"Black-Scholes-Artificial Neural Network": A novel option pricing model

Purpose: This study conducts a comparative study of various options pricing models and introduces a new model. Research methodology: This paper reviews eight option pricing models, including the Black-Scholes-Merton model (BSM), Monte Carlo simulation (MC), Heston, GARCH, Lattice, Jump Diffusion models (JDM), Normal Inverse Gaussian-Cox-Ingersoll-Ross Model, and a novel model called Black-Scholes-Artificial Neural Network (BSANN). The objective is to predict the European call and put options using a payoff calculation. The underlying asset is Khodro, a famous automobile producer company in Iran, for the last year. The daily prices were also used as historical data. The primary software used for the calculations and plots was MATLAB. An Excel option pricing toolbox was used to obtain more accurate and improved results. Results: Based on the results, it can be concluded that the proposed model, BS-ANN, provides the most accurate estimation with the lowest standard deviation. Limitations: There are several limitations to be considered when choosing an underlying asset. An important factor is the availability of sufficient data on the number of shared transactions. Another limitation of this study is the absence of trading halts. Additionally, caution is crucial when selecting an appropriate number of estimated parameters. Contribution: By utilizing the presented model, researchers, individuals, investors, and stock market analysts interested in trading can enhance their estimations. Novelty: The most significant novelty of this study is the presentation of a hybrid model incorporating unique features.


Introduction
These options have become extremely popular.There are two main reasons for this finding.First, options are interesting to investors because of their potential for speculation and hedging (Jiang & Pan, 2022).Second, there is an organized and structured way to determine their value.Thus, they can be purchased and sold confidently.
Options are securities whose values are derived from other securities, known as "underlying assets" (Parameswaran, 2022).There is a lot of data on the underlying assets.Pricing an option can be a time consuming task.Thus, it makes sense to use fast and accurate option pricing models (Almeida, Fan, Freire, & Tang, 2023).Options provide investors with a payoff that depends on the value of the underlying asset (Pucci di Benisichi 2019).The payoff is obtained by calculating the difference between the initial and future prices (Guo 2022).Therefore, we should analyze the future prices of the underlying asset with multiple possible outcomes.
Generally, there are two options: 1.Call option, 2. Put option.An option to buy a security is called a "call" option, while an option to sell is a "put" option (Guides, 2018).Many rules exist regarding how and when an option can be exercised.One of the major sorting options is to divide them into four categories: 1. European, 2. American, 3. Asian, and 4. African.Vanilla and 4. Exotic.The "European option" can be exercised on a specific future date for a price agreed upon in the contract.An "American option" can be exercised on any day prior to a specified future date.The options mentioned above are generally referred to as "vanilla options" to indicate that they are standardized and considered less complex compared to "exotic options" (Almeida et al., 2023).In Part Three, which focuses on the methodology, the subject of options is developed and expanded.
There are many studies on option-pricing models.However, in this article, we have attempted to examine a diverse range of option pricing models, showing impressive results and demonstrating the calculation of payoffs.In addition, we present a hybrid model.The main goal of this study is to introduce a hybrid model.While conducting a comparative study of different option pricing models with improved codes, the second goal is to determine the best model based on its error and accompanying plots.An important stock, Khodro, has been selected as the underlying asset.The main contribution of this article is that, on one hand, we have attempted to present a hybrid model for option pricing.However, the code has been improved.Different software programs, such as MATLAB and Excel, were used to improve the results.
Artificial intelligence (AI) has emerged as an important concept in recent studies.It has several characteristics that differentiate it from other methods, such as econometric, mathematical, and statistical models (Kase & Laka, 2019).These characteristics include the ability to accelerate calculations, compatibility with intricate data structures (El Fallahi et al., 2022), and enhancements through training and learning.Researchers are continuously striving to uncover new applications of AI in various fields, such as medicine, finance, economics, and transportation.
Consequently, we have attempted to utilize artificial neural networks (ANN) as a method for option pricing.A hybrid model known as the Black-Scholes-Artificial Neural Network (BSANN) was used to predict Khodro option prices.More details about this method are provided in the methodology section.
As is evident, like all other models or methods, options have various parameters and variables, such as strike price and expiration date (time to maturity).Therefore, if these parameters are carefully adjusted, taking into account economic factors such as the inflation rate and risk factors such as political and interest rate risks, it could be possible to make a fair and reasonable prediction.Now that you understand the status of options compared to other investment approaches, it is time to examine the concepts in options, such as option categories and related keywords.Bermudan Option An option that can only be exercised during a predefined portion of its lifespan Table 1 shows two types of options: call options and put options, along with various option categories including Vanilla, American, and European options.
The remainder of this paper is organized as follows.The second section is dedicated to a review of the literature.The third part describes the methodology and applied models, including the statistical population, period of time, and data statistics.Part Four focuses on the findings, results, and experimental processes.The final section consists of conclusions and remarks.For more information, please refer to the Appendix.

Literature review
There are several methods, such as mathematical models and computational algorithms, for recognizing the behavior of asset prices that offer an understanding of fairness (Das et al., 2023;Mayes and Govender, 2019;Tudor, 2022) The Black-Merton-Scholes formula assumes that all investors exist in a risk-neutral world, where the expected return is equal to the risk-free interest rate The most popular models, such as Black-Scholes-Merton and Monte Carlo simulations, use the same factors, such as time to maturity, strike price, and interest rate, in option pricing (Parameswaran, 2022).However, volatility is one of the major factors that make these models unique to each other.Volatility is a major issue in financial mathematics.First, it helps us understand price dynamics; second, it is the only factor that cannot be seen; third, volatility has always been an interesting issue in a wide range of research areas (Wang, Cheng, Yin, & Yu, 2022).Black and Scholes (1973) presented the first complete equilibrium option-pricing model.In the same year, Merton attempted to redefine the Black and Scholes model and made some modifications.The jump-diffusion model, suggested by Merton, is a stochastic process that involves both jumps and diffusion.Merton modified the BSM to account for dividends and variable interest rates (Parameswaran, 2022).One popular model is the Monte Carlo option pricing model, which utilizes Monte Carlo simulation and processes to calculate the valuation of an option.Boyle (Trinh & Hanzon, 2022) performed the first calculation of option pricing using Monte Carlo.In finance, the lattice model is used for the valuation of options with discrete-time model assumptions.The simplest lattice model is the binomial option pricing model (BM), which was developed in 1979 by Cox-Ross-Rubinstein and is commonly referred to as the CRR model (He, Coolen, & Coolen-Maturi, 2021).Some option pricing models assume constant volatility, whereas many financial time series are characterized by time-varying volatility.The GARCH models proposed by Engle (1982), Dana (2016), andBollerslev (1986) consider the dynamics of returns.Another model is the Levy process (Bertoin, 1992;Levy, 1992), which is a stochastic process with independent and stationary increments (Velasquez, 2020).
Every option-pricing model includes various assumptions and hypotheses.For example, the B-S model contains the assumptions considered (Karagozoglu, 2022).However, this remains a question.Are these assumptions true in real-life?Therefore, these assumptions must be modified.Owing to changes in these assumptions, different models have emerged.One of the assumptions in Black-Scholes-Merton (B-S-M) is that asset price returns follow a lognormal distribution.While most financial data exhibit skewness, kurtosis, and term structures, implied volatility is also observed (Bali et al., 2019).Therefore, some researchers recognize the volatility of asset returns as a hidden stochastic process and conclude that the B-S-M model cannot be used for stochastic volatility.As a result, they changed their methods and started using other processes, such as Levy, Monte Carlo simulation, and Fourier.
Recently, there has been considerable interest in the development of artificial neural networks (ANNs) for solving various problems (Roy, 2023).Neural networks, which are capable of learning relationships from data, represent a class of robust nonlinear models inspired by the neural architecture of the brain.Theoretical advances along with hardware and software innovations have addressed previous limitations in the implementation of machine learning, making neural network methods accessible to a diverse range of disciplines (Qin & Chiang, 2019).
The following table (Table 3) displays various research studies and models of option pricing, including their methodologies and results.

Models and formulas
Different models and methods have been used to simulate option prices.To summarize and condense the content, we have omitted these formulas.Instead, we refer to each model as an article.These models include the Monte Carlo simulation (M-C), Black-Scholes option pricing model (BSM), binomial option pricing model (BOPM), Merton's Jump Diffusion option pricing model, Heston option pricing model, NIG-CIR option pricing model, GARCH option pricing model, genetic algorithm (GA), and BSANN option pricing model.
To evaluate option prices using a Monte Carlo simulation, this method generates different possible paths for the underlying asset in a risk-neutral environment.Then, the value of the option is calculated for each path.Finally, it takes the average of all the option values and calculates their present values.The Monte Carlo method employs three steps to price an option (Trinh and Hanzon, 2022).1. Calculate the potential futures prices of underlying asset(s).2. Calculates the payoff of the option for each potential underlying price path 3. Discount the payoff back to today and average them to determine the expected price This process begins by computing the B-S numbers using input parameters, such as the strike price and interest rate.Subsequently, a random variable called epsilon was used as an input number.Finally, the option value and payoff were computed.The Black-Scholes model is primarily used to calculate the theoretical value of European-style options.It cannot be applied to American-style options because it can be performed before the maturity date.
The binomial option pricing model uses an iterative procedure that allows for the specification of nodes or points in time between the valuation date and the option's expiration date (Yeh & Lien, 2020).The binomial model is best represented using binomial trees, which are diagrams that illustrate the payoff and value of the option at different nodes throughout its lifespan.BOPM has several assumptions (He 2019).The binomial pricing model traces the evolution of the option price in discrete time under a riskneutral measure.This measure ensures that the discounted price process is martingale.
The jump-diffusion model, introduced in 1976 by Robert Merton, is a model for stock price behavior that incorporates small day-to-day "diffusive" movements together with larger, randomly occurring "jumps" (Chowdhury & Jeyasreedharan, 2019).The inclusion of jumps allows for more realistic "crash" scenarios, rendering the standard dynamic replication hedging approach of the Black-Scholes model ineffective.This causes option prices to increase compared with the Black-Scholes model, which depends on investors' risk aversion.This equation is based on the B-S formula.
The stock price is full of volatility and the interest rate is not fixed.Therefore, it would be preferable to use a model that considers this condition.One model that rejects constant volatility and uses a stochastic process is the Heston stochastic volatility model (Heston, 1993).This model assumes that the volatility is arbitrary.It is an extension of the Black-Scholes stochastic differential equation (SDE), where the volatility is derived from the Cox-Ingersoll-Ross (CIR) process (Cui et al., 2017).This model assumes that the distribution of asset returns is nonlognormal.
The NIG-CIR option pricing model was developed by Carr et al. in 2003.This is a combination of these two words.NIG stands for Normal Inverse Gaussian and CIR stands for the Cox-Ingersoll-Ross model (Kovachev, 2014).The model is defined under the risk-neutral measure, which was first introduced by Duan (1995) with a locally risk-neutral valuation relationship (LRNVR), in which the conditional variances and model parameters remain the same under both physical and risk-neutral measures.Duan's LRNVR has been widely used by finance researchers and practitioners to price options under the GARCH framework.However, as pointed out by Barone-Adesi (Jayaraman, 2022), empirical evidence shows that the restriction in Duan's LRNVR resulted in poor pricing and hedging performance.
Since the groundbreaking research by Bollerslev (1986) and Engle (1982), the GARCH volatility model family has been extensively utilized in empirical asset pricing and financial risk management.This is primarily because the likelihood function of asset returns in GARCH models can often be expressed in a closed form using observed data.Model parameters can be estimated using the maximum likelihood estimation (MLE) method (Smith, 2019), which is often a challenging task for most stochastic volatility models.Motivated by the success of GARCH models in fitting asset returns, Duan (1995) pioneered the application of GARCH models to SPX option pricing by proposing LRNVR.Duan et al. 's LRNVR has been reported in many studies.
One of the main applicable analytical software programs is MATLAB.Many script codes were prepared in the file.Thus, by accessing these m.files, it is possible to tune, that is, change and improve them based on your problem.In this study, MATLAB R2017a was used as the primary analytical tool.Another important software is the Option Pricing Excel add-in software.By adding this feature, input variables can be inserted, and it is possible to see the final output for different option groups, such as Asian and European groups.However, other software is used, such as the MG Soft Exotic Options Calculator, FinOptionsXL, and option-pricing packages.

Genetic algorithm
The genetic algorithm is a global search technique that utilizes various operations, including coding, decoding, mutation, and crossover, to discover an approximate optimal solution (Gen & Lin, 2023).Mutation and crossover are operators that create new and optimal solutions, respectively.Binary coding was used, and 17 bits of chromosomes were utilized.Of these, 12 bits are used to represent the existence ("1") or non-existence ("0") of the input variables, whereas the remaining five bits (2 5 =32) are used to determine the number of neurons in the hidden layers.The GA parameters are presented in Table 4.We used 70% of the data for training, and the remaining portion was used for the validation and testing datasets.Initially, we used 100 epochs and then increased it to 1000.The GA process is as follows.

Figure 1. Genetic Algorithm Process
In this study, certain stock market indicators (specifically, peer stocks in the automobile industry) were utilized as input variables to enhance the predictive capability.We must normalize the data before using AI techniques.As mentioned earlier, the GA has been used for feature selection.However, it is a heuristic algorithm for optimizing and obtaining the best possible solution.
In this study, we attempt to predict the price of Khodro options as the dependent variable.Daily data were used.The considered indicators are listed in Table 5. Target variable These indicators were adopted from the TSETMC database using the TSECLIENT software.features.This process is then iterated in the next layer, which is the hidden layer.However, in this case, the new weights and biases must pass through a linear activation function.Finally, the output layer exists.Before performing any calculations, the data were normalized for several reasons.The most important reason for this is data scaling.

S i ̃= (S i −S min )
S max −S min The Levenberg-Marquardt (LM) method is used as an optimization algorithm to minimize the error.First, the learning rate is set to 0.01.If we do not observe any improvement in the learning of the network, it increases to 0.1.The diagram below illustrates the research process and flowchart.
To calculate the option price, we need to consider the following parameters using the Black-Scholes (B-S) models.Historical data shows that stock prices range from 100 to 600.Time to maturity was considered to be approximately two years.The minimum and maximum risk-free rates are 0.1 and 0.2, respectively.The volatility based on inflation ranges from 0.4 0.5.Finally, the option price is between 0 and 0.9, based on normalized data.
For more information about calculations and formulas using the Black-Scholes model, please refer to section 3-2-2.

Result and discussion
First, we determine parameters such as the price of the underlying asset (current stock price), strike price, interest rate, volatility, and time to expire.We are going to price an option for a stock in the automobile industry with the symbol "Khodro" for a one-year period, which means the time to maturity is one year.The stock price is 400 Toman (or 4000 Rials), and the hypothetical strike prices for the call option and put option are 200 and 600 Toman, respectively.According to central bank reports, the interest rate, which represents the risk-free rate, is 20%.The stock market volatility was 42%.
We consider a strike price that is 50% different from the current price of the underlying asset, for various reasons.
1.The inflation rate was high.At the time of writing the paper, the inflation rate was 42% according to reports from the statistical center.2. Devaluation of the national currency 3.There is high risk associated with investing in stocks and options.Therefore, taking high risks increases the expected profit.4. Increasing the value of parallel markets, such as cryptocurrencies, foreign currencies, automobiles, housing, etc.

Monte Carlo Option Pricing Simulation
First, we should establish the parameters in various dimensions, such as strike price and volatility.We created an m-file in MATLAB with specified parameters including the input price, number of iterations, and asset paths.To accomplish this, we utilized a preexisting m-file or script.We then changed or adjusted the scale according to the desired outcomes.The underlying asset price for the initial days does not exhibit much volatility or increase significantly.During this time, the underlying assets increased to 2000 and 2500.The following table shows the value of stock options due to ROI during different months.The value of the stock options was divided into three parts, with ROI of 20%, 22%, and 25%.After a year, the value of stock options increased by more than 5000 Rials for both call and put options.For a call option, the estimated value of the options is 3,366.44,after 12 months, with a 20% growth rate.In a put option, the options are estimated to be worth 2,966.44 after 12 months, with a 20% growth rate.
Finally, the payoff is calculated for four positions: long, short, long, and short.The following figures show this: Table 10.Some details about long call, short call, long put and short put positions

Black-Scholes Option Pricing Model (BSM)
First, the parameters are set.Thus, it was possible to obtain results.Table11 shows the parameters and results.-9,133.19 -8,748.95 -8,360.83 -7,968.79 -7,572.79 -7,172.79 -6,772.79 -6,368.79 -5,960.75 -5,548.63 -5,132 It should be noted that all parameters and their values were adapted from various research papers and statistics.For example, volatility and interest rates are determined by the inflation rate, which is set by the central bank.In all tables, the appropriate parameters must be set, and the results can be seen automatically.However, interpreting these results is important.From the formulas, it is clear that gamma has the same value for both calls and puts, and the same applies to vega for call and put options.This can be seen directly from put-call parity because the difference between a put and a call is a forward contract, which is linear in and independent of ( a forward contract has zero gamma and zero vega).
In practice, some sensitivities are often expressed in scaled-down terms to align with the potential magnitude of the parameter variations.For example, it is often reported to be divided by 10,000 (1 basis point rate change), 100 (1 vol point change), and 365 or 252 (1-day decay based on either calendar days or trading days per year).A 1-year call option with an exercise price (strike price) of 200 on stock trading at 400.Determine whether you should buy the option if the annual risk-free rate is 20% and the annual standard deviation of stock returns is 42%.
The value of the call option needs to be determined using the Black-Scholes option pricing model and then compared with the current price of the option.If an option is fairly priced, it can be purchased.
We first need to find  1 and  2 : As per the model, the option value is higher than the strike price.Therefore, the call option is invalid.
If we have the current value of the call premium, stock price, exercise price, time to maturity, and riskfree rate, we can calculate implied volatility.After finding the call and put options, we calculate the payoff.Figure 14 and 15 present the payoffs and behavior of the option during the considered period.
Figure 14 shows the value of the call option with an underlying price of 400 Toman and strike price of 200.If the value of the option is greater than 200, the option is valid.If the price is below 200, the value is zero, and no option is applied.Figure 15 shows the value of the put option with an underlying price of 400 and strike price of 600.If the value of the option is less than 600, then the option is valid.If it is greater than 600, the value is zero, and no option is applied.

Binomial Option Pricing (BOP)
Several steps were taken to calculate the option price using the BOP model.To simplify matters, we make several assumptions: 1 .Input in blue cells.The output appears in yellow cells.
2 .The macro uses a binomial tree to price standard options.
3 .The assumptions are the GBM and risk-neutral valuation.4 .No tax or transaction costs were included.5 .The maximum number of steps is 255 .6.For reference, please refer to Hull J. (2000).
The parameters considered for calculating the call option are as follows: Here, is an example of the binomial options pricing model for a 20 period () call option.Let's say the current stock price () is 400.The strike price () of the option is 200.The option expires (time to maturity) in one year.At the end of the year, the stock price will either rise to 2616 or fall to 61.We assume that there is a 98% chance that it will rise to 2616 and a 2% chance that it will fall to 61.The interest rate () is 20%.If the stock rises to 2616, the value of the option will be 2417.This is Figure 14.Call Option price Figure 15.Put Option price because the option value is equal to the stock price minus strike price.There is a 98% chance that the option will be worth 2417 at the end of the year.If the stock falls to 61, then the value of the option will be 0.This is because option value cannot be negative.The values were either positive or zero.
There is a 2% chance that the option will be worth nothing by the end of the year.We can then calculate the call option using the BOPM.Table 13.An example of calculating the price of a European call option using the Binomial Option Pricing Model (BOPM) with 20 steps.
The option price varied for each step.The value of the option at each step can be seen in the following steps:  We use the same steps to calculate the put option.The considered parameters are listed in Table 14.Here is an example of the binomial option pricing model for a 20 period () put option.Let's say the current stock price () is 400.The strike price () of the option is 600.The option expires (time to maturity) in one year.At the end of the year, the stock price will either rise to 2616 or fall to 61.We assume that there is a 38% chance that it will rise to 2616 and a 62% chance that it will fall to 61.The interest rate () is 20%.
If the stock rises to 2616, then the value of the option will be 0.This is because option value cannot be negative.The values were either positive or zero.There is a 38% chance that the option will be worth nothing by the end of the year.If the stock falls to 61, then the value of the option will be 539.This is because the option value equals the strike price minus stock price.There is a 62% chance that the option will be worth 539 by the end of the year.
The next step is to calculate the put option using a Binomial Option Pricing Model (BOPM).Table 15.An example of calculating the price of a European put option using the Binomial Option Pricing Model (BOPM) with 20 steps The put option price for each step is shown in Figure 17.We calculate the option price using the Binomial Options Pricing Model (BOPM) with a maximum of 255 iterations.The results are presented in the appendix (Table A1).A related macro is used for the calculation of BOPM in the appendix (Table A2).

Merton's Jump Diffusion Option Pricing Model
First, we begin this section by setting the parameters, as in other sections.The table below shows the primary and associated parameters.In the figure above, there are five main jumps: a jump at 400, two jumps between 400 and 500, a jump between 600 and 700, and a jump between 900 and 1000 price levels.
The following figures show the volatility surface of call options and put options in comparison to the Black-Scholes model using a spot price of 100 (on a 100 scale).For a more comprehensive and meaningful understanding of implied volatility, value probability density, and returns for future years, please refer to the appendix (Table A3) for additional results.

Heston Option Pricing Model
First, we considered the parameters and their corresponding values.The option price and standard error for the call and put options were 232.98 and 133.58, with standard errors of 0.3748 and 0.3786, respectively.
The following figures show the volatility surfaces of call options and put options on a logarithmic scale:

NIG-CIR Option Pricing Model
The parameters of the model are as follows.The option price and standard error for the call and put options are 233.175,and 131.1045 while the standard errors are 0.351193 and 0.3541466, respectively.
A figure of a normal inverse Gaussian process with certain parameters below shows skewness and kurtosis.
Figure 23.Normal inverse Gaussian process The figure displays excess kurtosis and skewness.

GARCH estimation model
First, similar to the other methods, we begin the computation by setting the parameters.The initial price (underlying asset price) was 400.Therefore, we set the strike price to 200 for the call option, and 600 for the put option.The results for the call and put options are shown in Figure 24 and 25, respectively.

Genetic Algorithm Feature Selection
As mentioned earlier, GA was used for feature selection.Model selection was applied to find a neural network with a topology that optimizes the error of the new data.There are two different types of algorithms for model selection: order-and input-selection algorithms.Order selection algorithms are used to determine the optimal number of hidden neurons in the network.Input selection algorithms are responsible for determining the optimal subset of the input variables.
The algorithm selected for order selection in this application was SA.This method was inspired by the metallurgical industry and utilizes stochastic principles.A graphical representation of the resulting deep architecture is presented below.It contains a scaling layer, a neural network, and an unscaling layer.Yellow circles represent scaling neurons, blue circles represent perceptron neurons, and red circles represent unscaled neurons.The number of inputs was 11 and the number of outputs was one.The complexity, represented by the number of hidden neurons, is 1.For a call option, all variables are considered as inputs, whereas for a put option, only seven variables are selected.However, the results may be more accurate than the others.An important issue is the accurate and precise setting of the parameters according to the model.To make informed decisions, it is essential to have accurate information about the underlying asset, including factors such as the volatility rate and strike price.Additionally, it is important to be familiar with economic conditions such as the inflation rate and rate of return.
Table 17 shows the results of option pricing using the different models considered.There are also some assumptions: the interest rate (risk-free-rate) is 0.2, the volatility () is 0.42 and the expiration period is one year (360 days).The predicted option prices are obtained for each model with different standard deviations.According to the above table, the BSANN model with the lowest standard deviation has the best predictability.However, the GARCH estimation with the highest standard deviation is a relatively unsuitable model for predicting options.
Finally, it is important to consider that the options carry more risks.On the other hand, taking more risk increases potential returns.Thus, before making any investment, it is important to consider your strategy and approach, which includes determining whether you are risk-takers or risk-averse.

Limitation
There are some limitations in different terms, such as the choice of underlying asset.It is important to pay attention to the availability of sufficient data on the number of share transactions and absence of trading halts.Additionally, it is crucial to be careful when selecting an appropriate number of estimated parameters.

Suggestion
As a recommendation, optimizing artificial intelligence (AI) using metaheuristic algorithms, such as the Chimp Optimization Algorithm (ChOA) and the Aquila Optimization Algorithm (AO), can be effective and applicable in option price forecasting.AI offers several advantages.1. Speed up calculations 2. Increasing accuracy 3. Uncomplicated AI methods do not have any assumptions or hypotheses, making them applicable to any problem.They are adaptable to data and complex structures.However, these algorithms can prevent the problem of getting stuck in the local minima or maxima.

Figure 3 .
Figure 3. Architecture of the proposed Neural Network

Figure 6 .
Figure 6.ROI of Stock Option Over Time (Call option)

Figure 8 .
Figure 8. ROI of Stock Option Over Time (Put option) determined the probability of a standardized normal distribution using the NORMSDIST function in Microsoft Excel.(1) and (2) 0.990, and 0.972, respectively.Once we have (1) and (2), we can plug-in the relevant numbers in the Black-Scholes formula (

Figure 16 .
Figure 16.Call Option price, using the BOPM

Figure 24 .
Figure 24.Call option calculation using GARCH model

Figure 28 .
Figure 28.Input selection results Figure 30.Regression The above figure shows the regression based on different datasets, including training, validation, and testing.All regressions have R-squared values greater than 99%, which is indicative of an interesting result.The network performance during each iteration and the best network performance are shown in the figure below:

Figure 31 .
Figure 31.Performance of the network during iterations

Table 4 .
GA parameters

Table 5 .
Economic indicators as input variables

Table 6
denotes the amount of data.The different parameters and more details regarding the ANN parameters are available in the following table:
Table 8 lists the parameters.

Table 9 .
Input variables and results Then, you can observe the simulation results after 100.000 iterations in the following figure:

Table 12 .
Input variables and results for pricing European call option using the BOPM model

Table 14 .
Input variables and results for pricing European put options using the BOPM model Figure 17.Put Option price, using the BOPM

Table 16
There are more details on these jumps and price levels.Both the drift and jump values follow a normal distribution.The vertical lines in the plot indicate the jump positions.These parameters are listed in the following table along with their corresponding equations.

Table 18 .
Input variables and the results

Table 19 .
Input variables and the results

Table 20 .
GARCH Model Parameters

Table 22
The following chart displays the error history of various subsets during the GA input-selection process.The blue line indicates the training error.The initial values for the call and put options are 0.0162654 and 0.55321, respectively.The final value after 100 generations was 0.00602875 for the call option and 0.22185 for the put option.The orange line represents the selection error.Its initial values were 0.0205194 and 0.51623, and the final values after 100 generations were 0.00833242 and 0.20453.
Order selection for call optionOrder selection for put option

Table 23 .
Input selection results

Table 25 .
Error tables Options

Loss functions Call option Put option Training Selection Testing Training Selection Testing Sum squared error
In this article, various models such as the Black-Scholes-Merton model, Monte Carlo simulation, Heston model, GARCH model, Lattice model, Jump Diffusion model, Normal Inverse Gaussian-Cox-Ingersoll-Ross model, and Black-Scholes-Artificial Neural Network (BSANN) are used for the prediction of European call and put options.We cannot conclude which model is better because each model has different qualifications and characteristics.For example, the NIG-CIR option pricing model has more parameters.Therefore, this condition can decrease the calculation speed and complicate it.

Table 26 .
Different forecasting models Performance (Call option) Performance (Put option)