatility estimator as an alternative has emerged a significant literatures on volatility models that incorporated high frequency data. One of the emerging theories for a comparative volatility estimator is the so called Realized Volatility. Realized volatility is referred as the volatility calculated using a short period time series or using higher frequency periods. In Andersen and Bollerslev (1998) showed that high frequency data can be used to compute daily realize volatility which showed a better true variance than the usual daily return variance. This concept is adopted in Andersen, Bollerslev, Diebold & Labys (2003) to forecast the daily stock volatility which found that the additional intraday information are provide better result in forecasting low volume and up market day. The application of realized volatility has also been employed by Taylor (2004) in parameters estimation for weekly volatility forecasting using realised volatility derived from daily data. An encouraging result were showed by using the smooth transition exponential smoothing method whereby the research used eight stock indices to compare the weekly volatility forecast of this method with other GARCH models (Taylor, 2004). The concept of realized volatility has been employed by many researchers in forecasting of many other financial assets such as foreign exchange rates, individual stocks, stock indices and etcetera. One of the early application of realized volatility concept has used spot exchange rates of Deutschemark-US dollar and Japanese Yen-US dollar to show the superiority of using intraday data as realized volatility measure. The sum of squared five-minute high frequency returns incorporated in the forecasting model proved to outperform the daily squared returns as a volatility measure (Andersen et al., 1998). Another similar study done by Martens (2001) has adopted realized volatility in forecasting daily exchange rate volatility using intraday returns. The results showed that using highest available frequency of intraday returns leads to superior daily volatility forecast. Furthermore, realized volatility approach has also been extended to studies for risk and return trade-off using high frequency data. In Bali et al. (2005), the research provided strong positive correlation between risk and return for stock market using high frequency data. The usage of daily realized which incorporated valuable information from intraday returns produce more accurate measure of market risk. In addition to this study, Tzang et al. (2009) as applied the realized volatility approach as a proxy for market volatility rather than squared daily returns to assess the efficiency of various model based volatility forecast. Finally, the findings from a research done by Andersen, Bollerslev, Diebold & Labys (2001) shown that realized volatility in certain conditions is free for measurement error and unbiased estimator for return volatility. The proven research has prompted many recent works in forecasting intra-day volatility to applied realized volatility for their studies. This can be observed in McMillan & Garcia (2009), Fuertes et al. (2009), Frijns et al.(2008) and Martens (2001). Many researchers exploit the advantage of realised volatility as an unbiased estimatorâ€Ÿs measure for intra-day data and also as a simplified way to incorporated additional information into other forecast models. McMillan et al. (2009) utilised realised volatility to capture intraday volatilities itself as opposed to most researchers that uses realised volatility for daily realised approach. The study showed Hyperbolic Generalized Autoregressive Conditional Heteroscedasity (HYGARCH) as the best forecast model of intra-day volatility.
The forecast models that are presented in this study include: Random Walk (RW) 30 days Moving Average (MA30) Exponentially Weighted Moving Average (EWMA) with =0.06 (RiskMetrics) Exponentially Smoothing with Î± optimised (ES) Integrated General Autoregressive Conditional Heteroskedastic using daily data (IGARCH) Exponentially Weighted Moving Average (Riskmetrics) on daily realised volatility calculated from intraday data. (EWMA-RV) Exponentially Smoothing with Î± optimised on daily realised volatility calculated from intraday data. (ES-RV) General Autoregressive Conditional Heteroskedasticity model with intraday data using realised volatility approach (INTRAGARCH) Integrated General Autoregressive Conditional Heteroskedasticity with intraday data using realised volatility approach (IGARCH) General Autoregressive Conditional Heteroskedasticity with daily realised volatility (RV-GARCH)
Various comparative forecast models are used in order to evaluate the performance of incorporating intraday data. This study used dataset from nine stock indices include Malaysia (FTSE-BMKLCI), Singapore (STI), Frankfurt-Germany (DAX30), Hong Kong (Hang Seng Index), London-United Kingdom (FTSE100), France (CAC40), Shanghai-China (SSE), Shenzhen-China (SZSE), and United States (S&P 100). These series consisted of daily closing prices and also the intraday hourly last price of their respective indices. The daily closing prices were retrieved using â€œDataStream Advance 4.0â€? and also from Yahoo Finance (https://finance.yahoo.com). Whereas, the hourly intraday last prices of these stock indices were retrieved from Bloomberg Terminal from Bursa Malaysia. Each stock index has their respective trading hourâ€™s last price which produced a different number of observations for each series. The total number of trading hours within the day differed among different stock index. However, the sample period used in this study spanned approximately for 300 trading days, from 15 October 2009 to 15 March 2011. In order to simplify the study, the focus is based on a one-step-ahead volatility forecast. The first 200 trading days log returns were applied to estimate the parameters for various forecast models which is known as the in-sample forecast. The remaining 100 trading days log returns were used for post-sample evaluation. This study aimed to forecast volatility in daily log returns for various forecasting methods and used daily realised volatility as proxy for actual volatility. The next subsections presented the data description and the 10 forecast methods which will be considered in the study.
This subsection describes the methodology to forecast the in-sample and out-sample performance of various forecast models. The forecast model includes Random Walk (RW), Moving Average, GARCH models, and Exponential smoothing techniques.
This project paper adopted the simple moving average of squared residuals from the recent past 30 daily observations which is labelled as MA30 and the Random Walk (RW) for the standard volatility forecast model as performance benchmark. The 30 day simple moving average is given by: Whereby, Îµ2 = (rt – Î¼)2 shown in the previous section. The moving average is able to smooth out the short running fluctuations and emphasize on the long run trends or cycles through a series of averaging different subsets of datasets. On the other hand, the Random Walk (RW) is explained as the forecast result is equal to the actual value of the recent period. The actual value in this study used is the squared residual denoted as, Îµ2t. The equation is as shown below:ï?¥ Tomorrowâ€™s forecasted value = yesterday actual value ()ï€½
There are many different GARCH models for forecasting volatility that can be included in this research. However, the consideration in this study is limited to 2 forecast GARCH models which are the GARCH and IGARCH for practicality. The GARCH models in this study have applied GARCH (1, 1) specifications. The three forecast model used were labelled as IGARCH, INTRA-IGARCH, and INTRA-GARCH models. The IGARCH model is estimated using daily residuals as daily data is easily obtained from the source mentioned above. The general IGARCH forecast model used is given by:
But, the parameter estimate generate by EVIEW 7 will be using the following expression:
However, the INTRA-IGARCH and INTRA-GARCH models used hourly residual data to estimate the forecast for daily realised volatility. The forecast for volatility of these models over an N-trading hours span period would be recognised as the forecast of daily volatility. The N trading hourâ€™s span period is dependent on the trading hours of a specified stock index. In order to calculate the daily realised volatility, the equation is for N trading hours in a day for a particular stock index is given by: Where period i is the higher frequency of hourly data and the Îµ2t, is the squared residual of the particular hour. For example, if KLCI index has a 7 trading hours per day, the realised daily volatility is calculated from the sum of squared residual of these 7 hours. Additionally, forecast models such as INTRA-IGARCH and INTRA-GARCH applied equation 3 to obtain the daily realised volatility by replacing the squared residual, Îµ2t with values that is forecasted using these models.
The GARCH model can be estimated using daily realised volatility which is derived from the hourly squared residual with equation 3. In order to apply RV for GARCH forecast model, equation 3 has to be modified to be squared root to be able to obtain the parameter estimates that is needed using EVIEW 6. The equation is as follow: As for this project paper, the GARCH model that used daily realised volatility as input data is labelled as RV-GARCH.
The forecast model for exponential smoothing method has been implemented into two approaches. The first is by using minimisation of equation 3 to optimise the parameter and it is labelled as ES for this project paper. The actual value (squared residual), Îµ2t is obtained from the daily data. The second approach which is said to be the better proxy variance forecast has applied equation 4 for the minimisation. The forecast model for this exponential smoothing method is termed as ES-RV which adopted daily realised volatility from hourly data. Apart from that, the study also considered the smoothing parameter Î± as a fixed value of 0.06 as recommended by RiskMetrics (1996) for model using daily data and daily realised volatility data derived from hourly data. The forecast model is termed as EWMA and EWMA-RV respectively. By using equation 2 as shown previously, the EWMA used daily squared residual as Îµ2t – 1 parameter input while the EWMA-RV used the daily realised volatility as the Îµ2t – 1 parameter input.
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