Long Run Forecast of the Covariance Matrix
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Table of Contents Abstract i 1. Introduction - 1 - 1.1. Research Background - 1 - 1.2. Research Objectives - 3 - 1.3. Research Approach and Scope - 3 - 1.4. Layout of the Report - 4 -

1. Introduction 2.1. Research Background
Volatility is an important concept in finance. Volatility modelling and forecasting finds usage in several core financial operations, for instance – many asset-pricing models use volatility as an estimation parameter for simple risk; several famous option pricing formulas such as Black-Scholes use volatility; volatility estimates and forecasts are crucial for portfolio management and also in hedging risk. Because of the importance of volatility, as can be seen from the examples above, the interest in modelling and forecasting volatility has increased many-fold in recent times, with a special emphasis on forecasting. There are several types of techniques available for forecasting volatility, with extraordinary diversity of procedure such as the Autoregressive Moving Average (ARMA) models, Autoregressive Conditional Heteroscedasticity (ARCH) models, Stochastic Volatility (SV) models, regime switching and threshold models. (Xiao and Aydemir, 2007:1) A broad division between the techniques is based on primary assumptions of constant variance i.e. homoscedastic e.g. AMA models, or non-constant variance i.e. heteroscedastic or time-varying e.g. GARCH models. The GARCH i.e. Generalized ARCH models introduced past conditional volatility as an explanatory variable of the forecast volatility in addition to volatility new that was already a part of the original ARCH model. The GARCH models have become the most widely used auto-regressive heteroscedastic models. (Labys, 2006:28)
ARCH models have the capability of capturing volatility clustering in financial data i.e. the tendency for large (small) wings in returns to be followed correspondingly by large (small) swings of random direction. The conditional variance here depends on a single observation. The ARCH model was extended to the generalized form i.e. GARCH by including lags i.e. allowing changes to occur more slowly. The univariate GARCH models have been extended in several directions chiefly the Integrated ARCH or IGARCH model where the current information remains important for forecasts of the conditional variance; Exponential ARCH model or EGARCH model where the conditional variance is constrained to be non-negative; the ARCH-in-Mean or ARCH-M model used in asset-pricing theories of CAPM, where based on function, conditional mean increases or decreases with an increase in the conditional variance; the Fractionally Integrated ARCH where the lag length delays hyperbolically rather than geometrically. (Xiao and Aydemir, 2007:4-10)
Univariate models mentioned above model the conditional variance of each series entirely independent of all the other series. This can be a limitation since the volatility of a portfolio may spread over different assets and also the covariance between the series as well as the variances between the individual series may be of interest in many cases. A classic example is the variation in currencies due to their possible dependence through cross-country conditional variances as well as multiple individual factors affecting each currency. In order to overcome these limitations researchers were led to investigate multivariate models. Multivariate models can be used to model conditional heteroscedasticity in multivariate time models, focusing on modelling and predicting the time-varying volatility of the multivariate time series. (Zivot and Wang, 2006:481) GARCH models have been vastly applied to multivariate problems in empirical finance. In fact multivariate ARCH or M-GARCH models appeared almost at the same time as univariate models. Multivariate models however suffer with the limitation of having too many parameters to be useful for modelling more than two assets jointly. A case in point was the multivariate GARCH model proposed by Bollerslev et. al. in 1988 who used the Capital Asset Pricing Model CAPM in the framework of conditional movements to come up with the VEC model. (Bauwens, Hafner, and Laurent, 2012:18) A natural consequence of such modelling attempts was that the focus of the researchers shifted towards the design of innovative multivariate models that can be estimated for larger dimensions. The present research too focuses on this challenge i.e. the possibility of extending various existing univariate models into corresponding multivariate models 2.2. Research Objectives
Literature generally suggests that regardless of the forecasting procedure, the short-term volatility forecasting will be more accurate than long-term volatility forecasts. This paper mainly addresses the possibility and feasibility of extending the univariate model to multivariate setting. In addition to this, the models will also be comparatively evaluated statistically based on their performance i.e. capability of forecasting. The models under focus are extending the Exponentially Weighted Moving Average EWMA – a special case of the GARCH model, where the current observations have a higher weight than the previous observations, to Multivariate-EWMA model; the Dynamic Conditional Correlation DCC model which is an extension of the M-GARCH model where the assumptions of constant correlations are not required; the BEKK model proposed by Engle and Kroner where the variance is dependent on the amount of currently available information; and the multivariate skew GARCH or S-GARCH mode. 2.3. Research Approach and Scope
As mentioned above, the present research aims to compare the forecasting capabilities of four multivariate models – M-EWMA, DCC, BEKK, and S-GARCH. First, the four models are implemented in the multivariate context using the DCC approach. Following this, the models will be evaluated by evaluating the accuracy, bias and information content of the models’ performance. Forecast accuracy will then be tested using the ordinary univariate measures such as Root Mean Squared Error (RMSE), the Mean Absolute Error (MAE) and the Heteroscedasticity-adjusted MSE (HMSE) of Bollerslev and Ghysels (1996). In addition to this, the popular Diebold and Mariano test for null hypothesis of equal predictive ability or unconditional predictability will also be used to compare the four models.
The data used for the purpose of analysis will be the forecasting of the actual exchange rate compared to the US dollars of the British Pound, Japanese Yen, and the Swiss Franc i.e. the GBP/USD, JPY/USD and CHF/USD. These exchange rates are sampled between the period – 07/2007 and 06/2012. This period is selected in order to include the financial crisis period, so that the capability of the models to forecast the exchange rate can be evaluated. 2.4. Layout of the Report
Chapter 2 presents a review of the relevant literature where the models to be evaluated are studied in depth in addition to reviewing the works of previous researchers in the context of the research focus. Chapter 3 describes the research methodology, where the mathematical basis behind the present research in addition to presenting the specific analysis methods/tools used for the present research. Chapter 4 describes the data used for the research and its properties. Chapter 5 presents the analysis of the research results. Chapter 6 presents a summary of the conclusions drawn from the research analysis and recommendations for future work in this area.

2. Literature Review 3.5. Understanding and forecasting volatility
Sinclair (2011) mentions that the standard mathematical definition of volatility is “the square root of variance” as can be expressed as below: s2=1Ni=1N(xi-x)2 Equation 1
In the above equation, xiis the logarithmic return; xis the mean return of the sample; and N is the sample size. While the above equation is technically the definition of variance, in finance it is very difficult to distinguish the mean returns i.e. drift from the variance – a central problem. Also the estimates of the mean return are very noisy especially for small samples. Hence, it is common practice in finance to set the mean return in the above equation to zero i.e. x=0. This would increase the accuracy of the measurement since a main source of noise is removed. (Sinclair, 2011:17) Thus the Equation 1 reduces to s2=1Ni=1N(xi)2 Equation 2
Regardless of the statistical significance of the measures described above, measuring and more importantly forecasting volatility is not like measuring common variables such as price. Volatility is a latent variable and therefore must be inferred by the movement of prices. If there are large fluctuations, it is easy to infer that the volatility is high, but it is difficult to ascertain the volatility in terms of a numerical value. In addition, instantaneous volatility is unobservable; it needs time to manifest. A primary reason for this is because it is difficult to infer if a large shock, positive or negative, is transitory or permanent. (Danielsson, 2011) For example both the equations 1 and 2 measure only the current volatility. Extending the volatility measurement to a longer period e.g. annual volatility requires adding an adjustment factor. The simplest measure will be to multiply the daily estimate by the number of days – not a very efficient estimate. Several researchers have proposed alternative means of measuring volatility, but the bias between the methods have no proved very encouraging. (Sinclair, 2011:30-31) 3.6.1. Basic volatility forecast models
Volatility measurement is complicated as seen above, but obviously not as much as volatility forecasting over a period of time. In fact, volatility modelling is many times considered as an art rather than science because of issues such as non-normalities, volatility clusters and structural breaks. The latent nature of volatility means that it must be forecast by a statistical model, a process that inevitably entails making strong assumptions. Also the presence of volatility clusters suggests that the long-term volatility forecast is much more complicated and uncertain than short-term forecasts. The researchers usually tend to assign higher weights to the most recent observations while formulating volatility forecasting models, as it is generally assumed that using the most recent observations to forecast volatility is more efficient for future predictions. (Danielsson, 2011:32-33)
The simplest forecasting method is to just assume that the next N days will be like past N. this method is commonly known as the moving window method. Needless to say this type of estimate will not hold true in case of the unexpected nature of a real world market with its unexpected booms and crashes. Also known as the moving average, the model keeps the sample size constant, adds the newest daily return every day and drops the oldest. (Alexander, 2008:129) The equation of the simple moving average model is as shown below: st2=1WEi=1WEyt-12 Equation 3
In the equation above, stus the volatility forecast for day t, yt is the observed return on day t, and WEis the length of the estimation window i.e. the number of observations used in the calculation. From the equation, it is clear that the most recent return used for forecasting volatility of day t is t-1. The time period of the forecast is determined by the frequency of the data. (Alexander, 2008:129) 3.6.2. Exponential weighted moving average model
The exponential weighted average EWMA model is essentially a simple extension of the historical extension of the moving average measure. Some limitations of the moving average model are: all the observations are assigned equal weight; the data points to use in the data window are an important decision but the procedure to select the points is entirely subjective and does not have any statistical significance; and the model does not work well under an extreme market shock condition. An improvement over the MA model is the exponential weighted moving average EWMA model where different weights are assigned to the observation, the most recent observations having the highest weight as they are considered to have the strongest impact on the forecast as compared to older data points. (Brooks, 2008:384) In the EWMA model, the weights associated with previous observations decline exponentially over time. The EWMA model was popularized by JP Morgan with their RiskMetrics software that used the EWMA methodology for forecasting volatility. The EWMA model is known to provide useful forecasts for volatility and correlation over the very short term such as next day/week; however its usage is limited in case of long-term forecasting. (Alexander, 2008:130) The EWMA model can be represented mathematically as below: st2=1-λλ(1-λWE)i=1WEλiyt-12 Equation 4

In the above equation, the term λ represents the weight attached to the observations, and the first part of the equation above ensures that the sum of weights is 1. The value of λ lies between 0 and 1 and essentially acts as the decay factor. The EWMA model reacts immediately following an unusually large return, and then the effect of this return on the volatility gradually diminishes. In the equation 4, the Model user has only one choice to make in the above equation – λ i.e. the smoothing constant. The problem obviously will occur if the choice is made on an ad-hoc basis as the forecasts depend crucially on λ. Again despite the importance of the term λ, there is not statistical procedure to explain how to choose it. (Danielsson, 2011:33)
Both the simple methods of volatility forecasting discussed above i.e. the moving average and exponentially weighted moving average models assume that the returns are independently and identically distributed i.e. i.i.d and that they are multivariate normally distributed. In case of moving average all the past returns are weighted equally while in EWMA the weights reduce exponentially as the observations go from most recent to past values. (Alexander, 2008:130) 3.6. A Review of Univariate ARCH model family
The Autoregressive Conditionally Heteroscedastic ARCH class of models are the most widespread model used in finance to forecast volatility. These models have a definite advantage over the moving average models discussed in the previous section, since MA class of models are based on the assumption that returns are independent and identically distributed i.e. i.i.d, and thus the volatility and correlation forecasts derived would be equal to the current estimates. But empirical evidence has proved that neither the i.i.d. assumption nor the normally distributed are realistic for real financial markets. (Alexander, 2008:130-131) It has been observed that the volatility changes over times, with periods when volatility is exceptionally high and periods when volatility is unusually low i.e. volatility exhibits the property of clustering. The clustering behaviour again is dependent on the frequency of the data, for instance annual data does not show much clustering but daily and intraday data shows high evidence of clustering. The ARCH class of models have the ability of capturing volatility clustering in financial data, which is why more and more financial practitioners prefer to base their forecasts on the GARCH models. (Xiao and Aydemir, 2007:4) 3.7.3. Basic ARCH and GARCH Models
The ARCH model was first developed by Engle in 1982 and then extended by Bollerslev in 1986 to the Generalized ARCH i.e. GARCH. The term autoregressive means that the model is time-series based with an autoregressive i.e. regression on itself form. The volatility cluster shows that the variance and mean is not constant and is conditional as opposed to unconditional variance which simple means sample variance. Conditional variance with time essentially means that the variance estimate is dependent on the set of information available at a particular time. The term conditional heteroscedasticity in ARCH means that the time variation in conditional variance is built into the model. In other words, the idea that large returns are likely to be followed by large returns in either direction is what is referred to as conditional heteroscedasticity. Thus the ARCH type models offer an alternative way of estimating the conditional variance of a variable more flexible than the MA or EWMA. (Sollis, 2012:68-69) The basic ARCH model given by Engle was a stochastic process where the variables have a conditional mean of zero and conditional variance given by a linear function of previous squared variables. The squared variables followed an autoregressive process. The basic ARCH (k) model when conditional mean is 0, can be written as below:
Yt=σtεt Equation 5 σt2=ω+i=1kαiYt-i2 Equation 6
(Danielsson, 2011:36)
In the above Equation 6, k represents the number of lags. In order to ensure that the variance is always positive: ω > 0 and αi ≥ 0. In the equation 5, εt are random variables with mean 0 and variance 1 and have either the standard normal or the following standardized Student-t distribution. If k=1, the equation 6 reduces to the ARCH (1) model, according to which the conditional variance of present day’s return is equal to a constant plus the return of the previous day, as shown below: (Danielsson, 2011:36) σt2=ω+αYt-i2 Equation 7
The ARCH model was extended by Bollerslev in 1986 to generalized ARCH or GARCH by including lags of the conditional variance in the model for the conditional variance. The GARCH model i.e. GARCH (p, q) can be represented as below: σt2=ω+i=1pαiYt-i2+j=1qβjσt-j2 Equation 8
(Danielsson, 2011:38)
In the above equation, Yt=σtεt. In the equation 8, εt are random variables with mean 0 and variance 1 and have either the standard normal or the following standardized Student-t distribution. The most commonly used form of GARCH is the GARCH (1, 1) usually used to fit time series and hence it is known as the workhorse of volatility models in finance. This is because even through the GARCH (1, 1) model has just one lagged square error and one autoregressive term, it is sufficient for financial purposes since it has infinite memory. The GARCH (1, 1) model can be represented as below: (Danielsson, 2011:38) σt2=ω+αYt-12+βσt-12 Equation 9
The EWMA model discussed is previous section can be considered as a special case of the GARCH (1, 1) model when ω = 0 and α + β = 1. (Scott, 2005:71) In all the ARCH & GARCH models discussed above (equation 5-9), the conditional mean of the asset return Yt is assumed to be 0. However, if the asset-return Yt has a conditional mean of μt with conditional variance σt2, the basic premise of the equation changes as below:
Yt-μt=εt=σtzt Equation 10
In the above equation, εt represents the residual at time t and zt is an unobservable random variable belonging to an i.i.d. process with mean equal to 0 and variance equal to 1. With this assumption, the equations 6 through 9 would change by the substituting Y with ε or with (Y-μ) – as desired by the model developer/user. 3.7.4. Variations of the GARCH model
In the equations above, the empirical estimates of the sum of the parameters α and β are often near one and sometimes the sum exceeds one if the parameters are not constrained. This led Engle and Bollerslev in 1986 to consider the integrated specification when α + β = 1. The resulting model is known as the Integrated GARCH or IGARCH model with only the parameter ω constrained to be non-negative. In the Integrated GARCH (p, q) model, the parameter change would be for the equation 8, where the following condition should be true: i=1pαi+j=1qβj=1 Equation 11
For the IGARCH (1, 1) model p = q = 1. Thus the above equation reduces to α + β = 1 or β = 1 – α. The IGARCH (1, 1) model can be written as below: σt2=ω+αYt-12+(1-α)σt-12 Equation 12 The ARCH-in-mean or ARCH-M model was suggested by Engle, Lillien and Robins in 1987, and is a variation of the ARCH model where the conditional volatility is present in conditional mean. The generalized extension of the model would be GARCH-M. The ARCH-M model estimates the time-varying risk premiums with time-varying variances and is specified as below:
Yt=μt+εt Equation 13 μt=γ+δσt Equation 14
In the equation 14, γ and δ are parameters, and μt & σt are the conditional mean and variance of the return Yt. The parameter δ can be interpreted as the price of risk and is thus assumed to be positive.
The GARCH model and its variations discussed till now have been linear. This mean that the conditional variance is assumed to be a linear function of the squared past values. The drawback of such a model would be that the conditional variance is dependent only on the modulus of the past variables; the positive and negative variations would have the same effect on the volatility. However, it has generally been observed that the increase in volatility due to a decrease in price is stronger than that resulting from a price increase of the same magnitude. The properties of nonlinearity and asymmetry have been thus been introduced in order to account for the sign of volatility in the model. (Francq and Zakoian, 2010:245-246) The most popular model of such type is the Exponential GARCH or EGARCH model. The EGARCH model was developed by Nelson in 1991 and allows for asymmetric responses of stock market volatility to negative and positive changes in the residuals of the mean equation. The conditional variance here is expressed in logarithm and the positivity constraints required for the usual GARCH models can be removed. (Zivot and Wang, 2006:241) The EGARCH (p, q) equation can be written as below: logσt2=ω+i=1pαiεt-i+γiεt-iσt-i+j=1qβjlogσt-j2 Equation 15
(Zivot and Wang, 2006:241)
When εt-i is positive, the total effect of εt-i is (1+γi)| εt-i |, and when εt-i is negative, the total effect of εt-i is (1-γi)|εt-i|. When the effect is negative, there is a larger impact on volatility and the value of γi would be expected to be negative. The EGARCH model is asymmetric as long as γi ≠ 0. The logarithmic transformation guarantees that variances will never become negative. (Verbeek, 2008:314)
Another GARCH variant capable of modelling leverage effects is the threshold GARCH or TGARCH model. The TGARCH model can be represented mathematically as below: logσt2=ω+i=1pαiεt-j2+i=1pγiSt-iεt-j2+j=1qβjlogσt-j2 Equation 16
(Zivot and Wang, 2006:242)
In the above equation, St-i = 1 if εt-i < 0, and St-i = 0 if εt-i ≥ 0. This means that depending on whether εt-i is above or below the threshold value of zero, ε2t-i has different effects on the conditional variance σ2t. When if εt-i is positive, the total effects are given by αiε2t-i and when εt-i is negative, the total effects are given by (αi + γi) ε2t-i. This means that γi is positive for bad news to have larger impacts. This model is also known as GJR model because Glosten, Jagannathan and Runkle proposed essentially the same model in 1993. (Zivot and Wang, 2006:242)
Component GARCH or CGARCH given by Engle and Lee in 1993, models the variance as the sum of several processes or components. In case of a two-component model, one component is used to capture the short-term response to shock and another to capture the long-term response. Thus allows the model to capture long memory effects with slow decay of volatility. The standard GARCH (1, 1) model can be written as: σt2=ω+αεt-i2-ω+β(σt-j2-ω) Equation 17 ω=ω1-α-β=σ2 Equation 18 In the above equation ω represents the unconditional variance or long-term volatility. Thus the usual GARCH has a mean reversion tendency towards ω – constant at all times. If the fixed ω is allowed to vary over time i.e. it is replaced by the varying level qt, it leads to the component ARCH or CGARCH (1, 1) model as below: σt2=qt+αεt-i2-qt-1+β(σt-j2-qt-1) Equation 19 qt=ω+ρqt-1+φ(εt-i2-σt-j2) Equation 20
The difference between the conditional variance and its trend, σt2-qt, is the transitory or short-run component of the conditional variance, while qt is the time varying long-run volatility of the intra-day range. The above equation is the two-component GARCH model. 3.7. Multivariate Models 3.8.5. Multivariate EWMA
The exponential weighting is done by using a smoothing constant λ and as this value increases more weight is placed on the past observations and the more smooth the series becomes. 1st order EWMA models can be written as below: σt2=(1-λ)εt-i2+λσt-i2 Equation 21 This equation can be extended to the multivariate EWMA model by defining the covariance matrix ∑t. The equation for M-EWMA gives the future covariance of the portfolio components ∑t 3.8.6.

3. Methodology

4. Data Description 5.8. Data Source
The data used for this research is the exchange rate with respect to the US Dollar. Three different currencies are used – British Pound GBP, Japanese Yen JPY, and Swiss Franc CHY. The analysis is conducted on the 5-year exchange rate for each of the three currencies, starting from July 1, 2007 to June 28, 2012. This data has been collected from the Bank of England archives and consists of 1260 data points for each of the three currencies or a total of 3780 (i.e. 1260 * 3) data points. The exchange rate used is also referred to as the spot exchange rate or spot rate. The spot rate is essentially the home price of a foreign currency, in this case in terms of US dollars. Spot exchange rate is essentially identifies the price at which currencies can be traded immediately. In contrast, the forward rates identify the price at which currencies can be traded at some future date. In the present research, the data represents the mean rate of buying and selling or middle market rate, observed by the Bank of England’s Foreign Exchange Desk in the London interbank market at approximately 4.00 pm. These rates cannot be termed as official rates, but are rather the rates at which Bank of England would execute the settlement of a currency deal after two working days. (Bank of England, 2012; Evans, 2011:4) 5.9. Data Characteristics
For the purpose of analysis, the collected data sample has been divided into two parts – the in-sample and out-of-sample period. The in-sample part is used for initial estimation of the volatility models, while the out-of-sample part is used for estimation-check purposes. As mentioned above, there are 1260 observations for each currency out of which the last 210 observations will be used for the out-of-sample evaluation. This means that the first 1050 observations for each currency is used for in-sample estimation purposes. In summary, the data from July 1, 2007 to August 24, 2011 is used as in-sample period, while the data from August 25, 2011 to June 28, 2012 is used as out-of sample period. The log of actual data is taken since the equation is based on log-returns of asset prices. The data from the Bank of England archives is the conversion in terms of the US dollars in terms of British Pound GBP, Japanese Yen JPY, and Swiss Franc CHY. The requirement is however the reverse i.e. the values of British Pound GBP, Japanese Yen JPY, and Swiss Franc CHY in terms of US dollars is required. Hence, in the SPSS file, the log of the returns is calculated by taking the reciprocal of the original data values collected. 5.10. Descriptive Statistics
The figure 1 below shows the descriptive statistics of all the 3780 (i.e. 1260 * 3) data points. The standard deviation from mean is 0.10868 for British Pound, 0.12746 for Japanese Yen, and 0.10423 for Swiss Franc. This means that the values of Japanese Yen varied the most from the mean, though not by a large factor, as compared to British Pound and the Swiss Franc, both of which had similar values for the standard deviation. A look at the skewness values show that the British Pound and Swiss Franc values show positive skewness i.e. a long right tail, while the Japnese Yen shows negative skewness, i.e. a long left tail. An analysis of the kurtosis values for all the three variables shows that the values show negative kurtosis. The kurtosis reported here is also known as excess kurtosis and gives an idea about the distribution of the data values. All the three data variables are Platykurtic i.e. have flatter data values and are more dispersed along the X-axis than the normal distribution. The figure 2 shows a detailed visual view of these characteristics. The rule of thumb for unacceptable values is that the values should not exceed ±1, which is true here, so the skewness and kurtosis are not beyond unacceptable limits

Figure 1 Descriptive Statistics of all the Observations

Figure 2 General Skewness and Kurtosis Characteristics of variables
(Bian, 2011:14-16)

5.11. 5. Results 6. Conclusion

References
Alexander, C., (2008), “Market Risk Analysis: Practical Financial Econometrics”, New Jersey: John Wiley & Sons
Bank of England, (2012), “Explanatory Notes: Spot Exchange Rates”, webpage accessed on 27th June 2012, http://www.bankofengland.co.uk/statistics/pages/iadb/notesiadb/Spot_rates.aspx
Bauwens, L., Hafner, C.M., Laurent, S., (2012), “Handbook of Volatility Models and Their Applications”, New Jersey: John Wiley & Sons
Bian, H., (2011), “SPSS Introduction II”, Fall 2011, Office Faculty for Excellence, http://core.ecu.edu/ofe/StatisticsResearch/SPSS%20Introduction%20II.pdf
Brooks, C., (2008), “Introductory Econometrics for Finance”, 2nd edition, Cambridge: Cambridge University Press
Danielsson, J., (2011), “Financial Risk Forecasting: The Theory and Practice of Forecasting Market Rick with Implementation in R and MATLAB”, Chichester, West Sussex: John Wiley & Sons
Evans, M.D.D., (2011), “Exchange-Rate Dynamics”, Princeton, New Jersey: Princeton University Press
Francq, C., Zakoian, J.M., (2010), “GARCH Models: Structure, Statistical Inference and Financial Applications”, Chichester, West Sussex: John Wiley & Sons
Labys, W.C., (2006), “Modelling and Forecasting Primary Commodity Prices”, Aldershott, Hampshire: Ashgate Publishing Limited
Scott, H.S., (2005), “Capital Adequacy Beyond Basel: Banking, Securities, and Insur ance”, New York: Oxford University Press
Sinclair, E., (2011), “Volatility Trading”, New Jersey: John Wiley & Sons
Sollis, R., (2012), “Empirical Finance for Finance and Banking”, Chichester, West Sussex: John Wiley & Sons
Verbeek, M., (2008), “A Guide to Modern Econometrics”, 3rd edition, Chichester, West Sussex: John Wiley & Sons
Xiao, L., Aydemir, A., (2007), “Volatility Modelling and Forecasting in Finance”, Knight, J.L., Satchell, S., (Eds) (2007), Forecasting Volatility in the Financial Markets, 3rd edition, Oxford: Butterworth-Heinemann
Zivot, E., Wang, J., (2006), “Modelling Financial Time Series with S-Plus, Volume 13”, 2nd edition, New York: Birkhäuser