Evaluating neural networks in a hypothetical option market by training on multiple variance estimators

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option market by training on multiple variance

estimators.

Author:

Mats Van Beelen 10446133

Andreas Rapp : Supervisor

Ling WeiKong : Second reader

Msc Econometrics Track: Financial Econometrics

University of Amsterdam

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Statement of originality

This document is written by Student [Mats van Beelen] who declares to take full respon-sibility for the contents of this document.

I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Abstract

The performance of hybrid models in option pricing is studied. Hybrid models which are neural networks, include a forecast of a GARCH model specification to enhance the forecasting performance of A GARCH model. Multiple variance(volatility) estimators are studied to set as target variables for training a neural network. The evaluation of the models is done in there performance in a hypothetical option market. It can be concluded that kernel variance estimators are not viable target variables for pricing 1-day options. The hybrid models for 15-day ahead forecasting show better performance. However, the overpricing agents should be take into consideration. Switching forecasting algorithm in time led to more profit compared to using a single forecasting algorithm.

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1 Introduction

A popular research topic in time series analysis is estimating and forecasting the volatility of stocks and their derivatives as the forecast of volatility plays a central role in financial decision making. In option pricing one needs a forecast of the volatility which is closest to the true value of the volatility of the underlying derivative. Unfortunately the true volatility is unobservable and one has to make an estimate of the volatility. Much research has led to a variety of variance(volatility) estimators. The problem that forecasters face now is how to evaluate two or more competing forecast alternatives.

In 1990 Engle, Hong and Kane devised a technique to rank forecasting algorithms based on their performance in a hypothetical option market. This framework has been used multiple times in research to evaluate a wide range of variance estimators. In this paper the framework of Engle et al. is adopted to evaluate the use of neural networks in option pricing and n particular which variance estimators are viable to train a neural network. Hajizadeh et al. (2012) constructed hybrid models which consist of a neural network with as extra input variable the forecast of a GARCH model. They showed that the mean squared error of a hybrid model decreases compared to the GARCH model on which they are based. Neural networks are based on finding a minimum in a loss function which could be the mean squared error. Therefore, there could be a bias in evaluating the hybrid models on the same target variables on which they are trained. This is another motivation to evaluate the forecasts methods in the framework of Engle et al.

The hypothetical option market consists of agents that all represent a forecasting volatility algorithm. The agents trade a straddle( put and call option) with one another based on the S&P500 index. There are seven factors that affect the price of an option; the stock price, strike price, time to maturity, type of option, interest rates, dividends and volatility. As most of these factors are the same for all competing algorithms, with the exception of the volatility forecast, all agents in the market produce different price

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estimations for the value of the option. Agents with low volatility forecasts will believe that agents with higher forecasts are overpricing call and put options and vice versa. Hence an overpricing(underpricing) agent wants to buy(sell) a straddle from a underpric-ing(overpricing) agent.

The hybrid models, which participate in the hypothetical option market, need a forecast of a GARCH model and an estimator for the unobservable volatility. The GARCH models which provide a forecast of the volatility are the standard GARCH model(Bollerslev ,1986) and two extensions of it, the EGARCH introduced by Neslon(1991) and GJR-GARCH of Glosten et al. (1993). The extensions attempt to capture the asymmetric effect in the influence of positive and negative returns on the volatility. Hajizadeh et al. (2012) used the daily realized variance estimator as target variables to train the hybrid models. Other target variables which are considered: the 5-minute squared returns(Andersen et al. ,2003), the 5-minute sub sampled returns returns as motivated by Bandi and Russel(2008) and three realized kernel variance estimators(Nielsen et al., 2008).

At last, two traders are introduced that switch between forecasting algorithms in time. Both traders switch algorithms on the believe that different forecasting techniques give better forecasts of the volatility for a given time period. The switching of algorithms is based on the best performing method that has the most accumulated profit for a period of time.

This paper will start in section 2 with the methodology of the GARCH models, neu-ral network and variance estimators. Section 3 describes the hypothetical option market and some remarks which need to be studied. Thereafter the set up of the neural network and hybrid models will be discussed in section 4. Along with a description of the data and sources which are used. The results can be found in section 5 which also addresses the remarks made in section 3. In section 6 a summarizing can be found. Tables and Figures are in the Appendix.

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2 Methodology

2.1 The GARCH models

The ARCH (Auto Regressive Conditional Heteroscedasticity) family was first introduced by Engle (1982). Thereafter, numerous extensions of the ARCH family have been devel-oped. In this paper the GARCH model as proposed by Bollerslev (1986), the EGARCH model developed by Nelson(1991) and the GJR-GARCH model (Glosten, Jagannathan & Runkle, 1993) are used. The latter two are an extension of the GARCH model which cap-tures the asymmetric behaviour of returns of volatility. The conditional variance which is estimated by the three models is given by V (rt|Ft−1) = V (at|Ft−1) = σ2t. Here rt denotes

the log returns and Ft−1 is the information set available at time t-1. Throughout this

section at is referred as the shock of the asset and estimated by at = σtt. Likewise, t

is a sequence of independently identical distributed(i.i.d.) random normal variables with zero mean and unit variance equal. Taking the square root of the variance yields the volatility.

The generalized GARCH(p,q) model, where the conditional variance depend on p(past shocks) and q(past conditional variances), can be represented as follows:

at= c, σt= α0+ p X i=1 αia2t−i+ q X j=1 βjσ2t−j (1)

where α0 > 0, αi ≥ 0, andβj ≥ 0. The GARCH model does not capture the asymmetric

behaviour in positive and negative shocks on the volatility, whereas the EGARCH and GJR-GARCH do.

Nelson 1991 proposes the exponential GARCH(p,q). The EGARCH model allows for asymmetric effects between positive and negative asset returns and is given by:

at= σtt, ln(σt2) = α0+ p X i=1 αi |at−i| + γiat−i σt−i + q X j=1 βjln(σt−j2 ) (2)

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Here the parameter γ signifies the leverage effect of at−i, t is again a sequence of i.i.d.

normal(0,1) random variables and there are no restrictions on the other parameters.

Another model which accounts for the leverage effect in the returns is the GJR-GARCH model, which is essentially the same model as the Truncated GJR-GARCH (TGJR-GARCH) model (Zakoian, 1994). The GJR-GARCH(p,q) model can be written as:

at= σtt, σt= α0+ p X i=1 (αi+ γiIt−i)a2t−i+ q X j=1 βjσ2t−j (3)

where It−i is an indicator function for which It−i = 1 if at−i < 0 and 0 otherwise. The

parameters αi, γi and βi are all non-negative.

The GARCH models above are all fitted for nine different combinations of lags which are for every GARCH model or an extension of it, lags equal to (1,1), (2,2) and (3,3). Forecasting the models is done by the rolling window method, where each time 1000 past observations are fitted to forecast 1-day ahead and 15-day ahead volatilities.

2.2 Neural networks

The foundation of neural networks is in research which attempt to represent the neuro-logical activity of the human brain mathematically. Already in 1943 McCulloch and Pitts wrote a paper to propose multiple theories that mathematical describe such a process. The use of machine learning techniques, such as a neural network, have become more widely adopted since. Hajizadeh et al. (2012) wrote a paper in which they enhance the forecast-ing performance of a GARCH model with the use of neural networks and constructed a so called hybrid model. This section provides the use and working of a neural network and is based on the literature of Bishop (2006).

The GARCH models as considered above for fitting and forecasting the volatility are based on a linear combination of fixed functions. Those function are called in the literature of machine learning: basis functions. Neural networks have a fixed amount

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of basis functions as well but are more adaptive to the observed data. In addition the basis function can be non-linear and therefore a neural network is more capable of finding non-linear patterns in the data.

The most common used neural network is the multilayer perceptron also known as the feed-forward neural network. A feed-forward neural network consists of three types of layers; input layer, hidden layer and output layer. In figure 1 a representation of a multilayer perceptron is presented.

Consider nonlinear basis function δj(x) and the model for linear regression takes

the form: y(x, w) = f ( M X j=1 wj(δj(x)) (4)

where f(·) is a nonlinear activation function (Bishop, 2006). A neural network extends this form of model by allowing the basis functions (δj(x)to be functions by themselves. The

parameters within those function can be adjusted, along with the weight coefficients wj

which makes them more adaptive. Let M be the number of hidden units, N the number of input variables and K the number of output variables(which in volatility forecasting will be set to one).

For aj = PN_i=0w(1)_ji xi, zj = h(aj) and ak =

PM

j=0w (2)

kjzj the model in (4)

be-comes: y(x, w) = f ( M X j=1 w(2)_kjh( N X i=0 w(1)_ji xi)) (5)

Here the subscripts (1) and (2) are indicative for the corresponding layer in which the parameters are, h(·) is a differentialable nonlinear activation function.

To train a neural network a matrix of input variables is needed along with pre estimated or observed values called target variables t. The network update the parameters in the function y(x, w) to find the minimum of an error function. The error function

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which is used in for the neural networks and also for the evaluation, is the mean squared error(MSE): M SE = PN i (y(x, w) − t)2 N (6)

To find a minimum in the error function, the gradient of the error function must be equal to zero. For evaluating the gradient of a feed-forward neural network, error backpropagation is used. Error backpropagation consists of two stages. Firstly, the errors are fed backwards through the network to evaluate the derivatives. In the second stage the derivatives are used to adjust the weights. The adjustment is mostly done by stochastic gradient descent(Rumelhart et al. 1986) or an extension of it. The update of the weights is mostly done in samples/batches. After the neural networks are trained, forecasts can be made for the volatility.

2.3 Realized variance and kernel estimators

In this paper, a proxy for the true unobservable volatility is needed as target variable for the neural network as discussed above. For this reason a few proxies for the volatility are studied on which the models are trained. At first, the proxy which Hajizadeh et al. (2011) used to train their neural network is adopted. Secondly the classical realized variance estimator (Andersen, Bollerslev, Diebold, and Labys , 2003) using 5 minute returns will be used as target variable. Furthermore the optimal sampling frequency for the 5 min realized variance as discussed in the paper of Bandi and Russel (2003, 2007) will be selected to train on. At last three kernel estimators of Nielsen et al. (2008) are used as target variables.

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2.3.1 Variance estimators based on sampling returns

Hajizadeh et al. (2011) calculates the daily realized volatility (RV ) for ption length n on day t by: RVt= v u u tn−1 t+n X i=t (Ri− ¯R)2 (7) ¯ R = n X i=1 Ri n (8)

where Ri = ln(_PP_i−1i ) is the logarithmic return.

Andersen et al. (2003) define the realized variance estimator as the sum of squared returns over a trading day. In the absence of noise the estimator is consistent as M goes to infinity, the estimator for 5-minute returns can be calculated by:

ˆ RV = M X j=1 r2_jδ (9)

For 5-minute returns M is equal to 72(6 · x · 60₁₂). Bandi and Russell (2003) and Zhang, Mykland and A¨ıt-Sahalia (2005) have shown that RV is inconsistent in the existence ofˆ market microstructure noise and that the noise increases as the frequency M increases. Bandi and Russel as a result derived a theory for MSE optimal sampling to estimate the variance. They adopted the framework of Engle et al. (1990) as well to show that the op-timal sampling of the 5/15/30-minute returns generate more profit than their counterpart without sub sampling. Accordingly the 5-minute sub sampled returns variance estimator of the Oxford-Man Institute library is used next to the 5-minute returns without sub sam-pling. Consequently, one would like to see that the hybrid models trained on the variance estimator based on the 5-minute sub sampling returns do it better compared to the one without sub sampling.

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2.3.2 Variance estimators based on Realized kernels

Bandi et al. (2006) use kernel estimators to show that they outperform the use of high frequency data in forecasting volatility. The microstructure noise which affects the variance forecast in high frequency data is addressed multiple times in the literature. In their paper they find that flat-top symmetric kernels perform better than other kernels for forecasting the volatility. In this paper three kernel variance estimators are used as target variables and for measuring forecasting performance. The Non-Flat Parzen, Flat-top Tukkey-Hanning2

and Two-Scale/Bartlett (Nielsen et al. 2008) kernel variance estimators are studied.

Consider a continuous time log-price process X, on period [0,t] and time gap δ > 0, then the Flat-top and Non-FLat-top realized kernels of Nielsen et al. (2008) are respec-tively given by:

K(Xδ) = γ0(Xδ) + H X h=1 k(h − 1 H ){γh(Xδ) + γ−h(Xδ)} (10) K(Xδ) = γ0(Xδ) + H X h=1 k(h H){γh(Xδ) + γ−h(Xδ)} (11)

Here k(x) is a weight function for x ∈ [0, 1]. They show that if k(0) = 1, k(1) = 0 and H = c0n

2

3 the kernel estimator is asymptotically mixed Gaussian. The hth realized

autocovariance is: γh(Xδ) = n X i=1 (Xδj − Xδ(j−1))(Xδ(j−h)− Xδ(j−h−1)) (12)

where n = _δt, and h = -H,...,-1,0,1,...,H. The jth frequency return is given by (Xδj −

Xδ(j−1)). The difference between K(Xδ) and γ0(Xδ) is the correction between the realized

kernel and the non observable realized variance and can be seen as the microstructure noise.

The difference in the kernel variance estimators are based on which weight func-tion k(x) of the kernel is used. The three different kernel weight funcfunc-tion that are used are:

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1. The non Pazen kernel estimator makes use of the following kernel weight function: k(x) =      1 − 6x2+ 6x3 if 0 ≤ x ≤ 1₂ 2(1 − x)3 if 1₂ ≤ x ≤ 1 (13)

2. The Bartlett estimator kernel is based on the weight function:

k(x) = 1 − x (14)

and has the same asymptotic distribution as the two scale kernel estimator.

3. The Tukkey-Hanning2 variance kernel estimator uses as kernel weight function:

k(x) = sin2(π

2(1 − x)

2₎ ₍₁₅₎

To estimate the 15-day volatility for the three kernel estimators and the 5-minute returns estimators, the average of the daily estimates has been taken for the fifteen days lifetime of the option.

All the proxies for the unobservable volatility as presented above will be used set target variables for the neural networks. In the next section the evaluation technique based on the hypothetical option market is discussed. Whereas in section 5 a description of the data can be found along with the set up of the neural networks.

3 Hypothetical option market

For evaluating the hybrid models and the GARCH family models on which the hybrid models are based, the framework of Engle, Hong, and Kane (1990) is adopted. Their framework allows us to rank different models based on their volatility forecast, but without measuring it on some proxy for realized volatility. Instead, the ranking is based on the returns the models obtain from trading with each other. In this section the evaluating framework of Engle et al. is presented.

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Engle et al. designed a hypothetical option market based on a $1 share of the NYSE index. In this paper the S&P500 index is used. In the previous section the models are discussed, each model represents an agent in the hypothetical option market. The agents uses the Black-Scholes formula (Black and Scholes, 1973) to price the call/put options: ct= S0N (d1) − Kte−rTN (d2) (16) pt= Ke−rTN (−d2) − S0N (−d1) (17) d1 = ln(S0 K) + (r + σ2 t 2 )T σt √ T (18) d2 = d1− σt √ T (19)

Here t corresponds to the day, S0 is the stock price, Kt is the strike price of the option,

pt and ct represent respectively the put and call price, N(·) is the normal cumulative

distribution, T is the maturity of the option and σtis the forecast/estimate of the volatility

of the underlying derivative.

The hypothetical option market has three trading stages which will be considered now:

1. Each agents calculates their call/put with the Black-Scholes pricing formula with their forecasted volatility.

2. The agents trade pair-wise. The price of the straddle is calculated by the mid prices of the agents put and call prices. An agent with a volatility forecast which gave a higher price will now believe that the straddle is under-priced. The agent with the lower volatility forecast will believe that the straddle is overpriced. Therefore the two agents want to trade. The agent with the higher volatility forecast will be the buyer in the market and the other agent will sell the straddle to that agent.

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Each day, the three stages will occur and at the end of the testing phase the profit or losses will be accumulated for each agent. The strike price for an option on a $1 share of the S&P500 index will be taken to be $1 + the risk free interest rate.

Engle et al. (1990) constructed the framework as represented above only for 1-day options, as it was rather about the technique. In their paper they made some remarks about the framework which will be discussed shortly. Firstly, when more than two agents participate in the experiment the winner is not guaranteed to be the best in any other sample of the same group. Also, the Jensen inequality occur in the valuation technique. Engle et al. addressed this issue to adjust the variance forecast upwards due to the fact of the non linearity of the Black-Scholes option pricing formula. To adjust the volatility forecast one needs an estimation of the forecasting error. They decided not to include an adjustment in their forecasts since the realized volatility is unobservable. Section 2.3 also discusses that variance estimators tend to have microstructure noise which make them less suitable for an adjustment. Lastly, to measure the forecasting method independently a calculation of the profits is made when trading at his own prices. The best ranking algorithms should also have the smallest accumulated profit(in absolute values). The daily profit for a trade is max(Rt− Kt, Kt− Rt) − (ct+ pt). Here Rt is the daily return of

the S&P500 and Kt is the strike price. In addition the variance of the daily profits needs

to be small as well for a good forecasting algorithm.

4 Neural network set up and data description

The neural network as presented in section 2 needs the following input: an error function, activation functions, the number of hidden neurons, input variables i.e. lagged returns and target variables to train the network. This section provides a set-up of the neural network along with the data description

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4.1 Set-up neural network

In general training a neural network is a lot of trial and error. Therefore while choosing the amount off hidden neurons, 10 showed good performance in decreasing the loss function. The corresponding activation function for the first layer is a relu function: max(0,x). Be-cause the target variable is a forecast for the volatility the aviation function in the output player is a linear function of the hidden variables which makes it in essence a (stochas-tic)regression but with more parameters which need to be updated and estimated.

In total there are nine GARCH models which the hybrid models are based on. In section 2, six proxies for the volatilities are presented.This result in a total of 54 hybrid models leading to a total of 63 models.

The GARCH models used to forecast volatility are fitted on lagged shocks and variances. In the same way the input variables for the neural network only consists of lagged variables of the returns and past estimates of the volatility. In order to select the amount of lags used for the returns and volatility the mean squared error is calculated for models with different amount of lags. Additionally the call and put prices of the forecasts were calculated. There is no clear correlation between the amount of lags and MSE, but for more than 6 lags the mean squared error increases. Also the call and put prices show more variance as the amount of lags increases , therefore 5 lags will be used of both the past returns and volatilities.

Hajizadeh et al. (2012) constructed a hybrid model, which consist of the above described neural network with lagged variables for the return and volatility and a forecast of a GARCH model. They construct fifteen and twenty day volatility returns and showed that the hybrid model improves in mean squared error.

The stochastic gradient descent as introduced by Rummelhart et al.(1986) has var-ious problems while training on a data set. The standard stochastic gradient descent implies a standard learning rate for all input variables and that the learning rate is

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de-creasing in training time. Later in the training phase the updating of the parameters is done by a small step in the opposite direction of the gradient and eventually for large data sets there is no progress more in the end of training. RMSprop a widely used learning rate in machine learning and first introduced by Tieleman(unpublished) and used in the lecture of Hinton (2012) to target these problems. The learning rate for every variable is differ-ent and is based on the previous squared gradidiffer-ents. It uses a moving average of squared gradients to normalize the gradient itself. This decreases the step for large gradients to avoid exploding, and increases the step for small gradients to avoid vanishing.

4.2 Data

The data were is made us of comes from three different sources. The S&P500 closing prices comes form the data bank of the Wall Street Journal(www.quotes.wsj.com). For the monthly rates is made us of the data available from the department of treasury of the United States(www.treasury.gov) and the Oxford Man Institute of Quantitative Finance has provides data for different daily volatility estimators. In this paper the 5-minute returns, 5-minute sub-sampled returns, the top-flat parson, the two-scale/Barlett and Turkey-Hanning2 are used.

For training the hybrid models and forecasting the volatility with the models, the data which is employed starts on 01/02/98 and ends on 03/28/18. There are first a 1000 observations needed for a forecast with a GARCH model. Hence the GARCH models start on 12/21/01 with forecasting volatilities and stop with forecasting till the last day of testing. As a result, the training phase of the hybrid models also start on 12/21/01. The training phase end on 06/19/14, the last 950 observations till 03/28/18 are used for testing.

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5 Results

Mean squared errors

Before evaluating the models in the framework as discussed in section 3, the mean squared errors are calculated. Each hybrid model with the dependent proxy for the volatility is compared with the GARCH model on which the hybrid model is based. The proxy for the volatility is t in equation (6) and the forecast of the model is y(x,w) in (6).

In table 1 the difference between the hybrid model and their GARCH counterpart are reported for the 1-day ahead forecasts of the volatility. It can be seen that most hybrid models increase in mean squared error with the exception of the EGARCH and GJR-GARCH trained on the daily realized variance. Table 2 shows the difference in mean squared error between the hybrid model and their GARCH counterpart for 15-day ahead forecasts. For these hybrid models it holds that they all improve in forecasting with respect to the proxy variable which is used.

Pair-wise trading by variance estimator

To address the remark made in section 3 in which the winner is not guaranteed to be the best in any other sample, first each group of hybrid models with respect to their target variable will trade pair wise with the GARCH models.

In table 3 to table 8 the profit(losses) are reported for each agent for trading 1-day straddles. Table 3 with the daily realized variance estimator shows that every hybrid model generate less profit than their GARCH counterpart. In table 4 using the 5-minute demonstrates that all the GARCH models perform better and the hybrid-EGARCH(3,3) as well. Tables 5 to 7 have each two hybrid models based on the standard GARCH models that perform better. The Turkey-Hanning2 variance estimator in table 8

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The hybrid EGARCH and GJR-GARCH model with kernel variance estimators all perform worse. In addition the hybrid-EGARCH(3,3) while using the 5-minute sub sampled returns realized variance estimator as target variable performs better as can be seen in table 5.It can be concluded that there is no proxy for the volatility that enhances the performance of all the hybrid models when set as target variable in the neural network.

Tables 9 to 14 shows the profit(losses) accumulated by each algorithm in the hypo-thetical option market for 15-day options. All hybrid models for each variance estimator perform better than their standard GARCH and GJR-GARCH on which the hybrid mod-els are based. In table 8, 11 and 12 the hybrid modmod-els trained on respectively, the daily realized variance, the Parzen kernel variance estimator and the Two-scale/Bartlett kernel variance estimator all outperform their GARCH counterpart. The two variance estimators based on the 5-minute returns and the Tukkey-Hanning2realized kernel variance estimator

in respectively table 10,11 and 14 show that the hybrid model based on EGARCH(2,2) loses on its counterpart.

For 15-day ahead forecasting it shows that in general the hybrid models do it better than the GARCH models on which they are based. In the next experiment each algorithm is allowed to trade pair wise with all available models. This results in 63 agents which participate in the hypothetical option market.

Rank order by total profit

In table 15 the total profit or losses are reported for the options with a maturity of a day, respectively table 16 shows their ranking. The best agents in the market are using the 5-minute returns variance estimator as target variable Thereafter the 5-minute sub-sampled returns show some good performance as well. The hybrid models based on the EGARCH(2,2) forecasts are all not performing well. The GARCH models all generate profit and have in general a good ranking. For the daily realized variance estimator and

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the kernel estimators it holds that they are mostly losing money, only the Parzen kernel estimator has some hybrid models that generates profit.

Table 17 shows the profit(losses) of the 63 models by trading straddles with a ma-turity of fifteen days. In table 18 one can find the respective ranking of the participating models. Firstly, it can be seen that the GARCH models now score much lower rank-ings.Secondly, the EGARCH(2,2) and the hybrid models based on it are all making a loss. Also all EGARCH models are not performing that well. Moreover, hybrid models trained on the daily realized variance are only making a profit in two out of nine models. The Parzen-GARCH(2,2) is the best performing agent. Every other variance estimator, has at least one or two hybrid models which are in the top ten best performing agents. Yet there is no clear evidence that a particular target variable generates more profit.

Average profits

In section 3 another remark has been discussed whether the best forecasting algorithm in the market(s) above also has the smallest accumulated profit(in absolute values) when trading at it own prices. In table 19 the average profits and their variance are presented for trading 1-day options. Table 20 shows this for the 15-day option market. A negative value in the table relates to the fact that the agent was on average overpricing the straddles, a positive sign means that the agent was overpricing the straddles.

The smallest average profits for straddles with a maturity of one day are the GARCH models. After the GARCH models, the two variance estimators based on the 5-minute returns perform well. The hybrid models trained on the kernel estimators rarely rank above average. Looking at table 16 and table 19, it can be seen that the findings are in line with each other. But that the GARCH models show better results in the av-erage profits. Two of the algorithms overprice on avav-erage the straddles while the others under price. The variance of the profits is in each model do not show significant difference

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among each other, except for the hybrid models with the daily realized variance as target variable.

Table 20 reports the average profits(losses) in cents for 15-day straddles along with their ranking and corresponding variances. The GARCH models as well as the hybrid models trained on the daily realized variances score most of the lowest rankings. The hybrid models based on the kernel variance estimators, especially the two-scale/Bartlett and Turkey-Hanning2 kernel estimators have most of the highest rankings. Comparing

the rankings of the variance estimators based on the 5-minute returns, the models rank on average higher. Nevertheless the results in tables 18 and 20 are not completely conclusive. Since all agents are overpricing the straddles, the whole market is misinformed about the true price. This can be a result of the Jensen inequality as spoken about in section 3 as well, but Engle et al. (1990) suggested that the forecasts actually should be adjusted upwards. As the agents were already overpricing the straddles(Black-Scholes formula is increasing with the volatility) adjusting the volatility upwards will only increase the “loss”. In the variance of the profits only the GARCH based models have for each model the lowest variance compared to all the hybrid models, the rest of the agents show similar variances.

Comparing the mean squared errors in tables 1 and 2 with the profits(losses) in tables 15 and 17, the hybrid models that decrease the most in mean squared error(daily realized variance estimators) perform worse than any other hybrid model. But there is no other clear correlation between the decreasing of mean squared errors and the total accumulated profit. This gives an indication that the selected proxies for the volatility as target variables do not describe the underlying behaviour of the volatility fully of the underlying derivative. Whereas the daily realized returns variance estimator performed the worst among them all, while set as target variable.

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Switching algorithms

Overall, the results for both trading 1-day options 15-day options assumes that in a com-petitive asset market there is significant profit to be made by switching between algorithms during time. To test this assumption, two different traders are introduced which can use every forecasting method from the above markets. First, Trader 1 has priced the options at time t with the forecasting method which has the most cumulative profit at time t-1. Secondly, Trader 2 believes that the best performing forecasting method depend on the cumulative profit of the last five days. Additionally a third trader enters the market to test wether the best performing model for each day also generates the smallest average profits when trading. The trader is all knowing and used the forecasting algorithm that accumulated the most profit each day.

Table 21 and table 22 report the total profit with pair-wise trading, average profit by trading at own prices and number of algorithms used for Trader 1, Trader 2 and Trader 3. All traders show that they outperformed the models in the above markets, see tables 15 and 17. Accordingly this motivates the switching of algorithms during time. Moreover, the average profits in cents of all traders for 1-day options are small compared to table 18. Nevertheless they are not smaller which could suggest that the market is slightly misinformed about the true price. The traders who trade straddles with a maturity of fifteen days also increases in profit. Still, the average profits of the trader do not increase. This error estimate indicates that the market here is also slightly misinformed.

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6 Conclusion

This paper adopts the technique of Engle et al. (1990) to compare volatility forecasts of hybrid neural networks based on GARCH models. Each hybrid model and also the GARCH model which forecast they use, participate in the hypothetical option market. The agents trade on the believe that there volatility forecast better represents the underlying volatility behaviour of the derivative.

The models have been evaluated for trading 1-day and 15-day options on 1$ shares of the S&P500 index for the time period of 06/19/14 - 03/28/18. For 1 day options the best performing model is the hybrid neural network based on GARCH(1,1) trained with the 5-minute realized return variance estimator, overall this is a good estimator so set as target variable for the hybrid models. The hybrid-GARCH(2,2)model trained on the Parzen kernel estimator shows to accumulates the most profit by trading straddles with a maturity of fifteen days. In general the kernels are performing better in the 15-day option market compared to the 1-day. On the toher hand, the daily realized returns variance estimator performed the worst while set as target variable. It can be concluded that realized kernel variance estimators are not a good target variable for pricing 1-day options. Nonetheless, these kernel estimators show better performance in pricing options with a maturity of fifteen days.

The average profits of the agents while trading at own prices showed that the results were in line with the total profit generated form trading with each other. However mostly in the higher rankings, the group of hybrid models trained on a variance estimator and GARCH models change ranking.

Furthermore the results of Trader 1 and Trader 2 indicate that switching forecasting method in time generates more profit. Although, the average forecasting error(average profit trading at own prices) is increasing. As a result, the assumption that switching between algorithms during time lead to more profit need to be studied more carefully in

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a market in which there are better informed agents.

The overpricing agents in the 15-day option market should be addressed. Whereas, other option pricing formula could be used or adjusting the volatility forecasts downwards. Also while comparing the mean squared errors and the profits(losses) the enhanced hybrid models(increasing compared to the GARCH model) did not always led to generating more profit.

Subsequent research could include to estimate the upward bias of the 15-day ahead variance forecasts and testing with other option pricing formulas to overcome the over-pricing for the longer maturity options. In addition one could use other proxies which describes the unobservable volatility behaviour better to train the hybrid models.

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References

[1] Andersen, T., Bollerslev, T., Diebold, F.X. and Labys, P. (2003). Modelling and fore-casting realized volatility. Econometrica, Vol. 71, 2003 529-626

[2] Bandi,F. M. and Russel,J. R. (2008). Microstructure Noise, Realized Variance, and Optimal Sampling. The Review of Economic Studies, Volume 75, Issue 2, 1 April 2008 339–369

[3] Bandi,F.M., Russel,J.R. and Yang,C. R. (2008). Realized volatility forecasting and option pricing. Journal of Econometrics, Volume 147, Issue 1, november 2008 339–369

[4] Bishoph (2006). Machine learning and pattern recognition. literature used chapter 5 115- 133

[5] Black,F. Scholes,M. (1973). The pricing of options and corporate liabilities. The Jour-nal of Political Economy, vol. 81, No.3, 1973 637-654

[6] Bollerslev,T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of econometrics, vol. 31, No.3, April 1986 307-327

[7] Engle,R.F. (1982). Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation. Econometrica, Vol. 50, No.4, July 1982 987-1007

[8] Engle,R., Hong,C. and Kane,A. (1990). Valuation of variance forecasts with simulated option markets. Advances in Futures and Options Research, Vol. 6, 1992 393-416

[9] Glosten,L.R.Jagannathan.R and Runkle,D.E. (1993). On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks. Journal of Finance, vol. 48, No.5, 1993, 1779-1801

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[10] Hajizadeh,E., Seifi,A., Zarandi,M.H. and Turken, I.B. (2012). A hybrid modeling approach for forecasting the volatility of S&P 500 index return. Expert Systems with Applications 39, 2012 431-436

[11] McCulloch,W.S., Pitts,W.H. (1943). A logical Calculus of the ideas immanent in nervous activity. Bulletin of Mathematical Biophysics, Vol. 5, 1943 115- 133

[12] Nelson,D.B. (1991). Conditional Heteroskedasticity in Asset Returns: A New Ap-proach. Econometrica, vol. 59, No.2, 1991 347-370

[13] Nielsen,O.E., Hansen,P.R. Lunde,A. and Shephard,N. (2008). Desinign realized ker-nels to measure the es post variation of equity prices in the presence of noise. Econo-metrica, Vol. 76, No.6, November 2008 1481–1536

[14] Rummelhart,D.E., Hinton,G.E. and Williams,R.J. (1986). Learning representation by backpropagation errors. Nature, vol. 323, 9 October 1986 115- 133

[15] Tieleman,T. and Hinton,G. (2012). Divide the gradient by a running average of its recent magnitude. COURSERA: Neural Networks for Machine Learning, 6,a:e, 2012.

[16] Zakoian,J. (1994). Threshold heteroskedastic models. Journal of Economic Dynamics and Control, vol. 18, No.5, 1994 931-955

[17] Zhang,L., Mykland,A. and A¨ıt-Sahalia, Y. (2005). A Tale of Two Time Scales: Deter-mining Integrated Volatility With Noisy High-Frequency Data. Journal of the Ameri-can Statistical Association, Vol. 100, No.472, December 2005 1481–1536

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Appendix

GARCH EGARCH GJR-GARCH

Estimator (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Daily realized variance 3.63e-06 5.34e-06 3.43e-06 1.94e-06 1.87e-06 -1.70e-07 -1.03e-07 1.45e-06 2.16e-06

5-minute returns 5.42e-06 4.85e-06 5.36e-06 3.94e-06 3.60e-06 4.28e-06 4.40e-06 5.06e-06 5.09e-06 5-minute sub-sampled returns 4.92e-06 4.90e-06 4.83e-06 3.97e-06 3.37e-06 4.18e-06 4.52e-06 4.68e-06 4.82e-06 Parzen kernel 4.88e-06 5.22e-06 5.59e-06 3.93e-06 4.26e-06 4.79e-06 5.47e-06 4.68e-06 5.45e-06 Two-scale/Bartlett Kernel 8.26e-06 8.01e-06 8.31e-06 7.96e-06 8.42e-06 8.42e-06 9.07e-06 9.54e-06 9.72e-06 Tukkey-Hanning kernel2 8.30e-06 8.14e-06 8.43e-06 8.22e-06 8.50e-06 9.05e-06 9.37e-06 9.80e-06 1.00e-05

Table 1: Difference in mean squared errors between a hybrid model and their counterpart GARCH model for 1-day ahead forecasts.

Estimator (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Daily realized variance 1.66e-05 1.54e-05 1.67e-05 2.04e-05 2.14e-05 2.16e-05 2.06e-05 2.10e-05 2.22e-05

5-minute returns 1.36e-05 1.31e-05 1.34e-05 1.62e-05 1.66e-05 1.73e-05 1.75e-05 1.79e-05 1.79e-05 5-minute sub-sampled returns 1.32e-05 1.35e-05 1.36e-05 1.53e-05 1.52e-05 1.69e-05 1.77e-05 1.75e-05 1.78e-05 Parzen kernel 1.31e-05 1.27e-05 1.32e-05 1.51e-05 1.60e-05 1.68e-05 1.58e-05 1.75e-05 1.81e-05 Two-scale/Bartlett Kernel 1.40e-05 1.31e-05 1.41e-05 1.47e-05 1.62e-05 1.65e-05 1.87e-05 1.83e-05 1.90e-05 Tukkey-Hanning kernel2 1.33e-05 1.40e-05 1.37e-05 1.69e-05 1.66e-05 1.71e-05 1.82e-05 1.86e-05 1.89e-05

Table 2: Difference in mean squared errors between a hybrid model and their counterpart GARCH model for 15-day ahead forecasts.

(1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Hybrid model -5.53 -5.01 -6.59 -2.59 -5.99 -7.27 -3.14 -0.71 -3.33 GARCH model 0.97 -0.86 -1.20 9.10 8.54 5.11 8.25 4.61 5.64

Table 3: Total profit from trading 1 day options using the daily realized variance estimator as target variable

(1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Hybrid model 4.11 -1.38 0.87 3.86 -13.18 2.09 3.37 -4.70 -5.32 GARCH model -1.38 -3.25 -3.32 4.27 4.45 1.73 4.19 1.40 2.19

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GARCH EGARCH GJR-GARCH (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Hybrid model 4.08 -6.27 2.43 -0.70 -12.20 2.18 1.28 -9.13 1.22 GARCH model -0.57 -2.09 -1.88 4.69 4.87 2.12 4.84 2.25 2.89

Table 5: Total profit from trading 1 day options using the 5-minute sub-sampled realized variance estimator as target variable

(1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Hybrid model -0.03 -0.07 4.46 -4.77 -7.49 1.76 -5.53 -14.44 -1.50 GARCH model 0.54 -0.76 -0.48 5.71 5.69 3.29 5.97 3.45 4.20

Table 6: Total profit from trading 1 day options using the Parzen realized kernel variance estimator as target variable

(1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Hybrid model 3.75 -11.26 1.73 -6.57 0.67 -15.22 -4.67 -2.70 -7.26 GARCH model 2.47 1.19 1.05 7.14 7.04 4.84 7.32 4.87 5.60

Table 7: Total profit from trading 1 day options using the Two-scale/barlett realized kernel variance estimator as target variable

(1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Hybrid model -6.76 -2.37 4.34 -13.47 -0.41 -9.78 2.40 -12.71 -2.08 GARCH model 2.45 1.03 0.77 7.26 7.01 4.87 7.27 4.70 5.47

Table 8: Total profit from trading 1 day options using the Tukkey-Hanning2realized kernel variance estimator as

target variable

(1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Hybrid model -46.66 219.09 -41.11 107.44 118.86 102.25 191.01 -93.74 58.12 GARCH model -165.35 -129.05 -138.93 40.89 74.01 66.21 -93.14 -120.68 -146.76

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GARCH EGARCH GJR-GARCH (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Hybrid model 102.71 40.72 215.66 177.31 4.49 116.96 57.96 221.86 48.62 GARCH model -202.24 -164.97 -175.09 -8.11 22.89 13.71 -129.12 -156.99 -183.52

Table 10: Total profit from trading 15 day options using the 5-minute realized variance estimator as target variable

(1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Hybrid model 222.95 170.68 203.10 26.88 -2.74 70.50 158.05 55.59 67.12 GARCH model -200.37 -164.22 -174.41 -5.65 26.37 18.10 -129.67 -157.17 -183.46

Table 11: Total profit from trading 15 day options using the 5-minute sub-sampled realized variance estimator as target variable

(1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Hybrid model 199.63 232.18 159.55 37.51 50.38 90.86 -12.20 70.07 145.94 GARCH model -200.95 -164.82 -174.71 -6.29 25.15 16.28 -129.50 -156.06 -182.44

Table 12: Total profit from trading 15 day options using the Parzen realized kernel variance estimator as target variable

(1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Hybrid model 164.34 74.61 159.37 4.02 58.02 54.32 193.84 102.21 220.14 GARCH model -208.27 -172.94 -182.57 -12.41 19.58 12.95 -135.46 -162.47 -188.82

Table 13: Total profit from trading 15 day options using the Two-scale/Barlett realized kernel variance estimator as target variable

(1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Hybrid model 78.60 165.02 217.87 140.07 14.11 44.74 70.88 135.80 177.50 GARCH model -209.61 -172.92 -182.53 -15.16 16.44 5.81 -135.18 -162.03 -188.81

Table 14: Total profit from trading 15 day options using the Tukkey-Hanning2realized kernel variance estimator

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GARCH EGARCH GJR-GARCH Estimator (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Daily realized variance -7.68 -4.34 -15.72 -0.43 -9.82 -15.80 -4.70 -0.45 -2.51 5-minute returns 43.07 27.43 38.21 41.24 -16.13 34.23 40.80 23.05 20.98 5-minute sub-sampled returns 36.47 4.33 36.96 27.19 -18.40 32.34 30.58 -4.56 33.40 Parzen kernel 8.20 11.30 26.97 -4.48 -18.80 18.48 -7.03 -47.71 7.31 Two-scale/Bartlett kernel -4.30 -46.97 -11.15 -32.67 -13.52 -57.85 -29.71 -22.56 -36.29

Tukkey-Hanning2kernel -40.74 -23.07 1.23 -60.53 -18.69 -48.92 -11.61 -57.61 -26.58 GARCH family 13.07 9.79 10.30 25.06 26.56 21.95 25.95 22.14 22.69

Table 15: Total profit from trading 1 day options for each model competing against each other

Estimator (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Daily realized variance 40 35 45 31 41 46 38 32 33

5-minute returns 1 11 4 2 47 7 3 17 21

5-minute sub-sampled returns 6 29 5 12 48 9 10 37 8 Parzen kernel 27 24 13 36 50 22 39 59 28 Two-scale/Bartlett kernel 34 58 42 55 44 62 54 51 56 Tukkey-Hanning2kernel 57 52 30 63 49 60 43 61 53

GARCH family 23 26 25 16 14 20 15 19 18

Table 16: Ranking from trading 1 day options for each model competing against each other

Estimator (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Daily realized variance -685.18 304.71 -703.24 -317.32 -291.58 -302.18 131.10 -809.20 -422.55

5-minute returns 207.12 40.54 762.25 551.61 -106.37 275.41 69.18 795.56 55.39 5-minute sub-sampled returns 751.46 559.69 624.12 -316.08 -481.21 -117.27 528.01 -139.94 -100.75

Parzen kernel 557.24 829.89 361.77 -254.61 -211.25 54.02 -435.14 -35.87 330.65 Two-scale/Bartlett kernel 357.56 -165.44 415.46 -444.00 -218.37 -285.30 561.04 3.44 753.36 Tukkey-Hanning2kernel 49.18 523.98 814.73 439.35 -180.24 -77.71 130.68 375.41 577.90 GARCH family -738.20 -707.11 -717.25 -449.22 -406.28 -407.87 -700.93 -725.30 -762.28

Table 17: Total profit from trading 15 day options for each model competing against each other

Estimator (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Daily realized variance 55 21 57 47 44 45 24 63 50

5-minute returns 23 30 4 12 35 22 26 3 27 5-minute sub-sampled returns 6 10 7 46 54 36 13 37 34

Parzen kernel 11 1 18 42 40 28 51 32 20

Two-scale/Bartlett kernel 19 38 16 52 41 43 9 31 5 Tukkey-Hanning2kernel 29 14 2 15 39 33 25 17 8

GARCH family 61 58 59 53 48 49 56 60 62

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GARCH EGARCH GJR-GARCH Estimator (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Daily realized variance -0.12329 -0.12431 -0.16347 -0.11623 -0.12128 -0.11422 -0.12330 -0.12833 -0.12127

(· 10−3) (2.18) (1.93) (2.19) (2.00) (2.03) (2.12) (2.19) (2.00) (2.07) 5-minute returns +0.088413 +0.090414 +0.098618 +0.091615 +0.172451 +0.073710 +0.083811 +0.125132 +0.128334

(· 10−3) (2.69) (2.76) (2.69) (2.71) (2.67) (2.77) (2.73) (2.71) (2.69) 5-minute sub-sampled returns +0.096117 +0.148840 +0.096016 +0.129035 +0.174653 +0.086912 +0.100919 +0.164748 +0.105120

(· 10−3) (2.71) (2.70) (2.75) (2.65) (2.68) (2.77) (2.73) (2.67) (2.75) Parzen kernel ++0.11321 +0.13536 +0.11824 +0.14739 +0.18154 +0.11925 +0.16145 +0.20162 +0.12026 (· 10−3) (2.82) (2.75) (2.73) (2.77) (2.68) (2.77) (2.67) (2.67) (2.77) Two-scale/Bartlett kernel +0.16346 +0.19258 +0.15944 +0.18656 +0.15342 +0.19960 +0.16649 +0.16850 +0.18555 (· 10−3) (2.73) (2.73) (2.73) (2.73) (2.79) (2.77) (2.79) (2.78) (2.70) Tukkey-Hanning2kernel +0.19057 +0.14638 +0.14537 +0.20763 +0.15343 +0.19459 +0.14941 +0.20161 +0.17452 (· 10−3) (2.74) (2.86) (2.80) (2.71) (2.83) (2.71) (2.81) (2.73) (2.76) GARCH family -0.05766 -+0.05715 -0.05654 -0.05503 -0.05482 -0.05101 -0.06299 -0.06057 -0.06168 (· 10−3) (2.83) (2.84) (2.84) (2.67) (2.68) (2.72) (2.70) (2.69) (2.71)

Table 19: 1day: Average profits(losses) in cents by trading at own prices, ranking is presented bold and the profit variance is in parentheses.

Estimator (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) (1,1) (2,2) (3,3) Daily realized variance -1.9859 -1.5924 -1.9658 -1.8048 -1.8149 -1.8250 -1.6429 -2.0162 -1.8563

(· 10−2) (7.64) (8.55) (7.63) (7.93) (7.89) (7.90) (8.42) (7.54) (7.87) 5-minute returns -1.6733 -1.6836 -1.5212 -1.5823 -1.7345 -1.6530 -1.6937 -1.5213 -1.7040

(· 10−2) (7.84) (7.77) (8.26) (7.99) (7.62) (7.83) (7.78) (8.18) (7.73) 5-minute sub-sampled returns -1.5316 -1.5925 -1.5619 -1.7243 -1.7646 -1.7141 -1.5927 -1.7244 -1.7039

(· 10−2) (8.10) (8.08) (8.12) (7.65) (7.46) (7.66) (8.07) (7.66) (7.75) Parzen kernel -1.5517 -1.458 -1.5821 -1.6835 -1.6732 -1.6938 -1.8554 -1.6734 -1.6228 (· 10−2) (8.11) (8.48) (8.08) (7.74) (7.74) (7.68) (7.39) (7.81) (7.96) Two-scale/Bartlett kernel -1.447 -1.5314 -1.4710 -1.7142 -1.5618 -1.6531 -1.435 -1.5720 -1.362 (· 10−2) (8.36) (8.12) (8.28) (7.66) (7.95) (7.78) (8.44) (8.05) (8.71) Tukkey-Hanning2kernel -1.4711 -1.436 -1.331 -1.434 -1.5926 -1.5822 -1.5315 -1.469 -1.413 (· 10−2) (8.25) (8.45) (8.77) (8.42) (7.94) (7.96) (8.16) (8.30) (8.50) GARCH family -1.9657 -1.9555 -1.9555 -1.8352 -1.8251 -1.8047 -2.0163 -2.0160 -2.0161 (· 10−2) (7.13) (7.16) (7.15) (7.31) (7.32) (7.40) (6.97) (7.00) (6.99)

Table 20: 15day: Average profits(losses) in cents by trading at own prices, ranking is presented bold and the profit variance is in parentheses.

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Total profit Average profit Number of algorithms

Trader 1 44.10 0.0907 10 Trader 2 44.43 0.0641 63 Trader 3 (163.05) 0.0782 63

Table 21: Summary Trader 1,2 and 3 trading 1-day options. Total profits of pair wise trading in dollars, average profits reported in cents.

Total profit Average profit Number of algorithms

Trader 1 824.39 -1.55 11 Trader 2 1295.26 -1.57 61 Trader 3 (1392.00) -1.58 61

Table 22: Summary Trader 1,2 and 3 trading 15-day options. Total profits of pair wise trading in dollars, average profits reported in cents.

Evaluating neural networks in a hypothetical option market by training on multiple variance estimators

option market by training on multiple variance

estimators.

Statement of originality

Contents

1

Introduction

2

Methodology

3

Hypothetical option market

4

Neural network set up and data description

5

Results

6

Conclusion

References

Appendix