Forecasting accuracy of historical and extreme-value volatility using
an implied volatility index as benchmark.
R.K. van Harten (1477870) Supervisor: drs. A.J. Meesters
University of Groningen Faculty of Economics
October 2009
Abstract
This paper examines the accuracy of forecasted volatility based on standard historical and extreme-value volatility. The contribution of this study to the literature is that the benchmark which contains the future realized volatility is an independent volatility index and not a volatility estimate derived from one of the volatility estimation models used in most previous studies. The dataset contains daily open, high, low and closing prices from the S&P500 and the DAX index for a time period ranging from 1 January 2008 to 31 December 2008. Results derived from the forecasting performance indicate that in times of major market movements, like in the year 2008, none of the volatility estimation models are able to forecast future volatility with a low bias. Compared to previous studies, the historical close-to-close based volatility has the lowest error of all volatility estimation models.
JEL Classifications: C52, C53
Acknowledgements
I Introduction
Volatility estimation is of central importance for option pricing, investment decisions and risk management (Poon and Granger, 2003). In the last 25 years many attempts have been made to improve the accuracy of the classical volatility estimate used in the Black-Scholes (1973) option pricing formula, see for instance Parkinson (1980) and Rogers Satchell (1991). Volatility is traditional estimated as the squared sample standard deviation of returns over a certain period. This kind of volatility is called historical volatility. While it is not difficult to estimate historical volatility it is not a highly accurate measure for the true future volatility, i.e. the daily open, high and low prices should also be taken into account when estimating volatility. Therefore, the interest in volatility estimation and forecasting has steadily increased during the last decade (Poon and Granger, 2003). Besides the historical volatility it is also possible to derive volatility from option prices implied by the market, this kind of volatility is called implied volatility. In the frictionless market of Black and Scholes (1973), all prices are observed without errors and every option price can be inverted to find the unique implied volatility which is consistent with the observed prices when all other parameters are known.
Andersen and Bollerslev (1998) investigated the modeling and forecasting of historical and extreme-value volatility in more detail and confirmed the presence of a high level of error in the forecasting accuracy using the historical volatility method. Forecasted volatility based on high frequency data, the so called extreme-value volatility, is regarded as the best approximation of the actual volatility (Andersen et al, 2001 and 2003). Many of these extreme-value volatility estimators, like the Parkinson (1980), the Garman and Klass (1980) and the Rogers and Satchell (1991), model are easy to implement. Because, the use of high frequency data in these models one only requires the daily open, close, high and low price, which are readily available in the public domain.
The purpose of this research is to identify the most accurate model to forecast future volatility. The accuracy of the models is assessed by comparing the forecasted volatilities with actual volatility obtained from the volatility index. If a forecasting model is not significantly different from the assumed actual volatility, which is defined as the implied volatility in this study, than implied volatility does not contain additional information compared to the forecasted volatility. If this is the case, implied volatility is not considered to be superior to extreme-value volatility.
This study contributes to the existing literature by investigating the ability of extreme-value volatility estimators to forecast future realized volatility, using an independent benchmark as realized volatility. Most previous studies like Beckers (1983) and Wiggins (1991) among others, estimate volatility using the historical or extreme-value based volatility. The problem with these empirical studies and with estimation realized volatility in general is that actual volatility is not directly observable in the market. However, if you want to test the performance of your volatility forecast you need a benchmark which must mirror the actual volatility in the most accurate manner. The lack of observable realized volatility is solved in these previous studies with a benchmark which contains historical or extreme-value based volatility. By using these kind of benchmarks where its value is derived from the same methods to forecast future volatility, realized and forecasted volatility are of course highly correlated and therefore it is skeptical to state, based on the forecasting errors, that the forecasting volatility models are a good approximation for future realized volatility.
This study is interesting because volatility is one of the key inputs for option pricing and risk management. Therefore, volatility is a cornerstone in responsible financial decision making. Especially during the ongoing credit crunch, a fine-tuned volatility forecast is of great importance. Financial institutions could have saved billions if they were more able to anticipate on the rapidly changing financial markets.
The remainder of this article is organized as follows. Section 2 introduces the extreme-value estimators and other theoretical backgrounds used in this study. Section 3 presents the data description. In section 4 the methodology used will be explained in more detail. Empirical results are analyzed in section 5. Finally, conclusions and recommendations for further research are provided in section 6.
II Theoretical framework
Past research has shown that risk estimators are the key inputs for empirical financial economics, see for instance the studies done by Parkinson (1980) and Rogers and Stachell (1991). As already mentioned briefly in the introduction, the foundation for the modern option pricing theory was laid in 1973 by Black and Scholes. In their formula the expected variance of the stock over options remaining life is required as an input parameter to determine the option price. Yet, the expected variance or volatility of the underlying is not directly observable. However, knowing all other input parameters from the Black and Scholes option pricing formula and the option price given by the market, it is possible to calculate the required volatility implied by the market (Natenberg, 1994). Based on the fact that the market is never wrong, implied volatility obtained from option prices should be equal to the volatility related to the underlying asset. Therefore, the benchmark used in the study is obtained from multiple option series to mimic the realized volatility in the market as close as possible.
Therefore, Parkinson developed the Parkinson Model (from now on I refer to the Parkinson model using the abbreviation: P-Model) which uses the daily high and low price to calculate volatility. Garman and Klass (1980) extended the Parkinson model with their own Garman and Klass Model (from now on I refer to the Garman and Klass model using the abbreviation: GK-Model). In the GK-Model the daily open and close price are also taken into account. Assuming that trading is continuous and always monitored these EV estimators are at least five times more efficient than the close-to-close estimators (Beckers, 1983). As one might expect, EV models provide a more efficient volatility forecast for future realized volatility than the standard close-to-close models (Wiggins, 1991).
Early empirical studies on the P-Model and GK-Model by for instance Beckers (1983) and Wiggins (1991) confirmed the efficiency of the EV estimators. Beckers (1983) investigated the ability of the historical and the P-model estimators to predict future quarterly volatility using data from 208 listed stocks during January 1973 till March 1983. Beckers (1983) concluded that the high and low prices contain information which is unavailable in the closing prices and are therefore useful for prediction future volatility. In a later study done by Wiggins (1991) the performance of both the P-model and the GK-model was investigated using high, low, open and closing prices for virtually all NYSE and AMEX listed stocks over the period January 1973 - January 1986. He found that both volatility estimation models are much more efficient in estimating volatility than the historical close to close volatility.
The strength of these forecasting models was examined using historical volatility as benchmark. In a later study by Wiggins (1992), the S&P500 futures prices were used in EV volatility models. Wiggins (1992) found evidence that these EV models are not only a more efficient forecast than historical based volatility, but also outperforming implied volatility estimates in the sense that is a more efficient forecast for future realized volatility. A major drawback in these abovementioned studies is that the benchmark which contains historical based volatility does not automatically displays the realized volatility.
Vipul and Jacob (2007) found supporting evidence of a well performing RS-model in times of severe price oscillations, they used high-frequency data from the Nifty (India) Index for a period of 5 years from January 2001, through December 2005. In contrast to these finding, Bali and Weinbaum (2005) found that only the P-model and GK-model volatility estimators outperformed the historical close-to-close volatility. In their study they used four sets of high-frequency data, S&P500 index futures and three currencies, for a period from December 1986 through August 2003. In all cases the GK-model outperformed the historical close to close volatility and the other EV estimators.
Poon and Granger (2003) show that interest in financial market modeling and specifically the field of forecasting volatility increased progressive over the last two decades. More often, implied volatility derived from related option prices to the underlying asset or autoregressive methods are used to forecast future volatility. A shortcoming when using implied volatility is that not all assets have a related option series. Therefore, if you want to calculate implied volatility you need options of the underlying asset or price index. Vipul and Jacob (2007) also mention in their study that for calculating the implied volatility, the option market should be reasonably developed. However, not all option markets are developed, especially not those in emerging markets.
To counter these implied volatility shortcomings, extreme-volatility models can be an accurate alternative, because these volatility estimation models only require daily open, high, low and closing prices.
In addition to the three models introduced above there have been more attempts to model volatility. Kunitome (1992) developed a model where the estimator is based on a number of daily price observations. This model is often ignored, because of the limited availability of tick-level data. The estimator of Yang and Zhang (2000) is also often ignored because is does not provide a daily volatility estimate.
It is erroneous to evaluate the performance of volatility forecast made by the biased estimator and then evaluate the forecast with realized volatility obtained from the same biased estimation model. By doing this you only analyze whether the volatility has changed over time. This study contributes to a fair performance evolution by using an independent benchmark. The benchmark which is used to test the accuracy of the volatility estimation models is a volatility index. The value of this index is obtained from the option series related to the underlying asset/ index. This so called, implied volatility is the volatility applicable in the market. Under the assumption that the market is never wrong, this implied volatility should be the best method to obtain the realized volatility. After determining whether the most accurate volatility estimation model is also not drastically different to the realized volatility (benchmark) obtained from the volatility index, the model can than be of great use to estimate volatility over assets and indices where implied volatility is not available in the market, due to the missing options series of the underlying asset or indices.
A range of standard forecasting and performance models, used in most of the empirical studies, is also employed in this research to determine which model is best for forecasting future volatility see for instance Vipul and Jacob (2007 and 2008) and Poon and Granger (2003). The forecasting models are; Random Walk, Simple Moving Average and Exponentially Weighted Moving Average. The performance evaluation of the different volatility estimators will be evaluated using RMSE, MAE and MAPE. All these models will be explained in more detail in the methodology section.
III Data
The data used in the empirical tests performed in this study consists of the daily high, low, open and closing price observations from 2 stock market indices over the period of 1 January 2008 through 31 December 2008. Especially this period is very interesting because the credit crunch was hitting the stock market brutal. Therefore, if the volatility estimators can forecast realized volatility accurate in this period than their overall performance is also guaranteed in less turbulent periods. Furthermore, it is fascinating to research whether those EV volatility estimators will hold their accuracy in these times of extreme market behavior or that all estimators will default in estimating volatility during this stress test period.
The investigated indices are the S&P500 and the DAX index. We have used these indices, because they both have a coupled volatility index. In evaluating the performance of the forecasted volatility the volatility index is used as a benchmark for the actual observed volatility in the market. The volatility indices used are the VIX and the VDAX related to the S&P500, respectively the DAX index. All data is obtained from Datastream. The descriptive statistics for the used price indices are displayed in table 1 and 2. Each individual column displays the daily open, high, low and closing price.
Table 1:
Summary S&P 500 Index statistics of daily extreme log-returns including the daily open, high, low and closing prices for a period of 2 January 2008 through 31 December 2008.
OPEN HIGH LOW CLOSE
Table 2:
Summary DAX Index statistics of daily extreme log-returns for a period of 2 January 2008 through 30 December 2008.
OPEN HIGH LOW CLOSE
Mean -0.08% 1.11% -1.57% -0.20% Median -0.02% 0.72% -1.18% -0.14% Maximum 3.75% 10.80% 3.75% 10.80% Minimum -7.83% -4.89% -12.61% -7.43% Std. Dev. 0.0106 0.0184 0.0213 0.0238 Skewness -2.4622 2.0544 -1.9229 0.5436 Kurtosis 19.2841 11.8112 8.8729 8.1381 Observations 254 254 254 254
The descriptive statistics show that on average both indices have negative returns during the year 2008. This is caused by the severe credit crunch which was at its peak during the months September till November. The S&P500 index dropped from almost 1300 to just 900 points in the third quarter of 2008. The DAX index hit directly a cruel down swing of almost 20% in January and than dropped another 25% in the third quarter. Another interesting observation is the massive daily extreme high and low returns which indicate that there really was a lot of tension in the market. For both indices a maximum return during the day is around +11%. The maximum daily low return during the day was -10% for the S&P500 and more than -12% for the DAX. Those particular days were eventually the highest and lowest closing return for the year 2008 as well. Looking at the skewness and kurtosis numbers we can conclude that the distribution of the daily returns for both indices is not normal distributed. Therefore, the EV volatility estimators and especially the RS-model should perform better than the standard historical close-to-close volatility, because those models claim to perform better in a bullish or bearish market.
Figure 1: Overview development daily volatility for the S&P 500 and the DAX 0 0.01 0.02 0.03 0.04 0.05 0.06 0 1 /0 2 /2 0 0 8 0 1 /1 6 /2 0 0 8 0 1 /3 0 /2 0 0 8 0 2 /1 3 /2 0 0 8 0 2 /2 7 /2 0 0 8 0 3 /1 2 /2 0 0 8 0 3 /2 6 /2 0 0 8 0 4 /0 9 /2 0 0 8 0 4 /2 3 /2 0 0 8 0 5 /0 7 /2 0 0 8 0 5 /2 1 /2 0 0 8 0 6 /0 4 /2 0 0 8 0 6 /1 8 /2 0 0 8 0 7 /0 2 /2 0 0 8 0 7 /1 6 /2 0 0 8 0 7 /3 0 /2 0 0 8 0 8 /1 3 /2 0 0 8 0 8 /2 7 /2 0 0 8 0 9 /1 0 /2 0 0 8 0 9 /2 4 /2 0 0 8 1 0 /0 8 /2 0 0 8 1 0 /2 2 /2 0 0 8 1 1 /0 5 /2 0 0 8 1 1 /1 9 /2 0 0 8 1 2 /0 3 /2 0 0 8 1 2 /1 7 /2 0 0 8 1 2 /3 1 /2 0 0 8 Date V o la ti li ty VIX VDAX
As mentioned before the future realized volatility is obtained from two volatility indices: the VIX and the VDAX. The value of the VIX and the VDAX indices is quoted in terms of percentage points to the expected movement in respectively the S&P500 and the DAX Index.
To estimate the value of the VIX index involves calculating an approximation of "fair variance" for near-term and next-term options, weighting these two values to construct a constant 30-day variance, and then taking the square root to produce a value for the VIX. The reason the CBOE limits options to the two nearest months is because it is their goal for the VIX to estimate the implied volatility of what an at-the-money option on the S&P 500 would contain with 30 days left until expiration. For example, if the value of the VIX is 50, this value represents an expected annualized change of 50% over the next 30 calendar days. This means that the options markets is expecting an up or down change in the S&P500 of
% 43 . 14 12 % 50 =
in the next 30 calendar days. However, if you want to calculate the standard daily price move in business days than equation 1 is appropriate:
× = t c annulaized daily 365 1
σ
σ
(1)The same valuation method is applicable to the VDAX. To calculate the daily expected change in the underlying index the value of the volatility index is divided by the root of 252 trading days.
In the next section the volatility estimation models are explained in more detail. Also the forecasting and evaluation models will be discussed.
IV Methodology
This section summarizes the formulae used throughout this thesis. Also, the methods to estimate and forecast volatilities that have been applied in this study and the understanding of the benchmark are explained. The study is carried out in three steps. First, the volatility of a return index is estimated from historical prices using different volatility estimators. Second, the estimated volatilities from the first step are used for forecasting future volatility. Finally, the performance of the forecasted future volatilities are compared with the realized implied volatilities. Table 3 shows the used methods to estimate forecasted and evaluated the volatilities. These forecasting and performance models are commonly used in evaluation EV volatility forecasting. See for instance, Poon and Granger (2003) and Vipul and Jacob (2007).
In this section we primarily discuss different approaches to assess the forecasting performance of the EV volatility. The performance of the proposed methods combined with the EV volatility forecasts are compared with the future realized, implied volatility obtained from the volatility indices As mentioned in the data section the VDAX is set as a benchmark to compare the estimated volatilities on the DAX. The VIX is set as a benchmark for the estimated volatilities on the S&P 500.
Table 3:
Overview of the used models to estimate, forecast and evaluated daily volatility.
Estimation Models Forecasting Models Evaluation Models
Historical volatility (HV) RW RMSE
Parkison EV volatility (P) SMA MAE
Estimating historical volatility
For estimating historical volatility a fixed rolling window of 30 and 60 close-to-close prices is used. Therefore, no intraday data is needed for estimating historical volatility. I decided to use both a 30 and 60 days window, because the volatility index predicts the volatility for the next 30 days. In a less volatile market a wider window is appropriate to smooth out short term volatility fluctuations. Therefore, a rolling window of 60 days is also used as a robustness check and to counter these possible shortcomings if the rolling window is set too narrow . The model calculates the standard deviation of daily returns and information obtained during the trading day itself is discarded. Hull (2009) defines historical volatility as described by equation 2:
∑
= − − − = n t t hv r r n 1 2 1 1σ
(2) Where: = −1 ln t t t C Cr , the natural logarithm of the closing price of day t divided by the closing price
of t-1.
−
r is the average yield over a n-day period, in this case 30 and 60 days.
Estimation extreme-value volatility
Extreme-value models are based on the belief that it is possible to capture more effective estimator of volatility if you not only base the calculation on closing prices.
The model developed by Parkinson (1980) uses the daily high and low values instead of closing prices to estimate the variance. The Parkinson estimator of the volatility is given by equation 3:
(
)
2 1 2 ln 4 1∑
= − × = n t t t p H L nσ
(3)Because this model takes only the daily high and low price into account it is more likely that the P-model will overshoot the volatility in times of extreme market movements.
Garman and Klass (1980) developed a more extensive model by including also the open and close prices in their model. The GK model claims to have superior efficiency, meaning that it as minimum variance on the assumption that the process follows a geometric Brownian motion with zero drift, more information about the geometric Brownian motion can be find in Hull (2009). Their estimator is given by equation 4:
(
)
{
(
)(
[
)
(
)(
)}
(
)
2]
1 2 383 . 0 2 2 019 . 0 511 . 0 1 t t t t t t n t t t t t t t t gk O C O L O H O L H O C L H n − − − − − − + − − − =∑
=σ
(4)Where in addition to the symbols mentioned above the Otand Ct are respectively the log
transformed opening and closing prices during day t. The terms
(
H −t Ot)
and(
L −t Ot)
are respectively the normalized daily high and low during day t. The weights put to each part of the equation are taken directly from the optimization done by Garman and Klass (1980). The interested reader can find more information about the construction of GK-model in Garman and Klass (1980).The no-drift assumption is a good approximation when daily price fluctuations are small, which is usually the case for daily series. However there are often periods during which an asset process trends strongly, by which we mean that the drift is large relative to the volatility (Brandt and Kinley, 2002). The assumption of a driftless price process in the P and GK models could lead to an overestimation of volatility. This is likely to happen when a security price exhibits a certain bullish or bearish trend which is definitely the case in the used dataset. Rogers and Satchell (1991) developed an estimator for the volatility of a price process which is suitable for these non- zero drift movements. Their estimator is given by equation 5:
Forecasting of volatility
As mentioned above the forecasting methods include random walk (RW), simple moving average (SMA) and exponential weighted moving average (EWMA).
The RW is an economic theory in which market prices are assumed to be following a Markov process. A property of Markov-processes is that the last actual observation is the best predictor for the future value.
Therefore, the last actual observed volatility at day t ,
σ
t, is the best forecast for future realized volatility,σ
ˆ . See also equation 6: tRW
σ
ˆt =σ
t−1 (6)The SMA method uses subsets of the full dataset to analyze data intervals by creating series of unweighted averages. A moving average is commonly used to smooth out short-time fluctuations or highlight longer-term trends or cycles. In this study the moving average is set equally to a 5 day trading week. The SMA forecast of volatility is given by equation 7:
SMA
∑
= − = n i i t t n 1 1 ˆ σ σ (7)The EWMA method is characterised by the fact that this model allows one to calculate a value based on the previous day’s value. The EWMA model has an advantage over the RW and the SMA methods, because EWMA uses a fraction of its past value by factor
α
that makes the EWMA a good indicator of the history of the price movement. The model uses the latest observations with the highest weights in the volatility estimate. The forecasted volatility using the EWMA method is given in equation 8:EWMA ˆ
(
1)
1 ;0 1 1 2 1+ − ≤ ≤ =∑
+ = − − α σ α ασ σ n i i t t t n (8)Here, σˆ is the forecasted volatility for period t and t σt−1 and σt−2 are estimated volatilities corresponding to t-1 and its previous period t-2.
α
is obtained from an out of sample periodEvaluation of volatility forecast performance
The performance evaluation of the four volatility estimators will be done using a standard range of statistical methods. As mentioned in the first part of this section other academic studies used these methods extensively in the evaluation of the forecasting of certain predictors. The first two forecast error statistics depend on the scale of the dependent variable. These should be used as relative measures to compare forecasts for the same series across different models.
Root Mean Squared Error (RMSE)
The RMSE is a quadratic scoring rule which measures the difference between the forecast and actual observed volatility. These residuals are each squared and then averaged over the sample. Finally, the square root of the average is taken. Because all errors are squared before averaged, this model puts a relatively high weight on large error. Therefore, RMSE is most useful when large errors are discarded. The RMSE model is given by equation 9:
(
)
2 1 1∑
+ + = − = n t t a f n RMSE τ σ σ (9)Where σf andσa are respectively the forecasted and the actual daily volatility.
Mean Absolute Error (MAE)
The MAE is used for measuring the average difference between the forecast and the actual observed volatility, without taking their direction into consideration. The MAE is a linear score which means that all the individual differences are weighted equally in the average. Hence, RMSE and MAE are quite similar evaluation statistics except for the fact that relative large errors have less impact in the MAE model. To avoid the problem of positive and negative errors cancelling each other out only absolute values are taken into account. The MAE model is given by equation 10:
Both mentioned models are negatively-oriented scores. Hence, lower values means more precision by the forecasting models. Nevertheless, the MAE score will always be equal or smaller to the RMSE, a larger difference between those models means that there is a changing variance in the individual errors.
Mean Absolute Percentage Error (MAPE)
A major drawback of the RMSE and the MAE methods is that the relative size of the residual compared to the actual volatility is not taken into account. The MAPE method overcomes this problem by dividing the absolute residuals by the actual volatility before summing the absolute residuals. In this way, the MAPE takes into account the asymmetries in the residuals. The resulting measure can be interpreted as a percentage error, so that the results can be compared with the performance of other forecasting models independently of the scale of the different models. The MAPE model is given by equation 11, where the symbols are as above.
∑
+ + = − = n t t a a f n MAPE 1 1 τσ
σ
σ
(11)In the next section, empirical results, the obtained values from the abovementioned equations will be tested against a volatility index to analyze whether historical or extreme-value based volatilities are an accurate alternative to implied volatility. If none of the volatility estimators are excellent volatility forecasters compared to the implied volatility than the economical impact of the best performing volatility forecaster is analyzed.
V Empirical results
Tables 4 and 5 show the results obtained from the 4 different methods used for calculating daily volatility.
Table 4:
Summary S&P 500 Index statistics of daily volatility measured by different estimators (2 January- 31 December 2008). HV-30 HV-60 P-30 P-60 GK-30 GK-60 RS-30 RS-60 Mean 2.14% 2.00% 1.74% 1.64% 1.55% 1.47% 2.38% 2.26% Median 1.45% 1.40% 1.29% 1.24% 1.15% 1.17% 1.70% 1.74% Maximum 5.07% 4.66% 4.11% 3.69% 3.79% 3.26% 5.69% 5.08% Minimum 0.81% 0.99% 0.76% 0.83% 0.72% 0.76% 1.01% 1.12% Std. Dev. 0.0136 0.0116 0.0107 0.0091 0.0094 0.008 0.0149 0.0128 Skewness 1.1455 1.3202 1.1819 1.3257 1.2125 1.3308 1.1768 1.286 Kurtosis 2.6098 3.0657 2.7574 3.1086 2.8914 3.1494 2.7933 2.9828 Observations 253 253 253 253 253 253 253 253 Notes:
• HV, P, GK, RS are the volatilities based on the estimators of Historical Volatility, Parkinson, Garman and Klass and Rogers and Satchell, respectively.
• All volatilities are based on a moving-window of 30 or 60 days
Table 5:
Summary DAX statistics of daily volatility measured by different estimators (2 January- 30 December 2008) HV-30 HV-60 P-30 P-60 GK-30 GK-60 RS-30 RS-60 Mean 2.02% 1.90% 1.69% 1.58% 1.66% 1.56% 1.65% 1.55% Median 1.59% 1.55% 1.27% 1.28% 1.30% 1.29% 1.32% 1.28% Maximum 4.74% 4.11% 3.84% 3.31% 3.72% 3.22% 3.73% 3.21% Minimum 0.82% 0.89% 0.80% 0.74% 0.80% 0.74% 0.79% 0.75% Std. Dev. 0.0116 0.0095 0.009 0.0073 0.0085 0.007 0.0085 0.007 Skewness 1.1123 1.2094 1.2329 1.3252 1.2498 1.3518 1.3015 1.3767 Kurtosis 2.8713 3.1389 3.1438 3.4394 3.155 3.4643 3.2878 3.4681 Observations 254 254 254 254 254 254 254 254 Notes:
• HV, P, GK, RS are the volatilities based on the estimators of Historical Volatility, Parkinson, Garman and Klass and Rogers and Satchell, respectively.
The overall daily mean volatility for each model at least 1.5%. This relative extreme average volatility is of course the result of the examined credit crunch period. All volatilities based on an average of 60 trading days are intuitively lower than those obtained from a trading range of 30 days. To be precise each average volatility is around 0.10% lower than the 30 days volatility average. This can also be explained by a smoothing effect when estimating volatility over a longer period of time. Calculating historical volatility based on 30 and 60 days gives the highest daily volatility for the DAX index. This could indicate that volatility based on closing prices alone might overshoot the actual volatility. However, it is more likely that volatility derived from the Parkinson model would exceed the realized volatility, because this model takes the daily high and low price into account. Therefore, overshooting the volatility by the Model is in a period of extreme market movements not very unusual. Hence, the P-Model produces after the historical based volatility the highest daily volatility. Another interesting observation is that the volatility estimates obtained from all EV models for the DAX index demonstrate close related values. However, the mean and median volatility derived from the RS-model for the S&P500 index is severely larger than for the other used models.
Table 6 displays the performance of the various volatility estimation models using different forecasting and evaluation methods.
Table 6:
Estimation Performance of Various Estimators.
HV-30 HV-60 P-30 P-60 GK-30 GK-60 RS-30 RS-60 Panel: S&P 500 RMSE RW Mean 0.0026% 0.0030% 0.0021% 0.0043% 0.0037% 0.0063% 0.0043% 0.0039%
Std. dev. 0.0003% 0.0004% 0.0002% 0.0005% 0.0003% 0.0007% 0.0006% 0.0006% T 8.64 7.52 11.98 8.80 11.77 9.26 7.26 6.29 SMA Mean 0.0028% 0.0033% 0.0025% 0.0048% 0.0042% 0.0069% 0.0045% 0.0042% Std. dev. 0.0003% 0.0004% 0.0002% 0.0006% 0.0004% 0.0008% 0.0006% 0.0006% t 8.73 7.71 10.26 8.55 10.16 8.93 7.49 6.53 MAE RW Mean 0.3732% 0.3825% 0.3829% 0.4980% 0.5154% 0.6246% 0.4193% 0.3955% Std. dev. 0.0216% 0.0247% 0.0157% 0.0268% 0.0207% 0.0309% 0.0317% 0.0304% t 17.25 15.47 24.34 18.60 24.88 20.24 13.22 13.01 SMA Mean 0.3891% 0.4051% 0.4057% 0.5212% 0.5313% 0.6439% 0.4429% 0.4196% Std. dev. 0.0228% 0.0259% 0.0187% 0.0291% 0.0235% 0.0332% 0.0319% 0.0312% t 17.05 15.63 21.74 17.93 22.60 19.39 13.88 13.44 MAPE RW Mean 17.3269% 17.4281% 20.5946% 23.1346% 27.0985% 29.2790% 17.3720% 17.8126% Std. dev. 0.7021% 0.7843% 0.7632% 0.8331% 0.8462% 0.8540% 0.8977% 1.0040% t 24.68 22.22 26.99 27.77 32.02 34.28 19.35 17.74 SMA Mean 18.0838% 18.3208% 21.2653% 23.8639% 27.5175% 29.7906% 18.7026% 18.8921% Std. dev. 0.7488% 0.8040% 0.8240% 0.8818% 0.9045% 0.9118% 0.9125% 1.0310% t 24.15 22.79 25.81 27.06 30.42 32.67 20.50 18.32 Notes:
• HV, P, GK, RS are the volatilities based on the estimators of Historical Volatility, Parkinson, Garman and Klass and Rogers and Satchell, respectively. • All volatilities are based on a moving-window of 30 or 60 days.
Continuation table 6:
HV-30 HV-60 P-30 P-60 GK-30 GK-60 RS-30 RS-60 Panel: DAX RMSE RW Mean 0.0019% 0.0035% 0.0030% 0.0056% 0.0031% 0.0058% 0.0031% 0.0058%
Std. dev. 0.0002% 0.0004% 0.0004% 0.0008% 0.0004% 0.0008% 0.0004% 0.0008% t 10.48 7.83 7.14 7.11 7.28 7.28 7.45 7.40 SMA Mean 0.0024% 0.0041% 0.0036% 0.0063% 0.0036% 0.0064% 0.0036% 0.0064% Std. dev. 0.0002% 0.0005% 0.0005% 0.0009% 0.0005% 0.0009% 0.0005% 0.0009% t 10.29 7.65 6.72 7.03 6.77 7.18 6.83 7.26 MAE RW Mean 0.3609% 0.4370% 0.4090% 0.5262% 0.4129% 0.5358% 0.4159% 0.5386% Std. dev. 0.0157% 0.0251% 0.0226% 0.0335% 0.0231% 0.0340% 0.0229% 0.0339% t 22.99 17.42 18.07 15.71 17.84 15.74 18.15 15.88 SMA Mean 0.4019% 0.4666% 0.4299% 0.5513% 0.4298% 0.5592% 0.4329% 0.5611% Std. dev. 0.0180% 0.0274% 0.0263% 0.0358% 0.0266% 0.0362% 0.0263% 0.0361% t 22.27 17.05 16.35 15.40 16.13 15.45 16.47 15.55 MAPE RW Mean 18.3526% 20.5068% 20.5178% 23.2090% 20.1511% 23.3420% 20.1396% 23.4523% Std. dev. 0.6634% 0.7916% 1.0000% 0.8876% 0.7595% 0.9005% 0.7567% 0.9116% t 27.66 25.91 27.06 26.15 26.53 25.92 26.62 25.73 SMA Mean 19.8331% 21.7552% 21.0175% 24.1176% 20.5635% 24.1948% 20.5879% 24.2619% Std. dev. 0.6798% 0.8446% 0.8480% 0.9404% 0.8526% 0.9499% 0.8441% 0.9604% t 29.17 25.76 24.79 25.65 24.12 25.47 24.39 25.26 Notes:
• HV, P, GK, RS are the volatilities based on the estimators of Historical Volatility, Parkinson, Garman and Klass and Rogers and Satchell, respectively. • All volatilities are based on a moving-window of 30 or 60 days.
Another obvious observation is that the RW forecasting method is more accurate than the SMA Logically, the RW method is more accurate, because the EWMA estimator is 1 which implies the EWMA creates the most accurate forecast if all weight is put on the latest observation. A closer look at the MAPE tells us that the average forecasting error for the RW is between 20 and 24%. This number indicates that on average the forecasted volatility will over/ under shoot the future realized by at least 20%.
When there is no implied volatility available and volatility has to be estimated among one of these volatility estimation models the standard historical close-to-close volatility will outperform each of the EV volatility model on accuracy. However, the t value close to 29 tells us that it is highly doubtful that historical volatility can predict future realized volatility in the same way that implied volatility does.
Unless this significant deviation from the implied volatility, is historical volatility in any sense useful for estimating and forecasting future volatility when implied volatility is not available?
After deriving option prices from a simple option valuation method it is very unlikely that with 18.31% difference between the realized and the forecasted daily volatility you can price an option with one of the alternative volatility measures besides implied volatility. The economic impact of a 18.31% volatility difference in the forecasted volatility and the realized volatility will result in a price difference of ± € 0.25 based on an theoretical option price of € 1.18. These economical cost pin out that it is difficult to estimate volatility when implied volatility is not in hand. Appendix 2 explains how the values related to the economic impact are obtained.
VI Conclusion and further recommendations
The main objective of this study is to determine whether volatility forecasts based on historical or extreme-value volatility are an accurate alternative compared to implied volatility. The most accurate forecasting model can than be used for estimation volatility when implied volatility is not available in the market. Furthermore, if none of the abovementioned estimations model is able to forecast future realized volatility accurate than the economical impact is analyzed when using an alternative volatility estimation method
Several unexpected results emerged from this analysis. None of the extreme-value based volatility estimators could accurate forecast the future realized volatility. Even more remarkable, the EV based volatility estimators should at least outperform the volatility forecast based on historical close-to-close volatility according to most previous studies, which is definitely not the case for period that suffer from severe market movements. The results, within this stressful used time period, are appealing because in times of a strong bearish or bullish market sentiment, these extreme-value based volatility estimators should capture information from those observed high and low prices. Hence, those EV estimators should perform better than the historical based volatility. Especially, the RS-model was likely to perform well in these market sentiments, because this was the only estimation model which is suitable for these non- zero drift movements in the market. In contrast to previous research, see for instance the Becker (1983) and Wiggins (1991) Vipul and Jaco (2007) among others, the standard historical based volatility forecasts was performing with the lowest bias. However, this bias was still significantly different to the realized implied volatility. Also, the economical impact when using an inappropriate volatility estimation models during times of severe financial distress is enormous.
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Appendix 1 - Estiamtion EWMA-
α
First, the most efficient EWMA-
α
is obtained after minimizing the MAE. The out-of-sample obtained EWMA-α
estimators are presented in table 7:Table 7:
Out-of-sample EWMA-
α
estimations for the used indicesHV-30 HV-60 P-30 P-60 GK-30 GK-60 RS-30 RS-60 S&P500 EWMA RW 1 1 1 1 1 1 1 1 0.22% 0.24% 0.35% 0.37% 0.43% 0.45% 0.17% 0.18% SMA 1 1 1 1 1 1 1 1 0.24% 0.25% 0.35% 0.38% 0.44% 0.46% 0.19% 0.19% DAX EWMA RW 1 1 1 1 1 1 1 1 0.30% 0.29% 0.47% 0.47% 0.48% 0.49% 0.49% 0.50% SMA 1 1 1 1 1 1 1 1 0.31% 0.29% 0.47% 0.47% 0.49% 0.49% 0.50% 0.50% Notes:
• HV, P, GK, RS are the volatilities based on the estimators of Historical Volatility, Parkinson, Garman and Klass and Rogers and Satchell, respectively.
• All volatilities are based on a moving-window of 30 or 60 days. • Numbers in italic are the corresponding MAE minimizations.
Figure 3: Overview development daily volatility for the S&P500, over the year 2007 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 0 1 /0 3 /2 0 0 7 0 1 /1 7 /2 0 0 7 0 1 /3 1 /2 0 0 7 0 2 /1 4 /2 0 0 7 0 2 /2 8 /2 0 0 7 0 3 /1 4 /2 0 0 7 0 3 /2 8 /2 0 0 7 0 4 /1 1 /2 0 0 7 0 4 /2 5 /2 0 0 7 0 5 /0 9 /2 0 0 7 0 5 /2 3 /2 0 0 7 0 6 /0 6 /2 0 0 7 0 6 /2 0 /2 0 0 7 0 7 /0 4 /2 0 0 7 0 7 /1 8 /2 0 0 7 0 8 /0 1 /2 0 0 7 0 8 /1 5 /2 0 0 7 0 8 /2 9 /2 0 0 7 0 9 /1 2 /2 0 0 7 0 9 /2 6 /2 0 0 7 1 0 /1 0 /2 0 0 7 1 0 /2 4 /2 0 0 7 1 1 /0 7 /2 0 0 7 1 1 /2 1 /2 0 0 7 1 2 /0 5 /2 0 0 7 1 2 /1 9 /2 0 0 7 Date VIX HV-30 HV-60 P-30 P-60 GK-30 GK-60 RS-30 RS-60 Notes:
• HV, P, GK, RS are the volatilities based on the estimators of Historical Volatility, Parkinson, Garman and Klass and Rogers and Satchell, respectively.
• All volatilities are based on a moving-window of 30 or 60 days.
Figure 4: Overview devolopment daily volatility for the DAX, over the year 2007
0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 0 1 /0 2 /2 0 0 7 0 1 /1 6 /2 0 0 7 0 1 /3 0 /2 0 0 7 0 2 /1 3 /2 0 0 7 0 2 /2 7 /2 0 0 7 0 3 /1 3 /2 0 0 7 0 3 /2 7 /2 0 0 7 0 4 /1 0 /2 0 0 7 0 4 /2 4 /2 0 0 7 0 5 /0 8 /2 0 0 7 0 5 /2 2 /2 0 0 7 0 6 /0 5 /2 0 0 7 0 6 /1 9 /2 0 0 7 0 7 /0 3 /2 0 0 7 0 7 /1 7 /2 0 0 7 0 7 /3 1 /2 0 0 7 0 8 /1 4 /2 0 0 7 0 8 /2 8 /2 0 0 7 0 9 /1 1 /2 0 0 7 0 9 /2 5 /2 0 0 7 1 0 /0 9 /2 0 0 7 1 0 /2 3 /2 0 0 7 1 1 /0 6 /2 0 0 7 1 1 /2 0 /2 0 0 7 1 2 /0 4 /2 0 0 7 1 2 /1 8 /2 0 0 7 Date VDAX HV-30 HV-60 P-30 P-60 GK-30 GK-60 RS-30 RS-60 Notes:
• HV, P, GK, RS are the volatilities based on the estimators of Historical Volatility, Parkinson, Garman and Klass and Rogers and Satchell, respectively.
Appendix 2 – Economic impact
Standard option valuation formula for call option is given by equation 12:
( )
1( )
2) 0N d Ke N d S c= − −rT (12) Where: T T r K S d σ σ + + = 2 ln 2 0 1 and d2 =d1−σ TCurrent asset price, S0= € 45
Strike price, K = € 50
Annualized volatility,
σ
= 25% Annualized risk free rate, r = 3%Annualized time to maturity, T = 0.25 year
Using this example the call option is worth €1.18.
The Vega of an asset of derivatives, is the rate of change of value of the derivate with respect to the volatility of the underlying asset. Vega is given by the next equation 13:
( )
1 ' 0T
N
d
S
call=
×
ν
(13)Inserting the abovementioned parameters a 1% point volatility increase will results in an option price increase of € 0.054. In combination with a MAPE of 18.31% which is equal to a price increase/ decrease of:
247 . 0 € 054 . 0 € 25 % 31 . 18 × × =
