Deep Calibration With Artificial Neural Network: A Performance
  Comparison on Option Pricing Models
This paper explores Artificial Neural Network (ANN) as a model-free solution
for a calibration algorithm of option pricing models. We construct ANNs to
calibrate parameters for two well-known GARCH-type option pricing models:
Duan's GARCH and the classical tempered stable GARCH that significantly improve
upon the limitation of the Black-Scholes model but have suffered from
computation complexity. To mitigate this technical difficulty, we train ANNs
with a dataset generated by Monte Carlo Simulation (MCS) method and apply them
to calibrate optimal parameters. The performance results indicate that the ANN
approach consistently outperforms MCS and takes advantage of faster computation
times once trained. The Greeks of options are also discussed.
Young Shin Kim, Hyangju Kim, Jaehyung Choi
# Deep Calibration With Artificial Neural Network: A Performance Comparison on Option Pricing Models

###### Abstract

This paper explores Artificial Neural Network (ANN) as a model-free solution for a calibration algorithm of option pricing models. We construct ANNs to calibrate parameters for two well-known GARCH-type option pricing models: Duan's GARCH and the classical tempered stable GARCH that significantly improve upon the limitation of the Black-Scholes model but have suffered from computation complexity. To mitigate this technical difficulty, we train ANNs with a dataset generated by Monte Carlo Simulation (MCS) method and apply them to calibrate optimal parameters. The performance results indicate that the ANN approach consistently outperforms MCS and takes advantage of faster computation times once trained. The Greeks of options are also discussed.

_JEL classification:_ C15, C63, C65, G130

keywords: Deep calibration, Artificial Neural Network (ANN), Feedforward Neural Network (FNN), Option pricing models, GARCH model, Duan's GARCH model, Tempered stable model +
## 1 Introduction

Option pricing has been a centric topic in quantitative finance for the past few decades, with remarkable growth in option market. Since the seminal work of Black and Scholes (1973) and Merton (1973), the Black-Scholes model has remained the most fundamental model for option pricing. However, its restrictive assumptions, such as constant volatility or Geometric Brownian Motion (GBM), have been criticized for not reflecting the empirical characteristics of financial markets.

Many subsequent models have since been proposed to relax the assumptions of the Black-Scholes model. One successful approach is employing stochastic volatility under the generalized autoregressive conditional heteroskedastic (GARCH) framework. The early attempt was introduced by Engle and Mustafa (1992) focusing on implied conditional volatilities. Subsequently, Duan (1995) developed a more rigorous framework of the GARCH option pricing model using the locally risk-neutral valuation relationship that one-period ahead conditional variance remains constant under both the risk-neutral measure and the physical measure. The model extended later to the jump-diffusion model in Duan _et al._ (2004) and Duan _et al._ (2006). Duan's GARCH model assumes that the residuals follow a normal distribution, however, empirical evidence shows that the assumption of a normally distributed residual in the model does not properly describe asset return dynamics (Duan (1999) and Menn and Rachev (2009)).

Another approach, known as the Levy distribution, has been developed allowing for jumps, skewness, kurtosis, and heavy tails in underlying distribution to overcome the flaws of the GBM assumption. It is also referred to as a stable distribution, and its one well-known subclass is the classical tempered stable (CTS) distribution.1 The CTS distribution has been researched in several directions such as portfolio management (Tsuchida _et al._ (2012); Beck _et al._ (2013); Georgiev _et al._
(2015); Anand _et al._ (2016); Choi _et al._ (2021)) and momentum strategy (Choi _et al._ (2015)).

As a consolidation of these two approaches in the context of option pricing, Kim _et al._ (2008a), Kim _et al._ (2010a) and Kim _et al._ (2022) enhanced Duan's GARCH model by incorporating the classical tempered stable (CTS) distribution 2 and referred to it as the CTS-GARCH model. The CTS-GARCH model, which takes into account the non-normality in its innovation process, is considered one of the most advanced option pricing models as it addresses two key limitations of the Black-Scholes model simultaneously (Kim _et al._ (2010b)).

Footnote 2: The CTS distribution has been studied under different names including the _truncated Lévy flight_ by Koponen (1995), the _tempered stable_ by Barndorff-Nielsen and Levendorskii (2001), Barndorff-Nielsen and Shephard (2001) and Cont and Tankov (2004), the _KoBoL_ distribution by Boyarchenko and Levendorskii (2000), and the _CGMY_ by Carr _et al._ (2002). The KR distribution of Kim _et al._ (2008b) is an extension of the CTS distribution. Rosinski (2007) generalized CTS distribution, referred to as the tempered stable distribution.

However, these models are mostly high-dimensional, and this becomes problematic in the stage of parameter calibration; parameters need to be tuned, so as to find the best model output closest to the market price. Hence the Black-Scholes model still has been a practically useful model even with the superior performance of multidimensional models.

Recently, machine learning has shed a different light on the curse of dimensionality as a non-parametric or model-free solution. This is a rapidly evolving area with the maturation of computation power and technical advances in algorithms.

One of the main pillars of machine learning is Artificial Neural Network (ANN). ANN, also referred to as neural network or multilayer perceptions, has been introduced by McCulloch and Pitts (1943). ANNs are composed of the input layer, multiple hidden layers, and the output layer where each hidden layer performs a vector-to-vector or vector-to-scalar calculation to make the best output layer approximation. The parallel structure of hidden layers allows using multiprocessors, hence ANN is considerably faster than traditional algorithms at computation speed. With this advantage, ANN has been developed rapidly and different types of ANN have been proposed such as multilayer perceptron neural network, convolutional neural network, radial basis function neural network, recurrent neural network, modular neural network, and so on.
Many studies in finance also sought to find its ANN applications. In particular, ANN receives attention as a promising alternative for chronic computational difficulties in option pricing models as it is built upon multidimensional nonlinear models. The early attempts were applying ANN as a functional approximator for the Black-Scholes formula suggested by Malliaris and Salchenberger (1993) and Hutchinson _et al._ (1994). Since then, over a hundred papers have studied ANN for option pricing with various parameter inputs and performance measures (Ruf and Wang (2019)).

While most of the studies utilize ANN to estimate the option price based on parameters from the market data and measure their performance by out-of-sample tests, a different approach was proposed by Bayer _et al._ (2019) and Alaya _et al._ (2021), recently. Rather than focusing on the approximation of the pricing formula, ANN is applied to the calibration stage and finds model parameters in this approach. Since training sets are generated by Monte Carlo Simulation (MCS) method, hence a more sound dataset avoiding market incompleteness is available. Utilizing enough large datasets is another advantage.

This paper also focuses on ANN as an alternative calibration method. We generate training sets having 100,000 simulated vectors of parameters and prices using the MCS method for each of Duan's GARCH and CTS-GARCH models, and then train ANNs that consist of three hidden layers with twenty nodes for each layer. Specifically, a feedforward neural network is used, which does not include any cycles or loops. The trained ANNs are utilized in the calibration for S&P 500 index call and put option prices from every second Wednesday of each month between June 2021 and May 2022. We find that this approach not only presents a superior performance to the previous MCS method but performs remarkably faster. Additionally, the option Greeks are also computed by using the ANN method.

The remainder of this paper is organized as follows. We discuss the GARCH option pricing models in Section 2. Section 3 presents the construction and training methods for ANN. The performance of ANN in terms of calibration is investigated using empirical data in Section 4. Finally, Section 5 provides our conclusions.
## 2 Preliminaries

In this section, we revisit Duan's GARCH and CTS-GARCH option pricing models that feature _infinitely divisible_ innovations. An infinitely divisible random variable is a random variable that can be represented as an infinite sum of independent, identically distributed random variables. This concept plays a significant role in the study of stable distributions and option pricing models.

We begin by reviewing the CTS distribution as an example of an infinitely divisible distribution, followed by an overview of Duan's GARCH and CTS-GARCH option pricing models. Lastly, we outline European call and put option pricing using the MCS method for both GARCH option pricing models.

### CTS Distribution

\(X\) is referred to as the _Classical Tempered Stable distributed random variable_ and denoted by \(X\sim\text{CTS}(\alpha\), \(C\), \(\lambda_{+}\), \(\lambda_{-}\), \(m)\)(Rachev _et al._ (2011) and Kim _et al._ (2010b)) if the characteristic function of the distribution is expressed as

\[\begin{split}& E[e^{iuX}]=\phi_{\text{CTS}}(u;\alpha,C,\lambda_{+}, \lambda_{-},m)\\ &=\exp\left((m-C\Gamma(1-\alpha)(\lambda_{+}^{\alpha-1}-\lambda_{ -}^{\alpha-1}))iu-C\Gamma(-\alpha)\left((\lambda_{+}-iu)^{\alpha}-\lambda_{+}^ {\alpha}+(\lambda_{-}+iu)^{\alpha}-\lambda_{-}^{\alpha}\right)\right)\end{split} \tag{1}\]

where \(C_{+},C_{-},\lambda_{+},\lambda_{-}\) are positive, \(0<\alpha<2\), \(m\in\mathbb{R}\) and \(\Gamma\) is the gamma function.

If we substitute \(C=(\Gamma(2-\alpha)(\lambda_{+}^{\alpha-2}+\lambda_{-}^{\alpha-2}))^{-1}\) and \(m=0\) for \(Z\sim\text{CTS}(\alpha\), \(C\), \(\lambda_{+}\), \(\lambda_{-}\), \(m)\), then we have \(E[Z]=0\) and \(\text{var}(Z)=1\). In this case, the random variable \(Z\) is called the _standard CTS distribution_, and \(Z\sim\text{stdCTS}(\alpha\), \(\lambda_{+}\), \(\lambda_{-})\). The characteristic function of \(Z\) is given by

\[\begin{split}& E[e^{iuZ}]=\phi_{\text{stdCTS}}(u;\alpha,\lambda_{+}, \lambda_{-})\\ &=\exp\left(\frac{\lambda_{+}^{\alpha-1}-\lambda_{-}^{\alpha-1}}{ (\alpha-1)(\lambda_{+}^{\alpha-2}+\lambda_{-}^{\alpha-2})}iu+\frac{(\lambda_{+ }-iu)^{\alpha}-\lambda_{+}^{\alpha}+(\lambda_{-}+iu)^{\alpha}-\lambda_{-}^{ \alpha}}{\alpha(\alpha-1)(\lambda_{+}^{\alpha-2}+\lambda_{-}^{\alpha-2})} \right).\end{split} \tag{2}\]
We denote the function of log-Laplace transform of \(Z\) as follows (Kim _et al._ (2010b)):

\[l(x) :=\log\left(\phi_{\text{stdCTS}}(-ix;\alpha,\lambda_{+},\lambda_{-})\right) \tag{3}\] \[=\frac{x(\lambda_{+}^{\alpha-1}-\lambda_{-}^{\alpha-1})}{(\alpha- 1)(\lambda_{+}^{\alpha-2}+\lambda_{-}^{\alpha-2})}+\frac{(\lambda_{+}-x)^{ \alpha}-\lambda_{+}^{\alpha}+(\lambda_{-}+x)^{\alpha}-\lambda_{-}^{\alpha}}{ \alpha(\alpha-1)(\lambda_{+}^{\alpha-2}+\lambda_{-}^{\alpha-2})}.\]

### Duan's GARCH and CTS-GARCH Option Pricing Models

Let \((S_{t})_{t\in\{0,1,\cdots,T^{*}\}}\) be the underlying asset price process and \((y_{t})_{t\in\{0,1,2,\cdots,T^{*}\}}\) be the underlying asset log return process, where \(y_{t}=\log(\frac{S_{t}}{S_{t-1}})\) with \(y_{0}=0\), and \(T^{*}<\infty\) in the time horizon. Under the physical measure \(\mathbb{P}=\bigoplus_{t=1}^{T^{*}}\mathcal{P}_{t}\), \((y_{t})_{t\in\{0,1,2,\cdots,T^{*}\}}\) is supposed to follow the GARCH model:

\[\begin{cases}y_{t+1}=\mu_{t+1}+\sigma_{t+1}\epsilon_{t+1}\\ \sigma_{t+1}^{2}=\kappa+\xi\sigma_{t}^{2}\epsilon_{t}^{2}+\zeta\sigma_{t}^{2} \end{cases}\quad\text{ for }t\in\{0,1,2,\cdots,T^{*}-1\}, \tag{4}\]

where \(\mu_{t+1}\) is the daily expected return and \(\epsilon_{t+1}\) follows an infinitely divisible distribution. \(S_{0}\), \(\epsilon_{0}\) and \(\sigma_{0}\) are real constants with \(\epsilon_{0}=0\). The GARCH model involves parameters with \(\kappa\), \(\xi\), \(\zeta\) where \(\xi+\zeta<1\).

We next define \((R_{t})_{t\in\{1,2,\cdots,T^{*}\}}\) and \((d_{t})_{t\in\{1,2,\cdots,T^{*}\}}\) as the sequences of the daily risk-free rate of return and daily dividend rate of the underlying, respectively. Then, there is a risk-neutral measure \(\mathbb{Q}=\bigoplus_{t=1}^{T^{*}}\mathcal{Q}_{t}\) such that:

* \(\eta_{t+1}=\theta_{t+1}+\epsilon_{t+t}\), where \(\theta_{t+1}=\frac{\mu_{t+1}-R_{t+1}+d_{t+1}+w_{t+1}}{\sigma_{t+1}}\) is the market price of risk and \(\omega_{t+1}=\log E_{\mathbb{Q}}\left[e^{\sigma_{t+1}\epsilon_{t+1}}\right]\).
* \(\eta_{t+1}\) is also infinitely divisible under the measure \(\mathbb{Q}\).

By applying change of measures to Eq. (4), we obtain the risk-neutral price process under \(\mathbb{Q}\) as

\[\begin{cases}y_{t+1}=R_{t+1}-d_{t+1}-\omega_{t+1}+\sigma_{t+1}\eta_{t+1}\\ \sigma_{t+1}^{2}=\kappa+\xi\sigma_{t}^{2}(\eta_{t}-\theta_{t})^{2}+\zeta\sigma_ {t}^{2}\end{cases}\quad\text{ for }t\in\{0,1,2,\cdots,T^{*}-1\} \tag{5}\]
with \(\xi+\zeta<1\). To simplify the condition \(\xi+\zeta<1\), we define two parameters \(\psi\) and \(\gamma\) as \(\psi=\frac{\xi}{\zeta}\) and \(\gamma=\xi+\zeta\), respectively. Then we have

\[\sigma_{t}^{2}=\kappa+\frac{\gamma}{\psi+1}\left(\psi\sigma_{t-1}^{2}(\eta_{t-1 }-\theta_{t})^{2}+\sigma_{t-1}^{2}\right), \tag{6}\]

where \(\kappa,\psi,\gamma>0\), \(\eta_{0}=0\) and \(\sigma_{0}>0\).

Under the risk-neutral measure \(\mathbb{Q}\), the underlying asset price is defined as \(S_{t}=S_{0}e^{\sum_{j=1}^{t}y_{j}}\) for \(t\in\{1\), \(2\), \(\cdots\), \(T-1\), \(T\), \(\cdots\), \(T^{*}\}\). The European option with a payoff function \(H(S(T))\) at the maturity \(T\) with \(t\leq T\leq T^{*}\) is given by

\[E_{\mathbb{Q}}\left[e^{-\sum_{j=t+1}^{T}R_{j}}H(S(T))\ \bigg{|}\ \mathcal{F}_{t} \right]=E_{\mathbb{Q}}\left[e^{-\sum_{j=t+1}^{T}R_{j}}H(S_{t}e^{\sum_{j=t+1} ^{T}y_{j}})\ \bigg{|}\ \mathcal{F}_{t}\right]. \tag{7}\]

For example, European vanilla call and put price with strike price \(K\) and time to maturity \(T\) at time \(t=0\) are

\[\text{(Call)}=E_{\mathbb{Q}}\left[e^{-\sum_{j=1}^{T}R_{j}}\max\{S_{0}e^{\sum_{ j=1}^{T}y_{j}}-K,0\}\right] \tag{8}\]

and

\[\text{(Put)}=E_{\mathbb{Q}}\left[e^{-\sum_{j=1}^{T}R_{j}}\max\{K-S_{0}e^{\sum _{j=1}^{T}y_{j}},0\}\right], \tag{9}\]

respectively.

Moreover, let \(m\) be the moneyness defined as \(m=\frac{K}{S_{0}e^{\sum_{l=1}^{T}R_{t}}}\), then we have

\[\text{(Call)}=S_{0}V_{C}\text{ and (Put)}=S_{0}V_{P}, \tag{10}\]
where

\[V_{C}=E_{\mathbb{Q}}\left[\max\left\{\exp\left(\sum_{t=1}^{T}-d_{t}-w_{t}+\sigma_{t }\eta_{t}\right)-m,0\right\}\right] \tag{11}\]

and

\[V_{P}=E_{\mathbb{Q}}\left[\max\left\{m-\exp\left(\sum_{t=1}^{T}-d_{t}-w_{t}+ \sigma_{t}\eta_{t}\right),0\right\}\right]. \tag{12}\]

Now, we present two popular examples of the GARCH option pricing model with infinitely divisible innovations:

* _Duan's GARCH Model_(Duan (1995)): If we assume that \(\eta_{t}\)'s follow the standard Gaussian distribution, which is infinitely divisible, we obtain the Duan's GARCH option pricing model where \(w_{t}=\frac{\sigma_{t}^{2}}{2}\).
* _CTS-GARCH Model_(Kim _et al._ (2010a)): By assuming that \(\eta_{t}\)'s follow the standard CTS distribution, which is also infinitely divisible, we refer to the GARCH model as the CTS-GARCH option pricing model. Specifically, when we set \(\eta_{t}\sim\text{stdCTS}(\alpha,\lambda_{+},\lambda_{-})\) for all \(t\in\{1,2,\cdots\}\) under the measure \(\mathbb{Q}\), we have \[w_{t}=\log\left(\phi_{\text{stdCTS}}(-i\sigma_{t};\alpha,\lambda_{+},\lambda_ {-})\right)=l(\sigma_{t})\] (13) by Eq. (3).

### European Call and Put Option Pricing with the MCS Method

To simplify the model, we make the following assumptions for the remainder of this paper: the daily risk-free return \(R_{t}\) is a constant \(R\), \(\theta_{t}\) is a constant \(\theta\), and \(d_{t}=0\). We define \(t\in\{1\), \(2\), \(\cdots\), \(T\), \(\cdots\), \(T^{*}\}\) for \(0<T\leq T^{*}\) as a set of the time steps, where one step represents one business day. We also assume that there are 250 business days in a year, and we define the annual risk-free return as \(r=250\cdot R\). The year-fraction time is denoted by \(\tau=\frac{T}{250}\), and we define \(m\) as the moneyness,
where \(m=\frac{Ke^{-RT}}{S_{0}}=\frac{Ke^{-r\tau}}{S_{0}}\).

Based on these assumptions, we can generate a set of infinitely divisible random numbers \(\{\eta_{t,n}:t\in\{\)1,2,\(\cdots\), \(T^{*}\}\), \(n\in\{\)1, 2, \(\cdots\), \(N\}\}\) using the MCS method. Next, we apply the GARCH model to obtain the set of volatility \(\{\sigma_{t,n}:t\in\{\)1, 2, \(\cdots\), \(T^{*}\}\), \(n\in\{\)1, 2, \(\cdots\), \(N\}\}\) defined as

\[\sigma_{t,n}=\sqrt{\kappa+\frac{\gamma}{\psi+1}\left(\psi\sigma_{t-1,n}^{2}( \eta_{t-1}-\theta)^{2}+\sigma_{t-1,n}^{2}\right)}, \tag{14}\]

where \(\kappa,\psi,\gamma>0\), \(\eta_{0}=0\) and \(\sigma_{0,n}=\sigma_{0}\) for a constant \(\sigma_{0}\).

For a given time to maturity \(T\leq T^{*}\) and a moneyness \(m\), we approximate \(V_{C}\) and \(V_{P}\) as follows:

\[V_{C}\approx\hat{V}_{C}=\frac{1}{N}\sum_{n=1}^{N}\max\left\{\exp\left(\sum_{t =1}^{250\tau}-w_{t,n}+\sigma_{t,n}\eta_{t,n}\right)-m,0\right\} \tag{15}\]

and

\[V_{P}\approx\hat{V}_{P}=\frac{1}{N}\sum_{n=1}^{N}\max\left\{m-\exp\left(\sum_{ t=1}^{250\tau}-w_{t,n}+\sigma_{t,n}\eta_{t,n}\right),0\right\}. \tag{16}\]

From (15) and (16), call and put option values in Duan's GARCH and CTS-GARCH models can be calculated as follows:

* _Duan's GARCH model:_ We generate a set of standard Gaussian random numbers \(\{\eta_{t,n}:t\in\{\)1,2,\(\cdots\),\(T^{*}\}\), \(n\in\{\)1,2,\(\cdots\),\(N\}\}\) and set \(w_{t,n}=\frac{\sigma_{t,n}^{2}}{2}\). Then we obtain call and put option prices of Duan's GARCH model. In this case, we denote (15) and (16) as \[V_{C}^{Duan}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{0})=\hat{V}_{C}\quad\text {and}\quad V_{P}^{Duan}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{0})=\hat{V}_{ P},\] (17)
respectively. The call and put option prices can be approximated using MCS as \[\text{Call}^{Duan}(S_{0},K,\tau,r)\approx S_{0}V_{C}^{Duan}(m,\tau;\kappa,\psi, \gamma,\theta,\sigma_{0})\] (18) and \[\text{Put}^{Duan}(S_{0},K,\tau,r)\approx S_{0}V_{P}^{Duan}(m,\tau;\kappa,\psi, \gamma,\theta,\sigma_{0}).\] (19)
* _CTS-GARCH model:_ We simulate a set of standard CTS random numbers \(\{\eta_{t,n}:t\in\{1,2,\cdots,T^{*}\}\), \(n\in\{1,2,\cdots,N\}\}\) with parameters \((\alpha\), \(\lambda_{+}\), \(\lambda_{-})\). By setting \(w_{t,n}=l(\sigma_{t,n})\), we obtain call and put option prices of the CTS-GARCH model. Specifically, we use (15) and (16) to denote the call and put option values as \[V_{C}^{CTS}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{0},\alpha,\lambda_{+}, \lambda_{-})=\hat{V}_{C}\quad\text{and}\quad V_{P}^{CTS}(m,\tau;\kappa,\psi, \gamma,\theta,\sigma_{0},\alpha,\lambda_{+},\lambda_{-})=\hat{V}_{P},\] (20) respectively. Using MCS, we can obtain call and put option prices under the CTS-GARCH model such that \[\text{Call}^{CTS}(S_{0},K,\tau,r)\approx S_{0}V_{C}^{CTS}(m,\tau;\kappa,\psi, \gamma,\theta,\sigma_{0},\alpha,\lambda_{+},\lambda_{-})\] (21) and \[\text{Put}^{CTS}(S_{0},K,\tau,r)\approx S_{0}V_{P}^{CTS}(m,\tau;\kappa,\psi, \gamma,\theta,\sigma_{0},\alpha,\lambda_{+},\lambda_{-}).\] (22)

## 3 Artificial Neural Network

In this section, we construct two ANNs to calculate call and put option prices under Duan's GARCH model and CTS-GARCH model. More precisely, we design multi-layer ANNs to gen
erate similar results as the function values of \(V_{C}^{Duan}\), \(V_{P}^{Duan}\), \(V_{C}^{CTS}\), and \(V_{P}^{CTS}\). To facilitate the training, we take the logarithm of these four functions and define new functions as follows:

\[v_{C}^{Duan}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{0},\alpha, \lambda_{+},\lambda_{-}) =\log\big{(}V_{C}^{Duan}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{ 0},\alpha,\lambda_{+},\lambda_{-})\big{)} \tag{23}\] \[v_{P}^{Duan}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{0},\alpha, \lambda_{+},\lambda_{-}) =\log\big{(}V_{P}^{Duan}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{ 0},\alpha,\lambda_{+},\lambda_{-})\big{)}\] (24) \[v_{C}^{CTS}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{0},\alpha, \lambda_{+},\lambda_{-}) =\log\big{(}V_{C}^{CTS}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{ 0},\alpha,\lambda_{+},\lambda_{-})\big{)}\] (25) \[v_{P}^{CTS}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{0},\alpha, \lambda_{+},\lambda_{-}) =\log\big{(}V_{P}^{CTS}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{ 0},\alpha,\lambda_{+},\lambda_{-})\big{)} \tag{26}\]

### Generating Training Set

We first generate training sets for the four functions \(v_{C}^{Duan}\), \(v_{P}^{Duan}\), \(v_{C}^{CTS}\), and \(v_{P}^{CTS}\) using the MCS method as explained in the previous section. Since \(v_{C}^{Duan}\) and \(v_{P}^{Duan}\) have seven input parameters (\(m\), \(\tau\), \(\kappa\), \(\psi\), \(\gamma\), \(\theta\), \(\sigma_{0}\)), we consider seven nodes in the input layer. For \(v_{C}^{CTS}\), and \(v_{P}^{CTS}\), three additional input parameters (\(\alpha\), \(\lambda_{+}\)\(\lambda_{-}\)) are needed, in addition to the seven input parameters for \(v_{C}^{Duan}\) and \(v_{P}^{Duan}\). The range of the input parameters can be found in Table 1. More details on training set generation process for each model are as follows:

* _Duan's GARCH Model:_ We generate 100,000 seven-dimensional uniformly distributed random vectors for (\(m\), \(\tau\), \(\kappa\), \(\psi\), \(\gamma\), \(\lambda\), \(\sigma_{0}\)) with the boundary specified in Table 1. To avoid the clustering of the random vector, we use the Halton algorithm (Halton (1964)). Then we calculate 100,000 of \(v_{C}^{Duan}\) and \(v_{P}^{Duan}\), respectively, using MCS. In MCS, we use 20,000 sample paths based on the Duan's GARCH model with the seven-dimensional model parameters.
* _CTS-GARCH Model:_ We generate 100,000 ten-dimensional uniformly distributed random vectors for (\(m\), \(\tau\), \(\kappa\), \(\psi\), \(\gamma\), \(\theta\), \(\sigma_{0}\), \(\alpha\), \(\lambda_{+}\), \(\lambda_{-}\)) with the boundary specified in Table 1. Random numbers of \(\lambda_{+}\) and \(\lambda_{-}\) are generated by \(\lambda_{+}=\tan(\frac{u_{1}\pi}{2})+0.1\) and \(\lambda_{-}=\lambda_{+}=\tan(\frac{u_{2}\pi}{2})+0.1\), respectively, for uniform random numbers \(u_{1},u_{2}\in(0,1)\). We also use the Halton algorithm to generate uniform random vectors, as we did in the case of Duan's GARCH model. Then we calculate 100,000 of \(v_{C}^{CTS}\) and \(v_{P}^{CTS}\), respectively, using the MCS. In the MCS, we use
20,000 sample paths based on the CTS-GARCH model with the ten-dimensional model parameters.

### Training Multi-Layer ANNs

Using four training sets for \(v_{C}^{Duan}\), \(v_{P}^{Duan}\), \(v_{C}^{CTS}\), and \(v_{P}^{CTS}\), we train four multi-layer ANNs in this section. Each ANN consisted of three hidden layers with twenty nodes in each layer. The output is a single value, and the activation function of hidden layer nodes is the simple sigmoid function3, while the output activation function is the linear function4. The ANNs for \(v_{C}^{Duan}\) and \(v_{P}^{Duan}\) have seven input nodes, while the ANNs for \(v_{C}^{CTS}\), and \(v_{P}^{CTS}\) have ten input nodes. The architecture is depicted in Fig. 1. We denote the four ANNs corresponding to \(v_{C}^{Duan}\), \(v_{P}^{Duan}\), \(v_{C}^{CTS}\), and \(v_{P}^{CTS}\) as

Footnote 3: \(f(x)=\frac{1}{1+e^{-x}}\)

Footnote 4: \(f(x)=x\)

\[F_{C}^{Duan}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{0}), \tag{27}\] \[F_{P}^{Duan}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{0}),\] (28) \[F_{C}^{CTS}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{0},\alpha, \lambda_{+},\lambda_{-}),\] (29) \[\text{and}\;F_{P}^{CTS}(m,\tau;\kappa,\psi,\gamma,\theta,\sigma_{0 },\alpha,\lambda_{+},\lambda_{-}). \tag{30}\]

\begin{table}
\begin{tabular}{c c c} Parameter & Lower Bound & Upper Bound \\ \hline \(m\) & \(0.5\) & \(1.5\) \\ \(\tau\) & \(0.4\) & \(1\) \\ \(\kappa\) & \(0\) & \(1\cdot 10^{-5}\) \\ \(\psi\) & \(0.1\) & \(0.4\) \\ \(\gamma\) & \(0.5\) & \(0.9999\) \\ \(\theta\) & \(0\) & \(0.8\) \\ \(\sigma_{0}\) & \(1\cdot 10^{-6}\) & \(0.04\) \\ \hline \(\alpha\) & \(0.01\) & \(1.999\) \\ \(\lambda_{+}\) & \(0.1\) & \(\infty\) \\ \(\lambda_{-}\) & \(0.1\) & \(\infty\) \\ \hline \end{tabular}
\end{table}
Table 1: The range of input parameters
Additionally, we set the parameters \(\Theta_{Duan}=(\kappa\), \(\psi\), \(\gamma\), \(\theta\), \(\sigma_{0}\)) of Duan's GARCH model and \(\Theta_{CTS}=(\kappa\), \(\psi\), \(\gamma\), \(\theta\), \(\sigma_{0}\), \(\alpha\), \(\lambda_{+}\), \(\lambda_{-}\)) of CTS-GARCH model. Then, we have

\[\text{Call}_{ANN}^{Duan}(S_{0},K,\tau,r) =S_{0}\exp\left(F_{C}^{Duan}\left(\frac{Ke^{-r\tau}}{S_{0}},\tau; \Theta_{Duan}\right)\right) \tag{31}\] \[\text{Put}_{ANN}^{Duan}(S_{0},K,\tau,r) =S_{0}\exp\left(F_{P}^{Duan}\left(\frac{Ke^{-r\tau}}{S_{0}},\tau; \Theta_{Duan}\right)\right)\] (32) \[\text{Call}_{ANN}^{CTS}(S_{0},K,\tau,r) =S_{0}\exp\left(F_{C}^{CTS}\left(\frac{Ke^{-r\tau}}{S_{0}},\tau; \Theta_{CTS}\right)\right)\] (33) \[\text{Put}_{ANN}^{CTS}(S_{0},K,\tau,r) =S_{0}\exp\left(F_{P}^{CTS}\left(\frac{Ke^{-r\tau}}{S_{0}},\tau; \Theta_{CTS}\right)\right). \tag{34}\]

Subsequently, we obtain

\[\text{Call}^{Duan}(S_{0},K,\tau,r) \approx\text{Call}_{ANN}^{Duan}(S_{0},K,\tau,r), \tag{35}\] \[\text{and Put}^{Duan}(S_{0},K,\tau,r) \approx\text{Put}_{ANN}^{Duan}(S_{0},K,\tau,r), \tag{36}\]

Fig. 1: ANN structure of the calibration
under Duan's GARCH model, and

\[\text{Call}^{CTS}(S_{0},K,\tau,r) \approx\text{Call}^{CTS}_{ANN}(S_{0},K,\tau,r), \tag{37}\] \[\text{and}\ \text{Put}^{CTS}(S_{0},K,\tau,r) \approx\text{Put}^{CTS}_{ANN}(S_{0},K,\tau,r), \tag{38}\]

under CTS-GARCH model.

We train the four ANNs using functions using the Deep Learning toolbox in Matlab\({}^{TM}\). The default training algorithm for a function fitting network is Levenberg-Marquardt, as stated in the documentation5. The mean squared error (MSE) is presented as the performance measure of the network.

Footnote 5: [https://www.mathworks.com/help/deeplearning/ref/fitnet.html](https://www.mathworks.com/help/deeplearning/ref/fitnet.html)

Fig. 2 exhibits MSE for the epoch during the training of the four ANNs: \(F_{C}^{Duan}\), \(F_{P}^{Duan}\), \(F_{C}^{CTS}\), and \(F_{P}^{CTS}\). The minimum values of MSEs for \(F_{C}^{Duan}\) and \(F_{P}^{Duan}\) are 0.0169 and 0.0061 at epochs 84 and 178, respectively. For \(F_{C}^{CTS}\) and \(F_{P}^{CTS}\), the training was limited to 300 epochs, with minimum MSEs being 0.0541 and 0.0447 at epoch 300, respectively.

It is also noteworthy that the ANNs in this study are not intended for forecasting, but rather for finding better analytic approximations. Therefore, we do not utilize a validation set.

## 4 Calibration

In this section, we discuss the calibration of parameters using two different methods: the ANN method and the MCS method, with the MCS method serving as a benchmark method. We calibrate the parameters \(\Theta_{Duan}\) and \(\Theta_{CTS}\) using call and put option data of the S&P 500 index. For this investigation, we have selected 12 Wednesdays from the second week of each month, ranging from June 2021 to May 2022. The time to maturity of the option contracts varies between 7 and 90 days, and we exclude the options with zero bid prices or zero ask prices. Furthermore, we only calibrate parameters for out-of-the-money (OTM) options for both calls and puts.
## 4 Conclusion

Fig. 2: Number of epochs and mean squared error. Top left and right are the MSE of call and put prices for Duan’s GARCH model. Bottom left and right are the MSE of the CTS-GARCH model.
We first specify ANNs for OTM options as

\[F_{OTM}^{Duan}(m,\tau;\Theta_{Duan})=F_{P}^{Duan}(m,\tau;\Theta_{Duan})\cdot 1_{m <1}+F_{C}^{Duan}(m,\tau;\Theta_{Duan})\cdot 1_{m\geq 1} \tag{39}\]

and

\[F_{OTM}^{CTS}(m,\tau;\Theta_{CTS})=F_{P}^{CTS}(m,\tau;\Theta_{CTS})\cdot 1_{m <1}+F_{C}^{CTS}(m,\tau;\Theta_{CTS})\cdot 1_{m\geq 1}. \tag{40}\]

We calibrate parameters using the relative root mean square error (rel-RMSE) minimization method as follows:

\[\min_{\Theta_{Duan}} \left(\sum_{Kne^{-rT_{n}}<S_{0}}\left(\frac{F_{OTM}^{Duan}\left( \frac{K_{n}e^{-rT_{n}}}{S_{0}},\frac{T_{n}}{250};\Theta_{Duan}\right)-\log \left(\frac{P_{market}(K_{n},T_{n})}{S_{0}}\right)}{\log\left(\frac{P_{market}( K_{n},T_{n})}{S_{0}}\right)}\right)^{2}\right.\] \[+\sum_{Kne^{-rT_{n}}\geq S_{0}}\left(\frac{F_{OTM}^{Duan}\left( \frac{K_{n}e^{-rT_{n}}}{S_{0}},\frac{T_{n}}{250};\Theta_{Duan}\right)-\log \left(\frac{C_{market}(K_{n},T_{n})}{S_{0}}\right)}{\log\left(\frac{C_{market }(K_{n},T_{n})}{S_{0}}\right)}\right)^{2}\right)^{\frac{1}{2}} \tag{41}\]

and

\[\min_{\Theta_{CTS}} \left(\sum_{Kne^{-rT_{n}}<S_{0}}\left(\frac{F_{OTM}^{CTS}\left( \frac{K_{n}e^{-rT_{n}}}{S_{0}},\frac{T_{n}}{250};\Theta_{CTS}\right)-\log \left(\frac{P_{market}(K_{n},T_{n})}{S_{0}}\right)}{\log\left(\frac{P_{market }(K_{n},T_{n})}{S_{0}}\right)}\right)^{2}\right.\] \[+\sum_{Kne^{-rT_{n}}\geq S_{0}}\left(\frac{F_{OTM}^{CTS}\left( \frac{K_{n}e^{-rT_{n}}}{S_{0}},\frac{T_{n}}{250};\Theta_{CTS}\right)-\log \left(\frac{C_{market}(K_{n},T_{n})}{S_{0}}\right)}{\log\left(\frac{C_{market }(K_{n},T_{n})}{S_{0}}\right)}\right)^{2}\right)^{\frac{1}{2}}, \tag{42}\]

where \(S_{0}\) is the S&P 500 index price of the given Wednesday, and \(C_{market}(K_{n},T_{n})\) and \(P_{market}(K_{n},T_{n})\) are mid-prices of observed bid and ask prices for the call and put options with strike price \(K_{n}\) and
time to maturity \(T_{n}\)6.

Footnote 6: To solve the nonlinear optimization problems, we used the function lsqcurvefit() in Matlab\({}^{TM}\) with Trust-Region-Reflective Least Squares algorithm. See Coleman and Li (1994, 1996) and [https://www.mathworks.com/help/optim/ug/lsqcurvefit.html](https://www.mathworks.com/help/optim/ug/lsqcurvefit.html) for the details.

As a benchmark, we use the MCS method to calibrate the same parameters. Similar to the ANN cases, we define

\[v_{OTM}^{Duan}(m,\tau;\Theta_{Duan})=v_{C}^{Duan}(m,\tau;\Theta_{Duan})\cdot 1 _{m\geq 1}+v_{P}^{Duan}(m,\tau;\Theta_{Duan})\cdot 1_{m<1} \tag{43}\]

and

\[v_{OTM}^{CTS}(m,\tau;\Theta_{CTS})=v_{C}^{CTS}(m,\tau;\Theta_{CTS})\cdot 1_{m \geq 1}+v_{P}^{CTS}(m,\tau;\Theta_{CTS})\cdot 1_{m<1}. \tag{44}\]

We calibrate parameters using the MCS method as described in (41) and (42) by replacing \(F_{OTM}^{Duan}\) and \(F_{OTM}^{CTS}\) with \(v_{OTM}^{Duan}\) and \(v_{OTM}^{CTS}\), respectively. More details of the parameter calibration for the option pricing with the CTS-GARCH model are presented in Kim _et al._ (2010a), Kim _et al._ (2019), and Kim _et al._ (2022).

The calibrated parameters for Duan's GARCH model are provided in Table 2 and Table 3, while those for the CTS-GARCH model are provided in Table 4 and Table 5. Table 3 and 5 present the results obtained using the ANN method, while Table 2 and Table 4 present the results obtained using the MCS method. The rel-RMSE values are presented for performance analysis. Table 6 collects all rel-RMSEs for the two models (Duan's GARCH and CTS-GARCH) and both option pricing methods (MCS and ANN).

According to the table, we observe that:

* CTS-GARCH model with the ANN method shows the smallest rel-RMSE and therefore performed the best, with the exception of four cases on 11/10/2022, 12/8/2021, 3/9/2022, and 5/10/2022.
* Duan's GARCH model with the MCS method has the smallest rel-RMSE for the cases on 3/8/2022 and 5/10/2022.
* Duan's GARCH model with the ANN method has the smallest rel-RMSE for the cases on 11/10/2022 and 12/8/2022.
* CTS-GARCH model with the MCS method has the largest rel-RMSE values in this investigation.

Compared to Duan's GARCH model, CTS-GARCH model is more flexible as it has three more parameters. However, the complexity of the CTS-GARCH model makes model calibration with the MCS method inefficient. In this regard, the ANN offers an efficient alternative for model calibration, enabling the use of the CTS-GARCH model in practical applications.

In Fig. 3, the left column plots present the market prices and calibrated model prices of the Duan's GARCH model and CTS-GARCH model for OTM calls and puts on 6/9/2021. The right column plots exhibit the implied volatility curves for the market prices and Duan's GARCH and CTS-GARCH models with respect to the MCS method and ANN method. The first row plots of Fig. 3 show option prices and implied volatility for 9 days to maturity, the second row plots show 37 days to maturity, and the third row plots show 72 days to maturity. The curves suggest that the implied volatility curve of CTS-GARCH model prices with the ANN method is the closest to the market implied volatility curve compared to the other methods investigated in this study.

One of the advantages of using the ANN method is the ability to easily obtain the Greeks of options, which is not almost possible with the MCS method. For instance, Table 7 exhibits Delta (\(\Delta\)), Gamma (\(\Gamma\)), Theta (\(\Theta\)) and Rho (\(\rho\)) of at-the-money (ATM) calls and puts for the parameters calibrated on 6/9/2021 with maturities of 7, 37, and 72 days, respectively7. The Vega (\(\nu\)) is not
considered since the volatility is not constant but stochastic in this investigation. We calculate Greeks using the finance difference method for each variable.

It is widely acknowledged that the calculation time of the ANN method is remarkably faster than the MCS method. Table 8 presents the calculation time for OTM calls and puts using the MCS and ANN methods on Duan's GARCH and CTS-GARCH models, respectively. For example, we observed 558 prices of calls and puts on 6/9/2021. The MCS and ANN methods respectively take 0.13 and 0.085 seconds to calculate 558 calls and puts on Duan's GARCH model, and 0.3286 and 0.0187 seconds on the CTS-GARCH model. The number of call and put price observations (Obs.) varies each day and is shown in the Obs. column in Table 8. The calculation times using the ANN method of Duan's GARCH model are about 9 times faster than the MCS method. Similarly, in the CTS-GARCH model case, the MCS method takes approximately 20 times longer than the ANN method.

## 5 Conclusion

In this paper, we review Duan's GARCH model and the CTS-GARCH model for option pricing and investigate the use of ANNs to enhance calibration performance. To achieve this goal, we generate training sets for various model parameters and compute calls and puts prices using the MCS method. We then train a three-layer ANN with twenty nodes per layer using the generated

\begin{table}
\begin{tabular}{c|c c c c|c|c} \hline Date & \(\theta\) & \(\kappa\) & \(\xi\) & \(\zeta\) & \(\sigma_{0}\) & rel-RMSE \\ \hline
6/9/2021 & \(0.9293\) & \(1.68\cdot 10^{-6}\) & \(0.2778\) & \(0.5000\) & \(0.0085\) & \(0.3048\) \\
7/7/2021 & \(0.8984\) & \(1.21\cdot 10^{-6}\) & \(0.2931\) & \(0.5000\) & \(0.0068\) & \(0.3585\) \\
8/11/2021 & \(0.9513\) & \(1.05\cdot 10^{-6}\) & \(0.2847\) & \(0.5000\) & \(0.0057\) & \(0.3927\) \\
9/8/2021 & \(1.0877\) & \(9.88\cdot 10^{-7}\) & \(0.2492\) & \(0.5000\) & \(0.0065\) & \(0.4349\) \\
10/6/2021 & \(0.9826\) & \(2.70\cdot 10^{-6}\) & \(0.2584\) & \(0.5000\) & \(0.0094\) & \(0.2837\) \\
11/10/2021 & \(0.9905\) & \(1.65\cdot 10^{-6}\) & \(0.2681\) & \(0.5000\) & \(0.0075\) & \(0.2952\) \\
12/8/2021 & \(0.8833\) & \(2.27\cdot 10^{-6}\) & \(0.2943\) & \(0.5000\) & \(0.0081\) & \(0.2763\) \\
1/12/2022 & \(0.8447\) & \(2.13\cdot 10^{-6}\) & \(0.3000\) & \(0.5000\) & \(0.0073\) & \(0.2799\) \\
2/9/2022 & \(0.8745\) & \(2.94\cdot 10^{-6}\) & \(0.2869\) & \(0.5000\) & \(0.0091\) & \(0.2305\) \\
3/9/2022 & \(1.0237\) & \(5.27\cdot 10^{-6}\) & \(0.2464\) & \(0.5000\) & \(0.0158\) & \(0.1881\) \\
4/6/2022 & \(1.0867\) & \(1.47\cdot 10^{-6}\) & \(0.2434\) & \(0.5000\) & \(0.0092\) & \(0.3980\) \\
5/10/2022 & \(1.0220\) & \(3.97\cdot 10^{-6}\) & \(0.1918\) & \(0.5971\) & \(0.0176\) & \(0.2311\) \\ \hline \end{tabular}
\end{table}
Table 2: Calibration of Duan’s GARCH model parameters using the MCS method
\begin{table}
\begin{tabular}{c|c c c c c c c|c} \hline Date & \(\theta\) & \(\kappa\) & \(\xi\) & \(\zeta\) & \(\sigma_{0}\) & \(\alpha\) & \(\lambda_{+}\) & \(\lambda_{-}\) & rel-RMSE \\ \hline
6/9/2021 & \(0.5526\) & \(1.56\cdot 10^{-8}\) & \(0.0954\) & \(0.7906\) & \(0.0046\) & \(1.4202\) & \(0.1019\) & \(0.5492\) & \(0.1662\) \\
7/7/2021 & \(2.5598\) & \(4.65\cdot 10^{-6}\) & \(0.0631\) & \(0.5124\) & \(0.0065\) & \(1.0542\) & \(13.6212\) & \(0.1155\) & \(0.2372\) \\
8/11/2021 & \(1.2173\) & \(1.43\cdot 10^{-8}\) & \(0.1960\) & \(0.6349\) & \(0.0076\) & \(0.7306\) & \(46.0982\) & \(0.1629\) & \(0.2936\) \\
9/8/2021 & \(1.1724\) & \(6.67\cdot 10^{-10}\) & \(0.1582\) & \(0.7666\) & \(0.0068\) & \(0.8133\) & \(3.9428\) & \(0.1019\) & \(0.2781\) \\
10/6/2021 & \(0.5119\) & \(2.51\cdot 10^{-8}\) & \(0.0014\) & \(0.5016\) & \(0.0163\) & \(0.5610\) & \(13.7536\) & \(12.3532\) & \(0.2504\) \\
11/10/2021 & \(1.9605\) & \(4.93\cdot 10^{-8}\) & \(0.1048\) & \(0.5092\) & \(0.0079\) & \(0.6510\) & \(1.4461\) & \(4.1668\) & \(0.2701\) \\
12/8/2021 & \(0.0281\) & \(3.46\cdot 10^{-6}\) & \(0.2965\) & \(0.7125\) & \(0.0091\) & \(0.7452\) & \(31.5555\) & \(0.5439\) & \(0.2474\) \\
1/12/2022 & \(0.6766\) & \(2.27\cdot 10^{-6}\) & \(0.2979\) & \(0.5750\) & \(0.0075\) & \(0.5862\) & \(20.2640\) & \(3.5172\) & \(0.2233\) \\
2/9/2022 & \(0.4874\) & \(2.72\cdot 10^{-6}\) & \(0.2624\) & \(0.6696\) & \(0.0100\) & \(0.7577\) & \(27.9998\) & \(1.5475\) & \(0.2152\) \\
3/9/2022 & \(0.5884\) & \(2.91\cdot 10^{-6}\) & \(0.1191\) & \(0.8183\) & \(0.0134\) & \(0.6454\) & \(21.7075\) & \(4.0287\) & \(0.2684\) \\
4/6/2022 & \(1.5418\) & \(9.84\cdot 10^{-6}\) & \(0.1040\) & \(0.5271\) & \(0.0078\) & \(0.9238\) & \(21.6161\) & \(1.5811\) & \(0.3571\) \\
5/10/2022 & \(0.8144\) & \(3.15\cdot 10^{-8}\) & \(0.1669\) & \(0.7220\) & \(0.0178\) & \(0.7674\) & \(28.8533\) & \(4.4167\) & \(0.2756\) \\ \hline \end{tabular}
\end{table}
Table 4: Calibration of CTS-GARCH model parameters using the MCS method

\begin{table}
\begin{tabular}{c|c c c c c|c} \hline Date & \(\theta\) & \(\kappa\) & \(\xi\) & \(\zeta\) & \(\sigma_{0}\) & rel-RMSE \\ \hline
6/9/2021 & \(1.1103\) & \(4.64\cdot 10^{-7}\) & \(0.2258\) & \(0.5395\) & \(0.0064\) & \(0.1972\) \\
7/7/2021 & \(1.0196\) & \(6.32\cdot 10^{-7}\) & \(0.2354\) & \(0.5621\) & \(0.0049\) & \(0.2793\) \\
8/11/2021 & \(1.1729\) & \(1.74\cdot 10^{-7}\) & \(0.1994\) & \(0.5737\) & \(0.0056\) & \(0.3383\) \\
9/8/2021 & \(1.1810\) & \(2.89\cdot 10^{-7}\) & \(0.1927\) & \(0.5783\) & \(0.0064\) & \(0.4107\) \\
10/6/2021 & \(0.9015\) & \(8.75\cdot 10^{-7}\) & \(0.2115\) & \(0.6441\) & \(0.0074\) & \(0.2826\) \\
11/10/2021 & \(1.1827\) & \(4.99\cdot 10^{-7}\) & \(0.1968\) & \(0.5700\) & \(0.0066\) & \(0.2634\) \\
12/8/2021 & \(1.1276\) & \(7.31\cdot 10^{-7}\) & \(0.2238\) & \(0.5424\) & \(0.0060\) & \(0.2078\) \\
1/12/2022 & \(0.6377\) & \(4.46\cdot 10^{-7}\) & \(0.3000\) & \(0.6252\) & \(0.0047\) & \(0.3632\) \\
2/9/2022 & \(0.8367\) & \(6.29\cdot 10^{-7}\) & \(0.2081\) & \(0.6739\) & \(0.0076\) & \(0.2626\) \\
3/9/2022 & \(0.9301\) & \(1.64\cdot 10^{-6}\) & \(0.1905\) & \(0.6666\) & \(0.0121\) & \(0.2297\) \\
4/6/2022 & \(1.0649\) & \(5.37\cdot 10^{-7}\) & \(0.2088\) & \(0.5990\) & \(0.0070\) & \(0.3616\) \\
5/10/2022 & \(1.0130\) & \(1.67\cdot 10^{-6}\) & \(0.1506\) & \(0.7046\) & \(0.0141\) & \(0.2417\) \\ \hline \end{tabular}
\end{table}
Table 3: Calibration of Duan’s GARCH model parameters using the ANN method

\begin{table}
\begin{tabular}{c|c c c c c c|c} \hline Date & \(\theta\) & \(\kappa\) & \(\xi\) & \(\zeta\) & \(\sigma_{0}\) & \(\alpha\) & \(\lambda_{+}\) & \(\lambda_{-}\) & rel-RMSE \\ \hline
6/9/2021 & \(0.9386\) & \(1.69\cdot 10^{-6}\) & \(0.2805\) & \(0.5050\) & \(0.0086\) & \(1.9998\) & \(14.0138\) & \(10.0760\) & \(0.8819\) \\
7/7/2021 & \(0.9074\) & \(1.23\cdot 10^{-6}\) & \(0.2960\) & \(0.5050\) & \(0.0068\) & \(1.9921\) & \(11.2482\) & \(11.9459\) & \(0.9014\) \\
8/11/2021 & \(0.9608\) & \(1.06\cdot 10^{-6}\) & \(0.2876\) & \(0.5050\) & \(0.0057\)
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{Duan’s GARCH Model} & \multicolumn{2}{c}{CTS-GARCH Model} \\ \cline{2-5} Date & MCS & ANN & MCS & ANN \\ \hline
6/9/2021 & \(0.3048\) & \(0.1972\) & \(0.8819\) & **0.1662** \\
7/7/2021 & \(0.3585\) & \(0.2793\) & \(0.9014\) & **0.2372** \\
8/11/2021 & \(0.3927\) & \(0.3383\) & \(0.9211\) & **0.2936** \\
9/8/2021 & \(0.4349\) & \(0.4107\) & \(0.9113\) & **0.2781** \\
10/6/2021 & \(0.2837\) & \(0.2826\) & \(0.8716\) & **0.2504** \\
11/10/2021 & \(0.2952\) & **0.2634** & \(0.8938\) & \(0.2701\) \\
12/8/2021 & \(0.2763\) & **0.2078** & \(0.8832\) & \(0.2474\) \\
1/12/2022 & \(0.2799\) & \(0.3632\) & \(0.8971\) & **0.2233** \\
2/9/2022 & \(0.2305\) & \(0.2626\) & \(0.8822\) & **0.2152** \\
3/9/2022 & **0.1881** & \(0.2297\) & \(0.8142\) & \(0.2684\) \\
4/6/2022 & \(0.3980\) & \(0.3616\) & \(0.8864\) & **0.3571** \\
5/10/2022 & **0.2311** & \(0.2417\) & \(0.7809\) & \(0.2756\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: Relative RMSE values for model calibrations

\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline Call/Put & \(S_{0}\) & \(r\) & \(K\) & \(\tau\) & Model & \(\Delta\) & \(\Gamma\) & \(\Theta\) & \(\rho\) \\ \hline Call & \(4219.55\) & \(0.0020\) & \(4220\) & \(0.0360\) & Duan’s GARCH & \(0.4109\) & \(5.7074\cdot 10^{-3}\) & \(382.59\) & \(61.56\) \\  & & & & & & CTS-GARCH & \(0.2933\) & \(3.5068\cdot 10^{-3}\) & \(274.16\) & \(43.88\) \\  & & & & \(0.1480\) & Duan’s GARCH & \(0.6089\) & \(3.2794\cdot 10^{-3}\) & \(311.14\) & \(370.61\) \\  & & & & & CTS-GARCH & \(0.4744\) & \(2.4125\cdot 10^{-3}\) & \(312.57\) & \(288.33\) \\  & & & & \(0.2880\) & Duan’s GARCH & \(0.6656\) & \(2.2740\cdot 10^{-3}\) & \(181.63\) & \(780.57\) \\  & & & & & CTS-GARCH & \(0.5042\) & \(1.3049\cdot 10^{-3}\) & \(194.84\) & \(586.98\) \\ \hline Put & \(4219.55\) & \(0.0020\) & \(4215\) & \(0.0360\) & Duan’s GARCH & \(-0.2923\) & \(2.6660\cdot 10^{-3}\) & \(439.87\) & \(-45.64\) \\  & & & & & CTS-GARCH & \(-0.2196\) & \(1.7826\cdot 10^{-3}\) & \(389.37\) & \(-34.41\) \\  & & & & \(0.1480\) & Duan’s GARCH & \(-0.4410\) & \(2.0529\cdot 10^{-3}\) & \(442.75\) & \(-288.81\) \\  & & & & & CTS-GARCH & \(-0.2956\) & \(1.4555\cdot 10^{-3}\) & \(283.71\) & \(-194.58\) \\  & & & & \(0.2880\) & Duan’s GARCH & \(-0.4726\) & \(1.5977\cdot 10^{-3}\) & \(152.80\) & \(-613.73\) \\  & & & & CTS-GARCH & \(-0.2807\) & \(1.0474\cdot 10^{-3}\) & \(178.97\) & \(-369.67\) \\ \hline \hline \end{tabular}
\end{table}
Table 7: Greeks for ATM calls and puts on 6/9/2021
## 6 Conclusion

Figure 3: Calibrated call and put option prices and implied volatility curves at 6/9/2021
training set. Additionally, we create four ANNs for calls and puts under both Duan's GARCH model and the CTS-GARCH model. Finally, we demonstrate the effectiveness of the trained ANNs by using them to calibrate market option prices. The results show that the ANN is significantly faster than the MCS method once it has been trained and it outperforms the MCS method in terms of calibration performance. Furthermore, the ANN method allows us to calculate Greeks, which is not available with the MCS method.

\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & & \multicolumn{2}{c}{Duan’s GARCH Model} & \multicolumn{2}{c}{CTS-GARCH Model} \\ \cline{3-6} Date & Obs. & MCS (sec) & ANN (sec) & MCS (sec) & ANN (sec) \\ \hline
6/9/2021 & \(558\) & \(0.1300\) & \(0.0185\) & \(0.3286\) & \(0.0187\) \\
7/7/2021 & \(644\) & \(0.1387\) & \(0.0191\) & \(0.3411\) & \(0.0175\) \\
8/11/2021 & \(706\) & \(0.1339\) & \(0.0169\) & \(0.3025\) & \(0.0172\) \\
9/8/2021 & \(729\) & \(0.1518\) & \(0.0166\) & \(0.3363\) & \(0.0172\) \\
10/6/2021 & \(845\) & \(0.1422\) & \(0.0178\) & \(0.3390\) & \(0.0178\) \\
11/10/2021 & \(858\) & \(0.1463\) & \(0.0183\) & \(0.3229\) & \(0.0175\) \\
12/8/2021 & \(843\) & \(0.1538\) & \(0.0174\) & \(0.3342\) & \(0.0182\) \\
1/12/2022 & \(878\) & \(0.1598\) & \(0.0169\) & \(0.3356\) & \(0.0172\) \\
2/9/2022 & \(928\) & \(0.2017\) & \(0.0193\) & \(0.3539\) & \(0.0207\) \\
3/9/2022 & \(923\) & \(0.1625\) & \(0.0176\) & \(0.3351\) & \(0.0185\) \\
4/6/2022 & \(818\) & \(0.1410\) & \(0.0165\) & \(0.3285\) & \(0.0189\) \\
5/10/2022 & \(570\) & \(0.1211\) & \(0.0163\) & \(0.2978\) & \(0.0171\) \\ \hline \hline \end{tabular}
\end{table}
Table 8: Comparison of calculation time using the MCS and ANN methods for Duan’s GARCH and CTS-GARCH models