From volatility forecasts to option pricing : a comparison in economic value

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University of Amsterdam

Master Thesis

From Volatility Forecasts to Option Pricing:

A comparison in economic value

Author:

Pieter van Scherpenzeel

10673032

Supervisor:

Patrick Tuijp

Abstract

In this thesis we predict daily volatility with the use of high-frequency data of the S&P 500 index. The extended versions of the HAR-RV model of Corsi (2003) are used to model the realized volatility. We put focus on the prediction accuracy and the economic value of the models. To compare the economic value of the models, we create two hypothetical option markets, an option market with anonymous trading and an option market with a single market price, where traders make profits from trading straddles. We will show that the best performing model statistically, is also the most profitable model.

Keywords. Realized volatility, bipower variation, high-frequency data, forecasting volatil-ity, option pricing.

JEL classification: C58, G13, G17.

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1 Introduction

This paper focuses on volatility forecasting and option pricing. The models that we use are related to theoretical work on volatility forecasting of Corsi (2003) and Andersen, Boller-slev, and Diebold (2007). Several papers (e.g. Andersen, BollerBoller-slev, Diebold, and Labys (2003) and Koopman, Jungbacker, and Hol (2005)) have examined the prediction accuracy of autoregressive forecasting models for realized volatility. In contrast to these papers, we focus on the economic value of the volatility forecasting models. We set up two hypothetical option markets, one with anonymous trading and one with trading with a single market price, where options based on volatility forecasts are traded.

This thesis consists of two parts. First, the HAR-RV type of models of Andersen, Bollerslev, and Diebold (2007) are estimated and compared in predictive accuracy. With the obtained volatility forecasts of the different models, we then price options based on the Black-Scholes formula. Second, we compare the HAR-RV type of models in profitability and add more volatility forecast approaches to compare them in profitability. To do so, a hypothetical option market is set up where options are being traded. We create a market where each trader has its own volatility forecast model which he uses to price his option. Then, agents execute trades pair-wise.

Volatility forecasts are of paramount importance for many investment decisions. Many financial applications such as option pricing, asset allocation and risk management make use of volatility forecasts. Since the real volatility is unobservable, there is a vast literature relating the prediction of volatility. There are three types of volatility: realized volatility, rel-ative volatility and implied volatility. Realized volatility or historical volatility is estimated volatility that is based on past price movements. The relative volatility measures how a stock moves with respect to the whole market (for example how the Apple stock moves with respect to the S&P 500 index). The implied volatility cannot be computed from historical prices (as the realized volatility) but it is the market’s expectation of the future volatility of the stock price.

In early days the dominant approaches to model the volatility were the autoregressive condi-tional heteroskedasticity (ARCH) model of Engle (1982) and the generalized autoregressive conditional heteroskedasticity (GARCH) model of Bollerslev (1986). In recent years, the increase in availability of high-frequency data led to the development of realized volatility by

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Andersen, Bollerslev, Diebold, and Ebens (2001) and improved volatility measures at a daily frequency. Andersen, Bollerslev, Diebold, and Ebens (2001) defined realized volatility as the sum of all available intraday squared returns. Results of Andersen, Bollerslev, Diebold, and Labys (2003) suggest that simple autoregressive forecasting models for realized volatility outperform the GARCH model and related stochastic volatility models.

With the availability of high-frequency data and the results of realized volatility, forecasting models are constructed based on realized volatility (such as ARFIMA-RV by Barndorff-Nielsen (2002a)). Corsi (2003) proposes another model based on realized volatility: the Heterogenous Autoregressive model of Realized Volatility (HAR-RV). The results of Corsi (2003) show that the relatively easy to estimate HAR-RV model has comparable results with the ARFIMA model and outperforms the shorter-memory models.

Since the introduction of the HAR-RV model, many extensions of this model are introduced. One of these extensions is the HAR-RV-J model of Andersen, Bollerslev, and Diebold (2007), where they incorporate jumps to estimate the realized volatility. Another extensions is the HAR-RV-CJ model of Andersen, Bollerslev, and Diebold (2007), where they describe the realized volatility process by a combination of a ‘smooth and very slowly mean-reverting continuous sample path process and a much less persistent jump component’. By applying the related bipower variation measures and adopting the jump test of Barndorff-Nielsen, Graversen, and Shephard (2004), we obtain the HAR-RV-CJ model.

After estimating the HAR-RV, HAR-RV-J and HAR-RV-CJ models, we use the model of Bandi, Russell, and Yang (2008) to look at the relative option pricing performance of these models. with pricing options, it is possible to use a dollar metric in comparing volatility forecasts. Based on the methodology proposed by Engle, Hong, and Kane (1990), Bandi, Russell, and Yang (2008) propose a hypothetical option market in the following way. Each agent prices his option at the beginning of the day based on the Black-Scholes model. When there is one trader with an option price higher than the mid-point and one trader with an option price lower than the mid-point, the trader with the high option price will buy a call and a put option from the trader with the low option price. By determining the total profit of all the trades, we determine which (realized) volatility model is the most profitable for the option market. This approach is extended in the following way: all traders know all prices and with this information a single market price is determined. Now, traders have to decide

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whether to trade or not.

We use high-frequency data of the S&P 500 index over a 14 year horizon to predict volatil-ity. We estimate the HAR-RV, HAR-RV-J and HAR-RV-CJ models using Ordinary Least Squares (OLS). We find that in volatility forecast performance, determined by the root mean squared error (RMSE), the mean absolute error (MAE) and the Diebold-Mariano test statis-tic, the extended HAR-RV models outperform the HAR-RV model of Corsi (2003). In other words, the HAR-RV-J and HAR-RV-CJ models achieve an increase in prediction accuracy.

We also compute an option price-based dollar metric to determine the economic signifi-cance of performance differences. To transform volatility forecasts into option prices, we use the Black-Scholes option pricing framework. We find that the HAR-RV-CJ model is the most profitable of the three HAR-RV type models. We find that if we make a pair-wise comparison of the models of Andersen, Bollerslev, and Diebold (2007) that the HAR-RV-CJ model is the most profitable, so it pays to combine a continuous sample path and a jump process in the model. Further, it differs which horizon and transformed or not-transformed whether the HAR-RV model or the HAR-RV-J model is more profitable.

Another important implication of the pricing options based on volatility forecasts is that HAR-RV type models outperform the GARCH model and the only extended HAR-RV mod-els outperform the ARMA model. These results deviate from the prediction accuracy results of Andersen, Bollerslev, Diebold, and Labys (2003) and Koopman, Jungbacker, and Hol (2005) in that the HAR-RV models outperform the GARCH and ARMA models. Interest-ingly, the HAR-RV model does not outperform the ARMA model in economic value. Thus, presence of jumps in the model and the possibility of adding a continuous sample path com-ponent is crucial to become a more profitable trader.

This thesis is organized as follows. First, the related literature is presented in section 2 and then the high frequency data is described in section 3. In section 4 the empirical ap-proach is described. Section 5 presents the results which will be analyzed in section 6. Section 7 concludes.

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2 Related literature

This paper contributes to the existing literature on volatility forecasting and option pricing along several dimensions. First, the model that we use is related to the theoretical work on autoregressive (AR) type models in realized volatility forecasting. Corsi, Zumbach, Muller, and Dacorogna (2001) and Corsi (2003) suggest an additive cascade model of different volatil-ity components each generated by different market participants who take actions based on different time horizons of volatility. The volatility is parameterized as a linear function of lagged realized volatilities over different horizons. This leads to an AR-like model in realized volatilities. Starting with the work of Corsi et al. (2001, 2003), several papers extend the HAR-RV model.

One of the extensions is from Corsi and Reno (2009). They identify three endogenous components in the dynamics of market volatility: heterogeneity, leverage and jumps and find that each of the three components plays a significant role in forecasting volatility. The heterogeneity occurs because of the different time horizons. The leverage effect comes from the fact that volatility tends to increase more after a negative shock than after a positive shock of the same size.

The extension we use, is introduced by Andersen, Bollerslev, and Diebold (2007). They try to advance the realized volatility approach by implementing a separate measurement of the continuous sample variation and the jump part of the quadratic variation process. The approach of Andersen, Bollerslev, and Diebold (2007) builds on new theoretical results of Barndorff-Nielsen, Graversen, and Shephard (2004) and Barndorff-Nielsen and Shephard (2006) who define bipower variation. Barndoff-Nielsen, Shephard et al. (2004, 2006) con-struct bipower variation by summing cross-products of adjacent absolute returns. With the new technique, Andersen, Bollerslev, and Diebold (2007) make a simple and practical frame-work for measuring jumps in asset prices.

Second, our empirical results contribute to a rich literature that has empirically studied the prediction accuracy of realized volatility models and GARCH and ARMA-type models. Andersen, Bollerslev, Diebold, and Labys (2003), Koopman, Jungbacker, and Hol (2005) and Corsi (2003) find that realized volatility forecasts are more accurate than forecasts based on GARCH, ARMA-type models and related volatility models. In contrast to these papers, we focus on the economic value of the realized volatility models and GARCH and ARMA-type

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models in addition to the statistical forecast accuracy.

To study the differences in forecast accuracy, we compute the root mean squared error, the mean absolute error, and we use the the test introduced by Diebold and Mariano (2002). Diebold and Mariano (2002) propose a test with a null hypothesis of no difference in the accuracy of two forecasts. With loss associated to the forecast errors, we determine whether we reject the null hypothesis. In contrast to earlier tests, the loss function does not need to be quadratic or symmetric. Furthermore, forecast errors do not need to be Gaussian, may have a nonzero mean and serially correlated. We use this test statistic to compare the HAR-RV, HAR-RV-J and HAR-RV-CJ models and to determine which of the models performs best in forecasting accuracy.

Finally, to obtain option prices from volatility forecasts, we use the approach of Engle, Hong, and Kane (1990). In this paper Engle, Hong, and Kane (1990) propose a method to compare the profits and losses of option traders using different volatility forecasts. Each agent has its own method of pricing options (in this thesis, it means for example that agent 1 prices options with the HAR-RV model, agent 2 with the HAR-RV-J model and agent 3 with the HAR-RV-CJ model). Because each agent uses his own forecast, options are priced differently. These price differences lead to trades at the mid-point of the agents’ prices. To do so, we design a hypothetical option market and by trading straddles agents make profits and losses.

To design a hypothetical option market, we make use of the market proposed by Engle, Hong, and Kane (1990), where options are trade anonymously. Bandi, Russell, and Yang (2008) apply this method for high-frequency data. Since this approach has the consequence that traders with forecasts close to the median have the potential of making profits by mar-ket making, Bandi, Russell, and Yang (2008) propose another hypothetical option marmar-ket. This second type of hypothetical option market is an option market with a single market price. This setting is advantageous in two ways. First, traders with forecasts close to the median do not have the potential of market making anymore. Second, assuming a single market price is economically a more realistic argument. In contrast to the paper of Bandi, Russell, and Yang (2008) , we implement this second type of hypothetical option market and compare the results with the option market of Engle, Hong, and Kane (1990).

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3 Data

The asset we use, is the SPY of Bloomberg, an exchange traded fund that tracks the (SPDR) S&P 500 index. The S&P 500 is an American stock market index which gives the most reli-able image of the latest developments of the stock exchange. The S&P 500 index is composed of 500 selected stocks exchanges and spans over 25 separate industry groups.

The high-frequency dataset has trade information about all S&P 500 trades at a 1-second frequency. With 1-second frequency, it is possible that there were multiple trades during the same second. To overcome this problem, we take the median of the trades during the same second. To provide the 5-minute return data, as in Andersen, Bollerslev, and Diebold (2007), we take every 300th _return.

The S&P 500 high-frequency data is from the Trade and Quote database on Wharton Re-search Data Service. The data is from the 1st of September 1999 through the 31st of July 2013. Table 1 presents the summary statistics the 5-minute returns of the dataset. In Table

Mean Maximum Minimum Std.dev. Skewness Kurtosis

S&P 500* −0.000002 0.048974 -0.028838 0.001231 0.494316 31.302171

*_{The S&P 500 data comes from TAQ and has 273,000 observations.} Table 1: Data summary

2 we see the summary statistics for the the daily realized volatility RVt, the square root of the daily realized volatility RV_t1/2, its log, the jump component Jtand the square root of the jump component J_t1/2. On the next pages, Figure 1 presents the returns: log pt/pt−1, Fig-ure 2 the daily realized volatility RVt, Figure 3 the square root of the daily realized volatility RV_t1/2 and finally Figure 4 presents the square root of the jump component J_t1/2.

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RVt RV 1/2 t log(RVt) Jt J 1/2 t Mean 0.00012 0.00916 -4.20261 0.00001 0.00194 Standard deviation 0.00024 0.00024 0.44826 0.00001 0.00221 Minimum 0.00002 0.00173 -5.52495 0 0 Maximum 0.00600 0.07743 -2.22215 0.00104 0.03224 Skewness 9.78460 3.08033 0.47029 17.87616 3.14819 Kurtosis 155.71819 17.05477 3.31527 469.25719 23.10456

We define the realized volatility as: RV_t(d)=PM −1

j=0 r 2 t−j·∆

We define the jump component as: Jt= max [RVt− BVt, 0],

with BV_t(d)= (p2/π)−2PM −1

j=1 |rt−j·∆||rt−(j−1)·∆|, the standardized bipower variation.

Table 2: Summary statistics

Figure 1: 5-minute returns

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Figure 3: The standard deviation of the daily realized volatility

Figure 4: The standard deviation of the jump component

In Figure 1 we see the flash crash of the 6th of May 2010. On this day, the second largest intraday point swing and the largest one-day point decline occurred. In forty minutes time, the S&P 500 index dropped 3% and regained this 3% again.1

In Figure 3, we see that the square root of the daily realized volatility is relatively large. This relatively large square root reduces the forecast accuracy which is of relevance for the Diebold-Mariano test results and later on for the economic comparison of the models.

From the four Figures we see that the largest peaks in the daily realized volatilities (Fig-ure 2 and Fig(Fig-ure 3) are associated with the largest jumps (Fig(Fig-ure 4). In general, the realized

1_{The flash crash or the Crash of 2:45 was due to a large trade fund, which was aggresively selling future}

contracts. Because of the futures algorithms of smaller high-frequency trading firms, everyone in the market was selling their trades and this process was continuing for three minutes. To stop this continuous fall trading was paused for a short period of time. After this pause, trading stabilized and half an hour later markets were at the level of about an hour before.

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volatilities and jumps are higher in the latter half of the sample.

Finally, the largest realized volatility and one of the largest jumps was on October the 10th _{2008. This day is known as the day with the largest intraday point swings ever. In the} period around this day, the peak of the crisis, 14 of the top 20 largest intraday point swings occurred. Furthermore, it seems that there is serial correlation. To check this thought a Ljung-Box test and a Lagrange multiplier (LM) test (see the appendix 1 and appendix 2 for more information) are performed. This test gives the following result:

RVt RV 1/2 t log(RVt) Jt J 1/2 t LB5 6,090 10,107 11,186 100 256 LB10 10,318 18,192 20,352 377 654 LB15 13,883 25,256 28,517 479 905 LM5 57.9 262.4 1,483.3 23.3 77.5 LM10 75.1 294.5 1,519.6 41.0 126.5 LM15 82.8 302.4 1,540.7 45.6 135.3

All the p-values of the statistics are zero.

Table 3: Results of Ljung-Box and LM-test

From the results of the Ljung-Box test and the Lagrange-multiplier test, we see that the idea of serial correlation is confirmed. The p-values of the Ljung-Box test are all zero, so we always reject the null hypothesis and thus we reject that the data is not autocorrelated. Since the p-values of the Lagrange multiplier test are all zero too, so again we reject the null hypothesis and thus we reject that there is no serial correlation up to Q lags.

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4 Empirical approach

4.1 HAR-RV Models

Corsi (2003) proposes a new approach to model and forecast volatility. He uses that different market participants analyze past events and news with different time horizons, as proposed in the Heterogenous Market Hypothesis of Müller, Dacorogna, Davé, Pictet, Olsen, and Ward (1993). On the basis of this information, the market participants make their investment de-cision. With the Heterogenous Market Hypothesis, Müller, Dacorogna, Davé, Pictet, Olsen, and Ward (1993) try to explain the positive relationship between volatility and market par-ticipation.

Heterogeneity occurs from various characteristics such as differences in agents’ endowments, available information, prior beliefs, geographical location and risk profiles dissimilarity in processing information. Here, the heterogeneity occurs because of the difference in time horizons since each investor trades with its own frequency. There are the fast traders who trade daily or even more frequently (such as speculative traders), the medium fast traders who trade on a weekly basis (such as pension funds) and finally the slower traders who trade monthly (such as a central bank).

The HAR-RV model proposed by Corsi (2003) is an additive cascade of partial volatili-ties. Its roots lie in the HARCH model of Müller, Dacorogna, Davé, Olsen, Pictet, and von Weizsäcker (1997) and Dacorogna, Müller, Pictet, and Olsen (1997). The additive cascade of partial volatilities lead to an AR-like model in realized volatilities. This is where the term Heterogenous Autoregressive model of Realized Volatilities (HAR-RV) comes from.

4.1.1 HAR-RV model

In this paragraph, we explain the mathematical background of the HAR-RV model. The standard HAR-RV model considers the following continuous time process:

dp(t) = µ(t)dt + σ(t)dW (t), (1)

where p(t) is the logarithm of the asset price, µ(t) represents the time-varying drift com-ponent of the stock, dW (t) is a standard Brownian motion and σ(t) is a stochastic process independent of dW (t). For this process, the integrated volatility for day t is the integral of

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stochastic volatility process over the one-day interval [t − 1d, t]:

IV_t(d) = Z t

t−1d

σ2(ω)dω. (2)

As Andersen, Bollerslev, Diebold, and Labys (2003) show, the integrated variance of a Brow-nian motion is approximated by the realized volatility. The definition of the realized volatility over one day is:

RV_t(d) = M −1

X j=0

r2_t−j·∆, (3)

where ∆ = 1d/M and the intra-day returns at time interval ∆ are denoted by rt−j·∆ = p(t − j · ∆) − p(t − (j + 1) · ∆). When considering the realized volatility for a period longer than one day, we take the normalized sum of one-day realized volatilities. For example, the realized volatility for one week and one month is respectively:

RV_t(w) = 1 5 RV_t(d)+ RV_t−1d(d) + RV_t−2d(d) + RV_t−3d(d) + RV_t−4d(d) , RV_t(m) = 1 22 RV_t(d) + RV_t−1d(d) + RV_t−2d(d) + . . . + RV_t−21d(d) .

Then, as shown by Andersen, Bollerslev, Diebold, and Labys (2003), Andersen, Boller-slev, Diebold, and Ebens (2001), Barndorff-Nielsen (2002a), Barndorff-Nielsen and Shephard (2002b) and Comte and Renault (1998) it follows that in the absence of jumps:

RVt−→ Z t

t−1d

σ2(ω)dω,

so we see that in the absence of jump, the realized variation is a consistent estimator for the integrated volatility.

Now, we turn to the partial volatility. Every investor creates their own volatility com-ponent. We define the unobserved partial volatility as ˜σ_t(d), ˜σ(w)_t and ˜σ(m)_t for one day, one week and one month respectively. All of the partial volatilities have an AR(1)-like structure. We will see that the values are not lagged but rather the corresponding realized volatilities. We assume that the high-frequency return process is determined by the component with the highest frequency (one day) with ˜σ_t(d) = σ_t(d). Then, we define the return process as follows:

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where εt∼ N (0, 1).

The three unobservable volatility components are a function of the realized volatility at the same time horizon (AR(1) part) and the expectation of the long-term partial volatili-ties. For the monthly partial volatility, only the AR(1) part remains. This is because we assume that a high frequency trader (when determining their volatility expectation) will be influenced by lower frequency traders. This gives the following models:

˜ σ(d)_t+1d = c(d)+ φ(d)RV_t(d)+ γ(d)_Et[˜σ (w) t+1w] + ˜ω (d) t+1d, (5) ˜ σ_t+1w(w) = c(w)+ φ(w)RV_t(w)+ γ(w)_Et[˜σ (m) t+1m] + ˜ω (w) t+1w, (6) ˜ σ_t+1m(m) = c(m)+ φ(m)RV_t(m)+ ˜ω_t+1m(m) , (7)

where the RV’s are as described before and ˜ω_t+1j(j) (j = d, w, m) are the volatility innovations with a truncated left tail to ensure the positiveness of the volatilities.

With ˜σ(d)_t = σ_t(d)and with recursive substitutions it follows that equation (5) can be rewritten to:

σ_t+1d(d) = c + β(d)RV_t(d)+ β(w)RV_t(w)+ β(m)RV_t(m)+ ˜ω(d)_t+1d. (8)

In terms of realized volatilities, we assume that the daily volatility σ_t+1d(d) is written as the daily realized volatility plus an error, which occurs from data properties, such as microstructure effects:

σ_t+1d(d) = RV_t+1d(d) + ω(d)_t+1d. (9)

When we substitute equation (9) into equation (8) and let measurement errors be absorbed by the error term of the regression, we obtain the following equation:

RV_t+1d(d) = c + β(d)RV_t(d)+ β(w)RV_t(w)+ β(m)RV_t(m)+ ˜ω(d)_t+1d− ω(d)_t+1d2_. ₍₁₀₎ This equation has the characteristics of an autoregressive model but with realized volatilities. Because of the three different volatility components, equation (9) is the HAR(3)-RV-model but for simplicity we will call it the HAR-RV model.

2_{from now on, we will write this equation as: RV}

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Apart from the linear version, we also use two types of transformations of the HAR-RV model: a square-root transformation and a log transformation. We define these transforma-tions as:

(RVt+h)1/2 = β0+ βD(RVt)1/2+ βW(RVt−5,t)1/2+ βM(RVt−22,t)1/2+ εt+1,

log(RVt+h) = β0+ βDlog(RVt) + βWlog(RVt−5,t) + βMlog(RVt−22,t) + εt+1.

We use the method of Corsi (2003) to apply these transformations. We first take the daily square-root or log transformation and then we take the average.

Finally, Corsi (2003) shows that the long-memory HAR(3)-RV model outperforms the shorter-memory models. Furthermore, it has comparable results with the complicated ARFIMA model.

Estimation method

To estimate the parameters β(d), β(w) and β(m) we apply linear regression. Since we know from Table 3 that there is presence of serial correlation in the data, we use the Newey-West covariance correction for serial correlation. By estimating the HAR-RV model with OLS, the obtained coefficients will put a larger focus on fitting the high variances. This means that periods with low volatility are underweighted.

4.1.2 HAR-RV-J model

The HAR-RV model in the previous section is built on the assumption that a price process exhibits continuous sample path. Andersen, Bollerslev, and Diebold (2007) suggest that “price processes are best described by a combination of a smooth and very slowly mean-reverting continuous sample path process and a much less persistent jump component”. In that case, the assumption for the HAR-RV model is no longer appropriate. Now, we will take jumps into account. With jumps the continuous time process looks as follows:

dp(t) = µ(t)dt + σ(t)dW (t) + κ(t)dq(t) t = 0, . . . , T, (11) where p(t) is the logarithm of the asset price, µ(t) is a locally bounded variation process, W (t) is a standard Brownian motion and σ(t) is a stochastic process independent of W (t)

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and allowing for jumps in volatility. The additional q(t) is a counting process with intensity λ(t) and κ(t) = p(t) − p(t−) is the size of the discrete jumps in the process. Then, the quadratic variation for the return process reads:

[r, r]t= Z t 0 σ2(s)ds + X 0<s≤t κ2(s). (12)

We denote the ∆-period returns as rt,∆ = p(t) − p(t − ∆) and with these returns we compute the realized volatility:

RVt+1(∆) = 1/∆ X j=1

r2_t+j∆,∆. (13)

Now, as suggested by Comte and Renault (1998) and Barndorff-Nielsen and Shephard (2002b) the realized variation converges to (by the theory of quadratic variation):

RVt+1(∆) −→ Z t+1 t σ2(s)ds + X t−1<ω≤t κ2(s). (14)

These two components in the equation above are best constructed by separately measuring and modeling (Andersen, Bollerslev, Diebold, and Labys (2003)). The separate identification starts with defining the standardized bipower variation as introduced by Barndorff-Nielsen, Graversen, and Shephard (2004):

BVt+1(∆) = ( p 2/π)−2 1/∆ X j=2 |rt+j·∆,∆||rt+(j−1)·∆,∆|. (15)

Then Barndorff-Nielsen, Graversen, and Shephard (2004) show that for ∆ → 0, it follows that:

BVt+1(∆) −→ Z t+1

t

σ2(s)ds. (16)

When we combine equation (14) and equation (16) it holds that: RVt+1(∆) − BVt+1(∆) −→

X t−1<ω≤t

κ2(ω). (17)

The estimates of the squared jumps (κ2_{) in equation (17) could become negative. To ensure} that all the estimates of the squared jumps are positive, we use a truncation at zero as

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proposed by Barndorff-Nielsen, Graversen, and Shephard (2004):

Jt = max [RVt+1(∆) − BVt+1(∆), 0] . (18)

With the separate measurement of the jump component, the HAR-RV-J model is obtained: RVt,t+h= β0+βDRVt+βWRVt−w,t+βMRVt−m,t+βJ DJt+βJ WJt−w,t+βJ MJt−m,t+εt,t+h. (19)

Apart from the linear version, we also use two types of transformations of the HAR-RV-J model: a square-root transformation and a log transformation. We define these transforma-tions as:

(RVt,t+h)1/2 = β0+ βD(RVt)1/2+ βW(RVt−5,t)1/2+ βM(RVt−22,t)1/2+ βJ D(Jt)1/2 + βJ W(Jt−5,t)1/2+ βJ M(Jt−22,t)1/2+ εt,t+h,

log(RVt,t+h) = β0+ βDlog(RVt) + βWlog(RVt−5,t) + βMlog(RVt−22,t) + βJ D log(Jt) + βJ W log(Jt−5,t) + βJ M log(Jt−22,t) + εt,t+h.

We use the method of Corsi (2003) to apply these transformations. We first take the daily square-root or log transformation and then we take the average.

4.1.3 HAR-RV-CJ model

This HAR-RV-J model is further extended. The extension decomposes a method for sepa-rating the effects of the continuous sample path and jump portions of volatility. To ensure that the estimates of the squared jumps are positive, we use equation (18) for the HAR-RV-J model. But as we have seen in Figure 4, there are a significant number of small jumps. As Andersen, Bollerslev, and Diebold (2007) suggest, it may be expedient to treat small jumps as measurement errors, associating only the larger values of RVt(∆) − BVt(∆) with the jump component in the model. We provide a framework to do so.

To split the process into a continuous and a discontinuous part, Andersen, Bollerslev, and Diebold (2007) make use of Z-statistics and compare it with a certain critical value. We define the Z-statistic, as introduced by Huang and Tauchen (2005), as:

Zt−1(∆) = ∆−1/2 [RVt−1(∆) − BVt−1(∆)]/RVt+1(∆) (µ−4 1 + 2µ −2 1 − 5) max{1, T Qt+1(∆)BVt+1(∆)−2} 1/2, (20)

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where T Q stands for tripower quarticity. Huang and Tauchen (2005) provide a test based on tripower quarticity to determine whether the jumps are significant. We define the tripower quarticity as follows: T Qt−1(∆) = 1 ∆µ −3 4/3 1/∆ X j=3 |rt+j·∆,∆|4/3|rt+(j−1)·∆,∆|4/3|rt+(j−2)·∆,∆|4/3, (21) where µ4/3 = 22/3· Γ(7/6) · Γ(1/2)−1 = E(|Z|4/3).

Now, with the realizations of the Z-statistic, we determine whether the jumps are signif-icant. If Zt+1(∆)’s value is higher than a critical value Φα, the significant jumps are:

Jt+1,α(∆) = I[Zt+1(∆) > Φα] · [RVt+1(∆) − BVt+1(∆)], (22)

and the continuous sample path variability:

Ct+1,α(∆) = I[Zt+1(∆) ≤ Φα] · RVt+1(∆) + I[Zt+1(∆) > Φα] · BVt+1(∆), (23)

where we use the indicator function I. Equation (22) and (23) ensure that the total sum of these equations is equal to the total realized variation.

By defining the normalized multi-period jump and continuous sample path variability mea-sures as: Jt,t+h = 1 h[Jt+1+ Jt+2+ . . . + Jt+h], and Ct,t+h= 1 h[Ct+1+ Ct+2+ . . . + Ct+h],

where the jumps and continuous sample path components are defined as in equations (22) and (23) respectively.

Then, we define the HAR-RV-CJ as:

RVt,t+h= β0+βCDCt+βCWCt−w,t+βCMCt−m,t+βJ DJt+βJ WJt−w,t+βJ MJt−m,t+εt,t+h, (24)

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Again, we use two types of transformations of the HAR-RV-CJ model: a square-root trans-formation and a log transtrans-formation. We define these transtrans-formations as:

(RVt,t+h)1/2 = β0+ βCD(Ct)1/2+ βCW(Ct−5,t)1/2+ βCM(Ct−22,t)1/2+ βJ D(Jt)1/2 + βJ W(Jt−5,t)1/2+ βJ M(Jt−22,t)1/2+ εt,t+h,

log(RVt,t+h) = β0+ βCDlog(Ct) + βCWlog(Ct−5,t) + βCMlog(Ct−22,t) + βJ D log(Jt) + βJ W log(Jt−5,t) + βJ M log(Jt−22,t) + εt,t+h.

We first take the daily square-root or log transformation and then we take the average.

Estimation method

To estimate the parameters of the HAR-RV-J model: βD, βW, βM and βJ OLS is used. Again, we use OLS when estimating the parameters of the HAR-RV-CJ model: βCD, βCW, βCM, βJ D, βJ W and βJ M.

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4.2 Prediction accuracy

The volatility forecast performance of all the HAR-RV type models will be compared using the root mean squared error (RMSE) and the mean absolute error (MAE) one-step ahead forecast errors. These forecast errors give a first impression about the relative accuracy but this comparison is not a formal test. To determine the best performing model, a goodness of fit test will be used. Here, the Diebold-Mariano test statistic will be used. Below a more detailed description of the Diebold-Mariano test is given.

4.2.1 RMSE and MAE

To determine the RMSE and the MAE the dataset is split in two parts: an estimation part to construct the model and a prediction part. Let {yt} be the actual value and {ˆyt} be the forecasted value for one or more periods. Then, the RMSE and the MAE are defined as follows: RMSE = 1 T T X t=1 (yt− ˆyt)2 !1/2 MAE = 1 T T X t=1 |yt− ˆyt|, where T is the number of observations.

4.2.2 Diebold-Mariano test statistic

Diebold and Mariano (2002) propose a formal test of relative accuracy. For this test, series need to be forecasted. Let {yt} be the target series and yt+h|t1 and y

2

t+h|t be the two different forecasts (e.g. forecasts could be computed from an AR-model, MA-model, etc.) of yt+h based on the information. Then, the forecast errors are defined as:

ε1_t+h|t= yt+h− y1t+h|t ε2_t+h|t = yt+h− yt+h|t2 , the h-step forecasts are determined for t = t0, . . . , T .

The accuracy of the T forecasts is measured by a loss function: L(εi_t+h|t), for i = 1, 2,

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where a choice has to be made which loss function to use. The loss function is a function that takes a value of zero when no error is made, is nonnegative and is increasing in size as the errors become larger. Here, we use the squared error loss.

The Diebold-Mariano test is based on the differences in loss: dt= L(ε1t+h|t) − L(ε

2 t+h|t)

To determine which of the (in this case) two models predicts better (or if the forecast accuracy is equal), the null hypothesis is tested against the alternative:

H0 : E[L(ε1t+h|t)] = E[L(ε 2 t+h|t)] ⇔ E[dt] = 0, H1 : E[L(ε1t+h|t)] 6= E[L(ε 2 t+h|t)] ⇔ E[dt] 6= 0. We define the Diebold-Mariano test statistic as follows:

S = ¯ d q ( [LRV_d¯/T ) , with ¯d = _T1 0 PT t=t0dt and LRVd¯ = γ0 + 2 P∞

j=1cov(dt, dt−j). As shown by Diebold and Mariano (1995) it holds that under the null hypothesis:

S∼ N (0, 1).a

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4.3 Option pricing

To determine the economic significance of performance differences, we compute an option-based dollar metric. With the realized volatility estimates of the HAR-RV, HAR-RV-J and HAR-RV-CJ models as stated in equations (10), (17) and (22) respectively, we look at the at the relative option pricing performance of these models. To transform volatility forecasts into option prices, we use the Black-Scholes option pricing framework.

The Black-Scholes option pricing framework implicitly assumes a constant stock price volatil-ity, which contradicts to our time-varying volatility forecasts. But since we create hypothet-ical option markets where - as input for the Black-Scholes option pricing formula with one day to maturity - agents forecast one day ahead and assume them to be correct and constant over time, we solve this problem.

The Black-Scholes equation is a partial differential equation to describe the option price. We define the equation as follows:

δf δt + rS δf δS + 1 2σ 2_S2δ2f δS2 = rf. (25)

One solution to the differential equation above is the Black-Scholes formula for the prices of European call options:

C(S, t) = SΦ(d1) − Ke−r(T −t)Φ(d2), (26)

and the price of the put option follows by put-call parity:

P (S, t) = Ke−r(T −t)Φ(−d2) − SΦ(−d1), (27) where d1 = ln(S/K) + (r + σ2_{/2)(T − t)} σ√T − t , d2 = d1− σ √ T − t,

with Φ(·) the cumulative normal distribution, T − t the time to maturity, S the spot price of the underlying asset, K the strike price, r the continuously compounded risk-free rate and σ is the volatility of the returns of the underlying asset. Since we assume St= 1, K = 1 and

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r = 0, the call price in the Black-Scholes formula becomes: C(S, t) = Φ(d1) − Φ(d2), (28) and thus: C(S, t) = P (S, t) = 2Φ 1 2σt − 1, (29)

since with put-call parity:

C(S, t) + Ke−r(T −t) = P (S, t) + S ⇐⇒ C(S, t) + 1 = P (S, t) + 1 ⇐⇒ C(S, t) = P (S, t).

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4.3.1 Option market with Anonymous Trades

We have to design a hypothetical option market as introduced by Engle, Hong, and Kane (1990). In this case there are three or six agents, each assigned with one of the three or six models. Knowing his model, the agent trades one-day options on a $1 share of the S&P 500 index. As proposed by Bandi, Russell, and Yang (2008), the exercise price of the options is $1 with a risk-free rate of zero. The latter is a reasonable assumption since the daily risk-free rate is negligible. The agents use the Black-Scholes model to price the call option as in equation (29) (so all the agents use the same model to price the volatility).

Now, Engle, Hong, and Kane (1990) propose trades between agents in the following manner: • Every agent determines his Black-Scholes option price for a one-day call/put option on a$1 dollar share with respect to the model he uses (agents trade anonymously and do not know what the prices of other agents are).

• The agents execute trades pair-wise. Trades are settles at the mid-point of the higher and the lower price. So for traders i and j the price is: Pi,j = Pi+P₂ j.

Since the price of an option is increasing in the underlying asset variance, agents with lower variance forecasts will perceive the options to be overpriced. As a result, the agent will sell one put and one call (a straddle). This means that when an agent thinks the options are overpriced, he believes that both options will end up out of the money due to the expected low volatility.

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For every trade, the agent uses its variance forecast to delta-hedge his position. The hedge ratio δ of the option is Φ 1₂σt. This means that an agent who buys a call goes short on Φ 1₂σt shares of the stock. And an agent who buys a put goes long on 1−Φ 1₂σt shares of the stock. This gives a hedge ratio for the straddle of 1−2Φ 1₂σt, since the straddle is a portfolio of a put and a call option. Then, the profit of an agent who buys a straddle is:

|RS&P500 t | − 2Pi,j,t+ RS&P500t 1 − 2Φ 1 2σt , (31)

and for the agent who sells the straddle:

2Pi,j,t− |RS&P500t | − R S&P500 t 1 − 2Φ 1 2σt , (32) where RS&P500

t is the return for day t of the S&P 500 index

• At the end of the trading day, the profits and losses for each agent are determined. This approach has by design the consequence that traders with forecasts close to the median have the potential of making profits by market making. Agents may sell to those with higher prices and buy from those with lower prices, although their forecasts are inaccurate.

4.3.2 Option market with a Single Market Price

As Bandi, Russell, and Yang (2008) propose, there is another setup of comparing the models in profitability. Our second market setting considers a single market price for all transactions. Now, all agents know all prices and consequently a single market price is constructed. We assume that this market price is equal to the median price on a certain day (Pmedian). Now, different from the previous setting, traders are allowed to make decisions whether to trade or not. Consider the following case: both agents have a price above the median price. Then, the trader with the lower price forecast of the two has to sell his straddle although he thinks the straddle to be underpriced. So if we do not allow for decision making, then it could be that agents have a negative expected profit. To overcome this, agents only trade if and only if:

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Now, the profit of an agent who buys a straddle is:

|RS&P500_t | − 2Pmedian + RS&P500t

1 − 2Φ 1 2σt

, (34)

and for the agent who sells the straddle:

2Pmedian − |RS&P500t | − R S&P500 t 1 − 2Φ 1 2σt . (35)

This market setting is advantageous in two ways. The assumption of one market price is economically a more realistic argument. And traders with forecast close to the median do not have the potential of making profits by market making anymore. Now, it is not possible to sell to those with higher prices and buy from those with lower prices anymore.

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5 Results

This section presents the results for the regression of the RV, RV-J and HAR-RV-CJ models. We will look at the significance of the results and compare the forecasted realized volatilities of the models with the observed realized volatility. Then, we will look at the predictive accuracy of the models with the RMSE, MAE and Diebold-Mariano statistic. At last, the profitability of the models will be compared. We will do this by pricing options on the basis of the realized volatility forecasts.

5.1 HAR-RV model

From the regression results in Table 4 we see that all the realized volatility coefficients are significant. We see that the R2 _{is of the log transformation is the highest, so this framework} produces the most robust forecasts. In all three models the R2 _{is maximized at h = 5 which} means that the linear model is the most efficient when a weekly horizon is used. Furthermore, we see that for the linear model, the square-root transformation and the log transformation the daily coefficient is decreasing for the long-run regressions, so the daily realized volatility components becomes relatively less important. For the linear model, we see that the same holds for the weekly realized volatility component. For the square-root transformation and the log transformations, the monthly realized volatility component becomes relatively more important.

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R Vt+ h (R Vt+ h ) 1 / 2 log( R Vt+ h ) Horizon 1 5 22 1 5 22 1 5 22 β0 0.0000** 0.0000 *** 0.0000*** 0.0005*** 0.0009*** 0.0019*** -0.5060** * -0.7564*** -1.5651*** (0.0000) (0.0000) (0.0000) (0.0001) (0.0001) (0.0001) (0.0871) (0.0767) (0.0885) βD 0.2556*** 0.1952*** 0.1089* ** 0.4364*** 0.2921*** 0.1988*** 0.4261*** 0.2871 *** 0.1923*** (0.0203) (0.0148) (0.0014) (0.0199) (0.0178) (0.0204) (0.0198) (0.0174) (0.0201) βW 0.4322*** 0.3491*** 0.3329* ** 0.2978*** 0.3698*** 0.3536*** 0.3452*** 0.4007 *** 0.3265*** (0.0342) (0.0025) (0.0025) (0.0290) (0.0259) (0.0290) (0.0293) (0.0258) (0.0299) βM 0.2196*** 0.3105*** 0.27 20*** 0.1926*** 0.2449*** 0.2594*** 0.1818*** 0. 2331*** 0.3131*** (0.0307) (0.0024) (0.0022) (0.0230) (0.0205) (0.0232) (0.0230) (0.0202) (0.0235) R 2 0.5260 0.635 6 0.5607 0.7312 0.7614 0.6698 0.7761 0.8030 0.7093 R 2 adjusted 0.5256 0.635 3 0.5604 0.7309 0.7612 0.6695 0.7759 0.8029 0.7091 R Vt+ h = β0 + βD R Vt + βW R Vt− 5 ,t + βM R Vt− 22 ,t + εt,t + h (R Vt+ h ) 1 / 2= β0 + βD (R Vt ) 1 / 2+ βW (R Vt− 5 ,t ) 1 / 2+ βM (R Vt− 22 ,t ) 1 / 2+ εt+1 log( R Vt+ h ) = β0 + βD log( R Vt ) + βW log( R Vt− 5 ,t ) + βM log( R Vt− 22 ,t ) + εt+1 The HAR-R V v olatilit y forecast regressions are re p orted ab o v e for differen t horizons (daily , w eekly and mon thly). The non-robust standard errors are in paren theses and calculated b y using the in v erse Hessian. W e construct the daily realized v olatilities from the fiv e-min u te returns from the dataset whic h giv es a total of 3,499 daily observ ations. *** ** * indicate significancy at a 0.1%, 1% and 5% lev el resp ectiv ely . T able 4: HAR-R V Regression Results for Daily , W eekly and Mon thly Measures

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Figure 5: Observed and forecasted RV based on HAR-RV model

In Figure 5, the observed realized volatility is plotted against the forecasted realized volatility. We see that the forecasted realized volatility partly predicts the large outliers of the observed realized volatility. The forecasted realized volatility does jump but does not have the peaks the observed realized volatility has.

To see the fit of the observed and realized volatility, we use a scatterplot with the best fitted line (for horizon h = 1 and the linear model). Figure 6 supports the idea that the outliers of the observed realized volatility are partly predicted by the forecasted realized volatility. The largest volatility forecast is around 0.0030, where the largest peak of the observed realized volatilities is twice as big.

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5.2 HAR-RV-J model

The OLS estimates for the HAR-RV-J model are in Table 5. By comparing Table 4 and Table 5, we see that the R2 is increasing when the jump component is added. We see that the constant β0 is small for the linear model. Furthermore, the coefficients of the daily realized volatility βD and the weekly realized volatility βW are relatively large compared to the negative coefficient of the monthly realized volatility βM. This result confirms the idea of long memory in volatilities. The daily and weekly realized volatility components and the daily jump component are decreasing for the long-run regressions, so they become relatively less important. In the linear model all the jump components are significant as well.

When the three horizons are compared, we see that the R2 _{is maximized at h = 5 which} means that the linear model is the most efficient when a weekly horizon is used. All the co-efficients of the HAR-RV part are significant, which means an existence of highly persistent volatility dependence.

When we look at the regression results for the two non-linear models, we conclude dif-ferent things. First, we see that the R2 _{is maximized at h = 5 which means that the linear} model is the most efficient when a weekly horizon is used. For both the square-root trans-formation and the log transtrans-formation we see that the coefficients of the HAR-RV part are significant. However, some jump coefficients are insignificant for the transformed models. In other words, the predictability in the realized volatility regressions for the transformed models is mostly due to the HAR-RV part in the HAR-RV-J model. For both the transfor-mations, the daily realized volatility component becomes relatively less important and for the log transformation the monthly component becomes more important. When we look at the jump components, we see that the monthly jump component for the square-root transformation becomes more important, whether for the log transformation the daily jump component becomes less important.

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R Vt,t + h (R Vt,t + h ) 1 / 2 log( R Vt,t + h ) Horizon 1 5 22 1 5 2 2 1 5 2 2 β0 0.0000** 0.0000*** 0.0000*** 0.0007*** 0.0008* ** 0.0016*** -0.5080*** -0.8884*** -1.9306*** (0.0000) (0.0000) (0.0000) (0.0001) (0.0001) (0.0001) (0. 1357) (0.1193) (0.13 79) βD 0.3581*** 0.2388*** 0.1370*** 0.45 95*** 0.2984*** 0.2035*** 0.4397*** 0.2908*** 0.1975*** (0.0240) (0.0172) (0.0168) (0.0211) (0.0191) (0.0215) (0. 0203) (0.0179) (0.02 07) βW 0.4842*** 0.4250*** 0.2445*** 0.28 26*** 0.4070*** 0.3171*** 0.3580*** 0.4330*** 0.3301*** (0.0396) (0.0284) (0.0277) (0.0322) (0.0292) (0.0329) (0. 0309) (0.0271) (0.03 15) βM -0.0268 -0.0 620 -0.0596 0.1 675*** 0.1299*** 0.0924** 0.1550*** 0.1842*** 0.26 93*** (0.0446) (0.0320) (0.0312) (0.0323) (0.0293) (0.0329) (0. 0261) (0.0229) (0.02 67) βJ D -1.0450*** -0.5870*** -0.35 64*** -0.1610*** -0.0950*** -0.0784** -1017.8126*** -511.4646* -406.3630 (0.1144) (0.0820) (0.0799) (0.0282) (0.0256) (0.0287) (295.4121 ) (259.4102) (299.9 335) βJ W -1.7070*** -1.9690*** 0.52 81** -0.0606 -0.2255*** 0.1 351* -1365.7198 -2795.0701*** -365.7059 (0.2858) (0.2048) (0.1997) (0.0629) (0.0571) (0.0641) (782.6426) (687.3863) (795.3 127) βJ M 4.1170*** 6.3100*** 5.6910*** 0.1943* 0.5478*** 0.7158*** 2494.5609* 4614.7212*** 4150.8179*** (0.5675) (0.4066) (0.3966) (0.0990) (0.0898) (0.1009) (1203.0192) (1056.9182) (1222. 4617) R 2 0.5527 0.6692 0.5961 0.7328 0.7605 0.6708 0.7776 0.8049 0.7106 R 2 adjusted 0.5520 0.6687 0.5954 0.7323 0.7601 0.6702 0.7772 0.8046 0.7101 R Vt,t + h = β0 + βD R Vt + βW R Vt− 5 ,t + βM R Vt− 22 ,t + βJ D Jt + βJ W Jt− 5 ,t + βJ M Jt− 22 ,t + εt,t + h (R Vt,t + h ) 1 / 2= β0 + βD (R Vt ) 1 / 2+ βW (R Vt− 5 ,t ) 1 / 2+ βM (R Vt− 22 ,t ) 1 / 2+ βJ D (J t ) 1 / 2+ βJ W (J t− 5 ,t ) 1 / 2+ βJ M (J t− 22 ,t ) 1 / 2+ εt,t + h log( R Vt,t + h ) = β0 + βD log( R Vt ) + βW log( R Vt− 5 ,t ) + βM log( R Vt− 22 ,t ) + βJ D log( Jt ) + βJ W log( Jt− 5 ,t ) + βJ M log( Jt− 22 ,t ) + εt,t + h The HAR-R V-J v olatilit y forecast regressions are re p orted ab o v e for differen t horizons (daily , w eekly and mon thly). Th e non-robust standard errors are in paren theses and c alcul ate d b y using the in v erse Hessian. W e construct the daily realized v olatilities from the fiv e-min ute returns from the dataset whic h giv es a total of 3,499 daily observ ations. *** ** * indicate sign ificancy at a 0.1%, 1% and 5% lev el resp ectiv ely . T able 5: HAR-R V-J Regression Results for Daily , W eekly and Mon thly Measures

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5.3 HAR-RV-CJ model

The OLS estimates for the HAR-RV-J model are in Table 6. By comparing Table 4 and Table 6, we see that the R2 is increasing when the jump and continuous sample path com-ponents are added. When we compare the R2 in Table 5 and the R2 in Table 6, we hardly see any differences. This is in line with the conclusion of Andersen, Bollerslev, and Diebold (2007). Furthermore, the coefficients of the continuous sample path βCD and the continuous sample path βCW are relatively large compared to the negative coefficient of the monthly continuous sample path βCM. The daily and weekly continuous sample path components and the daily jump component are decreasing for the long-run regressions, so they become relatively less important. In the linear model all the jump components are significant as well. This is not in line with the conclusion of Andersen, Bollerslev, and Diebold (2007). In their results most of the jump coefficients are insignificant, so Andersen, Bollerslev, and Diebold (2007) conclude that ‘the predictability in the HAR-RV realized volatility regressions is al-most exclusively due to the continuous sample path components’. We conclude that both the jump components and the continuous sample path components affect the predictability in the HAR-RV realized volatility regressions. When the three horizons are compared, we see that the R2_{is maximized at h = 5 which means that the linear model is the most efficient} when a weekly horizon is used. All the coefficients of the HAR-RV part are significant, which supports the idea of Andersen, Bollerslev, and Diebold (2007) and means an existence of highly persistent volatility dependence.

When we look at the regression results for the two non-linear models, we conclude different things. First, we see that the R2 is maximized at h = 5 which means that the linear model is the most efficient when a weekly horizon is used. For both the square-root transformation and the log transformation we see that the coefficients of the continuous sample path are significant. Most jump coefficients are insignificant for the transformed models. In other words, the predictability in the realized volatility regressions for the transformed models is mostly due to the continuous sample path. So, for the transformations, the conclusion of Andersen, Bollerslev, and Diebold (2007) is supported. For both the transformations, the daily continuous sample path component becomes relatively less important and for the log transformation the monthly component becomes more important. Finally, when we compare the results in Appendix 3 with the results in Table 8, we see that the choice of α hardly has an influence on the coefficient estimates.

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R Vt,t + h (R Vt,t + h ) 1 / 2 log( R Vt,t + h ) Horizon 1 5 22 1 5 22 1 5 22 β0 0.0000** 0.0000*** 0.0000*** 0.0006*** 0.0008*** 0.0016*** -0.6511*** -1.0340*** -2.0911*** (0.0000) (0.0000) (0.00 00) (0.00 01) (0.0001) (0.0001) (0.1302) (0.1145) (0.1323) βC D 0.3580*** 0.2387*** 0.1356*** 0.4380*** 0.2863*** 0.1938*** 0.4285*** 0.2871*** 0.1943*** (0.0239) (0.0171) (0.01 66) (0.02 11) (0.0191) (0.0216) (0.0199) (0.0175) (0.0202) βC W 0.4725*** 0.4119*** 0.2422*** 0.3299*** 0.4394*** 0.3431*** 0.3339*** 0.4008*** 0.3115*** (0.0392) (0.0281) (0.02 73) (0.03 21) (0.0291) (0.0329) (0.0298) (0.0262) (0.0304) βC M -0.0112 -0. 0408 -0.0439 0.1402*** 0.1065*** 0.0724* 0.1678*** 0.1978*** 0.2682*** (0.0432) (0.0310) (0.03 01) (0.03 01) (0.0273) (0.0309) (0.0249) (0.0219) (0.0255) βJ D -0.6870*** -0.3462*** -0.2144** -0.0247 -0.0095 -0.0163 -419.8492 -130.6129 -150.0461 (0.1017) (0.0729) (0.07 10) (0.02 48) (0.0225) (0.0254) (288.9242) (253.8495) (293.2245) βJ W -1.1270*** -1.4450*** 0.77 92*** -0.0 511 -0.1775*** 0.1495* -403.5341 -16 53.0148* 433.0253 (0.2619) (0.1877) (0.18 27) (0.05 70) (0.0517) (0.0584) (754.0909) (662.5212) (765.6991) βJ M 3.9810*** 6.1500*** 5.6410*** 0.2390** 0.5951*** 0.7967*** 2068.0346 4327.0486*** 4534.9634* ** (0.5276) (0.3780) (0.36 81) (0.09 06) (0.0821) (0.0929) (1173.8 226) (1031.9653) (1191.9729) R 2 0.5525 0.6689 0.5971 0.7365 0.7645 0.6729 0.7 764 0.8037 0.7092 R 2 adjusted 0.5517 0.6683 0.5964 0.7360 0.7641 0.6724 0.7 760 0.8033 0.7087 R Vt,t + h = β0 + βC D Ct + βC W Ct− 5 ,t + βC M Ct− 22 ,t + βJ D Jt + βJ W Jt− 5 ,t + βJ M Jt− 22 ,t + εt,t + h (R Vt,t + h ) 1 / 2 = β0 + βC D (C t ) 1 / 2 + βC W (C t− 5 ,t ) 1 / 2 + βC M (C t− 22 ,t ) 1 / 2 + βJ D (J t ) 1 / 2 + βJ W (J t− 5 ,t ) 1 / 2 + βJ M (J t− 22 ,t ) 1 / 2 + εt,t + h log( R Vt,t + h ) = β0 + βC D log( Ct ) + βC W log( Ct− 5 ,t ) + βC M log( Ct− 22 ,t ) + βJ D log( Jt ) + βJ W log( Jt− 5 ,t ) + βJ M log( Jt− 22 ,t ) + εt,t + h The HAR-R V-CJ v olatilit y forecast regressions are re p orted ab o v e for differen t horizons (daily , w eekly and mon thly). The non-robust standard errors are in paren theses and c alcul ate d b y using the in v erse Hessian. W e construct the daily realized v olatilities from the fiv e-min ute returns from the dataset whic h giv es a total of 3,499 daily observ ations. *** ** * indicate significancy at a 0.1%, 1% and 5% lev el resp ectiv ely . The significan t v ariabilit y are from jumps and the con tin uous sample path equations (20) and (21) and w e use a c ritical v alue of α = 0 .001. T able 6: HAR-R V-CJ Regression Results for Daily , W eekly and Mon thly Measures

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5.4 MAE and RMSE

Horizon Model MAE RMSE

HAR-RV 0.00005 0.00017 h=1 HAR-RV-J 0.00005 0.00016 HAR-RV-CJ 0.00005 0.00016 HAR-RV 0.00005 0.00012 h=5 HAR-RV-J 0.00005 0.00011 HAR-RV-CJ 0.00005 0.00011 HAR-RV 0.00005 0.00012 h=22 HAR-RV-J 0.00005 0.00011 HAR-RV-CJ 0.00005 0.00011

Table 7: RMSE and MAE results

When we compare the RMSE and MAE results for the linear model, we see that (although small) the HAR-RV model is outperformed by the HAR-RV-J model and the HAR-RV-CJ model. Further, the HAR-RV-CJ model outperforms the HAR-RV-J model. So, we see that of the three models, the HAR-RV-CJ model predicts best. In other words, describing the price process by a combination of a continuous sample path process and a jump component gives better results in relative accuracy of the forecasts.

When we look at Appendix 4, where we present the RMSE and MAE results for the square root transformation and log transformation, we see that for the square root transformation the RV-CJ model outperforms the RV and RV-J models. Again, the HAR-RV-CJ model predicts best and from this result it follows that better prediction results are obtained when the continuous and jump components of realized volatility are regressed.

But when the results for the log transformation are compared, we see that the HAR-RV-J model outperforms the HAR-RV and HAR-RV-CJ models. So, for the log transformation, it follows that better prediction results are obtained when we just add the jump components to the model. Finally, we see that in absolute value the MAE and RMSE are relatively large for the log transformation.

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5.5 Diebold-Mariano

In Table 8 we present the Diebold-Mariano test results for the linear model. These numbers represent the differences in loss: dt= L(ε1t+h|t) − L(ε

2

t+h|t). To interpret the results correctly, we take an example. For h = 1 we find a Diebold-Mariano test statistic of 1.4865 for the HAR-RV and HAR-RV-J model. In this specific case, the first loss function L(ε1_t+h|t) is from the HAR-RV model and the second loss function L(ε2_t+h|t) is from the HAR-RV-J model.

Horizon HAR-RV HAR-RV-J HAR-RV-CJ

HAR-RV 1.4865 1.5345 (0.1372) (0.1250) h=1 HAR-RV-J -1.4865 -0.2127 (0.1372) (0.8315) HAR-RV-CJ -1.5345 0.2127 (0.1250) (0.8315) HAR-RV 1.1344 1.1640 (0.2567) (0.2445) h=5 HAR-RV-J -1.1344 -0.1399 (0.2567) (0.8887) HAR-RV-CJ -1.1640 0.1399 (0.2445) (0.8887) HAR-RV 0.7539 0.7920 (0.4510) (0.4284) h=22 HAR-RV-J -0.7539 0.3214 (0.4510) (0.7479) HAR-RV-CJ -0.7920 -0.3214 (0.4284) (0.7479)

The p-values are in parentheses. For all the tests, the linear model is used. In the appendix the results for the square-root and log transformation can be found. We use a 5% level of predictive accuracy.

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From the results in Table 8 we see that, since all the results are positive, the two ex-tended versions of the HAR-RV model dominate the HAR-RV model regardless the horizon. Furthermore, we see that for h = 1 and h = 5 the HAR-RV-CJ model is dominated by the HAR-RV-J model but this is not the case for h = 22. But since all the p-values are at least 0.1250, we conclude that for the linear model at a 95% confidence level we cannot reject the null hypothesis of equal forecast accuracy (no matter what horizon we choose).

When we look at the transformation results in Appendix 5, we again see that the two extended versions of the HAR-RV model dominate the HAR-RV model regardless the hori-zon and which transformation. For the square-root transformation, we see that for all the horizons the HAR-RV-CJ model dominates the HAR-RV-J model. Since the p-values of the HAR-RV test statistics versus the two extended models are all 0.000, we conclude that for the square-root transformation model at a 95% confidence level we can reject the null hypothesis of equal forecast accuracy (no matter what horizon we choose) and thus the extended models are an improvement in forecasting volatility. Since the p-values for the HAR-RV-J model versus the HAR-RV-CJ model are almost 1, we cannot draw this conclusion again. Now, we cannot reject the null hypothesis of equal forecast accuracy, no matter what horizon we choose.

For the log transformation we can only reject the null hypothesis for the HAR-RV model versus the HAR-RV-J model and thus that the HAR-RV-J model results in an improvement of forecasts.

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5.6 Option Pricing

To determine which model is the most profitable, we look at the average profits and the Sharpe ratios. Since the portfolio represents the entire investment, we use the Sharpe ratio. The Sharpe ratio for portfolio p, denoted by Sp, is given by:

Sp = ¯ rp− rf

σp

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where ¯rp is the expected portfolio return, rf is the risk-free rate (which we take to be zero) and with σp the standard deviation of the profits across different methods. Now, we define it as the average profits divided by the standard deviations of the profits. This Sharpe ratio tells us what the ratio of reward-to-volatility is. The higher an asset’s Sharpe ratio, the higher the return the investor will get per unit of risk.

Horizon Model Average Profit Sharpe Ratio Total Profit

HAR-RV -0.09671 -0.17153 -336.33929 h=1 HAR-RV-J -0.00994 -0.04407 -34.76066 HAR-RV-CJ 0.10893 0.27895 378.85937 HAR-RV -0.00332 -0.00872 -11.54661 h=5 HAR-RV-J -0.13358 -0.34815 -463.91563 HAR-RV-CJ 0.13859 0.35869 481.32331 HAR-RV 0.06222 0.10218 215.08488 h=22 HAR-RV-J -0.21457 -0.54285 -741.76556 HAR-RV-CJ 0.15258 0.42616 527.45262

From 10/15/2008 to 11/13/2008, there is no trade possible due to the normalization factor which causes all the call prices/put prices to be 1. Thus, we have a total of 3,487 observations.

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Square-root transformation

Horizon Model Average Profit Sharpe Ratio Total Profit HAR-RV 0.08735 0.17599 299.78410 h=1 HAR-RV-J -0.10888 -0.24468 -373.66834 HAR-RV-CJ 0.02153 0.03807 73.87880 HAR-RV -0.02797 -0.05236 -95.88449 h=5 HAR-RV-J -0.10313 -0.25176 -353.51743 HAR-RV-CJ 0.13110 0.28155 449.40772 HAR-RV -0.00461 -0.01269 -15.74234 h=22 HAR-RV-J 0.04037 0.12408 137.78430 HAR-RV-CJ -0.03575 -0.10090 -121.97536 Log transformation HAR-RV 0.09103 0.21994 316.59861 h=1 HAR-RV-J -0.39534 -0.99968 -1374.97588 HAR-RV-CJ 0.30431 0.72023 1058.37274 HAR-RV -0.49084 -1.04390 -1704.68655 h=5 HAR-RV-J 0.01531 0.09303 53.15791 HAR-RV-CJ 0.47554 1.17014 1651.53584 HAR-RV -0.60523 -0.18706 -2092.27141 h=22 HAR-RV-J 0.00067 0.00905 2.29718 HAR-RV-CJ 0.60456 1.43772 2089.96519

Table 10: Profits per Model and Horizon

These results provide evidence in favor of the idea put forward by Bandi, Russell, and Yang (2008). They state that if the forecast is accurate, the corresponding daily profit should be small in absolute value.

In Table 9 we see that the HAR-RV-CJ model is the most profitable no matter which horizon we choose. It is striking that an agent who trades with the HAR-RV-J model never makes any profits and for h = 5 and h = 22 performs worse than the HAR-RV model. So the idea of the predictive accuracy results, where the extended HAR-RV models outperform the HAR-RV model, is partly confirmed. Furthermore, we see that the ranking based on profits and on Sharpe ratios is the same.

When we turn to the results of the transformations, we see that for the log transformation the HAR-RV-CJ model is the most profitable. For the square-root transformation, the most profitable model depends on the horizon we choose. The square-root results are not in line with the predictive accuracy results, where we show that the extended models outperform

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the HAR-RV model. Again, there is no difference in rank based on profits and Sharpe ratios.

When we add an agent who trades with the maximum daily forecast, an agent who always trades with the minimum daily forecast and a mean trader, the trader with the maximum forecast always makes the most profit. Thus, the agents were underestimating the variance which also follows from Figure 6, Figure 8 and Figure 10. This underestimation leads to a relatively low price for the straddles. Furthermore, the mean trader hardly makes any profits.

Since the small amount of agents, there are hardly any profit differences when comparing the two types of hypothetical option markets. To overcome this problem we will introduce some volatility models to compare the results of the two types of option markets.

5.6.1 HAR-RV models versus other volatility models

As shown by Andersen, Bollerslev, Diebold, and Labys (2003), Koopman, Jungbacker, and Hol (2005) and Corsi (2003), realized volatility forecasts are more accurate than forecasts based on ARMA, AR and GARCH models. We will check whether this idea is reflected in the profitability of the models. Fleming, Kirby, and Ostdiek (2001) state that the accuracy of volatility forecast models does not answer the question of whether it brings higher eco-nomic value. First, a short introduction of the included models. These models are compared in profitability on the two types of hypothetical option markets.

ARMA-type model

The autoregressive moving average expresses the series xt as a function of past observations and past innovations. The ARMA(p,q) model is defined as:

xt = α1xt−1+ α2xt−2+ . . . + αpxt−p+ εt+ β1εt−1+ β2εt−2+ . . . + βqεt−q.

In the comparison of profitabilities the AR(1)-part xt = α1xt−1+ α2xt−2+ εt will be used as well.

GARCH model

The generalized autoregressive conditional heteroskedastic (GARCH) is an ARMA model for conditional variances. The GARCH(p,q) model is defined as:

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where

σ2_t = ω + α1a2t−1+ α2a2t−2+ . . . + αpat−p2 + β1σ2t−1+ β2σ2t−2+ . . . + βqσt−q2 .

Now, we let six agents trade on a hypothetical option market. The first option market is the market with anonymous trading. We see that with anonymous option trading, the HAR-RV-CJ model is the most profitable. Behind this model is the HAR-RV-J and ARMA model. The least profitable model is the GARCH model. So we see that the most accurate HAR-RV-CJ model (in terms of RMSE and MAE) outperforms the other less accurate mod-els economically. It is striking that the ARMA model is more profitable than the HAR-RV model, although the HAR-RV model is more accurate as shown by Corsi (2003). Further-more, since the GARCH model is the least profitable we conclude that agents who make up their volatility forecasts by using high-frequency data are more profitable. So the more information an agent has about the volatility, the more profitable their strategy is.

In the second hypothetical option market, we consider a single market price for all transac-tions whether in the previous setting pair-wise trades at mid prices took place. With the second type of market, traders with median forecasts do not have the potential to make profits of market making anymore.

Model Average Profit Sharpe Ratio Total Profit

HAR-RV -0.01977 -0.00289 -68.74329 HAR-RV-J 0.04717 0.09385 164.04857 HAR-RV-CJ 0.09434 0.23759 328.12874 AR(2) -0.02797 -0.03417 -89.13242 ARMA(2,2) 0.02828 0.04713 98.34280 GARCH(1,1) -0.12195 -0.19670 -424.15732

For the HAR-RV model and the extended versions, we use the linear model with horizon h = 1.

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Model Average Profit Sharpe Ratio Total Profit HAR-RV -0.01636 0.00339 -56.89349 HAR-RV-J 0.06943 0.15202 241.48732 HAR-RV-CJ 0.17196 0.44846 598.09328 AR(2) -0.05002 -0.06670 -173.98356 ARMA(2,2) 0.02910 0.04856 101.32895 GARCH(1,1) -0.20165 -0.32525 -701.35081

For the HAR-RV model and the extended versions, we use the linear model with horizon h = 1.

Table 12: Profits per Model for an Option Market with a Single Market Price

From the results above we see that there is no difference in ranking with the anonymous option trading. Still, the HAR-RV-CJ model is the most profitable and the GARCH model the least. But there is difference in the total profitability. The HAR-RV-CJ model is far more profitable than in the previous setting, whether an agent will lose a lot more money when forecasting volatility with the GARCH model. If we look at the position an agent with the GARCH model takes, we see that a GARCH trader underprices its straddle but close to the median. In the previous hypothetical option market, a GARCH trader was able to compensate this position, whether this is not possible in this setting. For both hypothetical option markets, we see that ranking based on profits and on Sharpe ratios is the same.

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6 Discussion

In the previous chapter, we present results for a number of different models and analyses. We estimate the coefficients of the HAR-RV, HAR-RV-J and HAR-RV-CJ model of Andersen, Bollerslev, and Diebold (2007). They find that the jump coefficients are insignificant and conclude that the predictability in the regressions is mostly due to the continuous sample path coefficients. In contrast, we find that for the linear model, both the jump components and the continuous sample path components are significant. For the square-root and log transformations, our results provide evidence in favor of the conclusion that the predictabil-ity in the regressions is mostly due to the continuous sample path coefficients.

We find that the root mean squared error (RMSE) and mean absolute error (MAE) for the extended HAR-RV models are smaller than for the HAR-RV model. This is in line with the results of Clements and Liao (2013) and C¸ elik and Ergin (2014). It follows that we obtain better prediction results when we use the extended HAR-RV models. Our results of the Diebold-Mariano test support what they find. However, for the linear model and for the log transformation we cannot reject the null hypothesis of equal forecast accuracy at a 95% confidence level. For the square-root transformation, we can reject the null hypothesis of equal forecast accuracy. Thus, for the square-root transformation, the extended models are an improvement in forecasting volatility.

We find that as expected, after transforming the volatility forecasts to option prices, the HAR-RV-CJ model is the most profitable when we make a pair-wise comparison. So, a separate measurement of the continuous sample path component and the jump component gives a trader the most profit. However unexpectedly, the HAR-RV-J only outperforms the HAR-RV model profit-wise for certain horizons and transformations. So, the large number of small nonzero jumps in the HAR-RV-J model (not treated as measurement errors as in the HAR-RV-CJ model) are not advantageous for the economic value of this model.

Finally, when we compare the realized volatility forecasts with the GARCH and ARMA models, we find that the HAR-RV-CJ model is the most profitable and the GARCH model the least. So, traders who compute their volatility forecasts by using using high-frequency data are more profitable. This is in line with Gomber and Haferkorn (2013) where the prof-itability of high-frequency trading is shown. However, it is counterintuitive that the ARMA model outperforms the HAR-RV model, which provides evidence for the idea of Fleming,

From volatility forecasts to option pricing : a comparison in economic value

University of Amsterdam

Master Thesis