ANN model for Call Options Pricing using S&P 100 Market Data

GEORGIOS RIGOPOULOS

Department of Economics,

National and Kapodistrian University of Athens,

Athens,

GREECE

Abstract: - Stock options pricing is of key importance for markets and traders and is largely based on

theoretical models like the Black-Scholes model. However, developments in machine learning open a novel,

data-driven, perspective in contrast to the theoretical ones. This work explores the feasibility of artificial neural

network model utilization for call option pricing, using the traditional Black-Scholes model as a benchmark. A

multilayer perceptron model is trained to learn the Black-Scholes function and tested in real market call options

data originating from thirty-five S&P 100 stocks. Findings demonstrate that artificial neural networks perform

relatively well with market data and can be a valid data-driven approach for call option pricing, competitive to

Black-Scholes. A unique contribution of this study is that testing data is not derived from the same distribution

as training data, something common in existing works with similar models. Although further exploration and

experimentation are required to reach the required robustness and become less ad hoc and data sensitive, data-

driven pricing using artificial neural networks is a promising approach and can play a substantial role in option

pricing.

Key-Words: - option pricing, artificial neural network, multilayer perceptron, call options, Black-Scholes

model, machine learning, deep learning.

Received: March 8, 2023. Revised: November 4, 2023. Accepted: December 6, 2023. Published: January 25, 2024.

1 Introduction

Stock options comprise a dynamic domain in

financial research and numerous models have been

proposed during the past few decades for their

efficient pricing. The older and more traditional

models rely mainly on theory, while recent

developments in pricing include models that rely on

data. The major characteristic and benefit of the

conventional models, like the seminal Black-

Scholes model, is that they offer closed-form

mathematical solutions, something that makes them

very valuable approaches for pricing in market

conditions. They are very popular due to their speed,

flexibility, and accuracy, however, they are quite

limited, as only some option types can be supported

and only if specific assumptions and parameter

values are met. On the other hand, alternative

models exist, relying on numerical procedures, like

Monte Carlo simulation and the Binomial model.

These approaches follow theoretical constructs and

simulate the behavior of underlying assets to

estimate option prices. These models can support a

wider set of option types compared to the limited

Black-Scholes model, [1], [2], [3], [4]. The key

limitation of conventional approaches is that they

are theoretical abstractions and, as such, they are not

able to capture the entire complexity and dynamics

of the underlying process market mechanisms. So,

they inevitably have limitations in assumptions, or

their parameters and they do not perform accurately

or promptly in every setting or option type.

Following the developments in machine

learning and artificial neural networks, researchers

in the domain proposed novel option pricing models

that rely on data instead of theory. These models use

either empirical or artificially generated data for

training and testing and they do not require any

theoretical construct beforehand. Artificial neural

network approaches are the most representative

approach from this group of methods, and they can

be competitive with conventional models in many

option types. Machine learning models, and

especially artificial neural networks, can model any

kind of nonlinear behavior and interaction, without

the need of underlying theoretical abstraction.

However, their approach makes it very hard to

generate an explainable model, something that is an

inherent feature of artificial neural networks. So,

their black-box behavior, even if the model is

successful in pricing, remains an issue for

researchers. Also, machine learning models require

large volumes of data to be trained accurately,

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which is not always feasible. Additionally, as they

rely on the training approach and dataset, they tend

to be domain and data-specific, rather than universal

as compared to the traditional models. So, even if

machine learning-based models are becoming a

competitive alternative to traditional pricing

methods, further research is necessary to offer more

robust and widely used approaches, [5], [6], [7], [8],

[9].

Following the above, this work explores the

feasibility of using artificial neural networks in

option pricing, using the traditional Black-Scholes

model as a benchmark. Relevant approaches

demonstrate artificial neural networks trained to

learn the Black-Scholes function, but very few

works use market data to test the models. The

majority use simulated or generated datasets for

both training and prediction. In this work, we train

the network using artificially generated data of

around 6.5 million instances, and then we apply the

testing in real market option data from thirty-five

S&P100 stocks. So, testing data is not derived from

the same distribution as training, and we can

examine model performance in real market data,

adding thus a unique contribution to existing

research.

The structure of the paper is as follows. In the

next sections, key background information for

options and their pricing is presented. The Black-

Scholes method is also presented briefly in the

section along with key terminology of artificial

neural networks. Several key relevant publications

on machine learning models and their applicability

in pricing are also discussed. In section three, we

present the method and the datasets, followed by

results and a discussion on findings. The work

concludes with some discussion of the findings and

next steps. Overall, the key outcome from the

present work is that artificial neural networks can

play a substantial role in option pricing, although

further exploration and experimentation are needed

to reach the required robustness and become less ad

hoc and data-sensitive as a method.

2 Background

2.1 Options Basics

In general, an asset’s present value is linked to its

expected cash flow. However, some assets, called

options, depend on underlying assets, derive their

value from them, and their cash flows depend on the

occurrence of specific events. So, the expected cash

flows approach cannot be used to estimate their

value. For this reason, alternative methods have

been developed to price them fairly. Options are

financial instruments used either for risk reduction

and hedging or as investments following market

trends of the underlying assets, [1].

An option is a contract between two parties for a

specific quantity of an underlying asset, with an

expiration date (maturity date). The holder of the

contract has the right, but not the obligation, to buy

or sell the specified quantity of the asset at a

specified price (strike price), either at the maturity

or earlier. If an option is exercised by the holder, it

expires without any further obligation. Concerning

the right to buy or sell the underlying asset, options

are distinguished into call and put options.

Call options offer the right to buy a specified

quantity of the underlying asset at the strike price,

either on maturity or any time before. If the option

is not exercised until the expiration date, it expires

without any benefit or further obligation for the

holder. The holder pays a price to purchase the

option expecting a benefit if the price of the

underlying asset is higher than the strike price. In

this case, the holder exercises the option at strike

price and buys the underlying asset at this price,

instead of the higher market price. The difference is

the gross investment profit. If the asset price is

lower than the strike price at maturity or earlier, the

option is not exercised. So, the net profit is the

difference between the gross profit and the call

purchase price, if the option is exercised.

Put options offer the right to sell a specified

quantity of the underlying asset, at strike price,

again either at maturity or earlier. A put option has a

price paid by the investor who expects a profit in

case the price of the underlying asset is less than the

strike price of the option. If the underlying asset has

a price lower than the strike price of the put option

on maturity or before, the option is exercised and

the option holder sells the underlying asset at a

higher price compared to the market value, which

comprises the gross profit of the investment. In case

the underlying asset has a price higher than the

strike price, the option is left to expire. The net

profit again comprises the difference between the

gross profit and the put option purchase price, [2].

Options can be also classified in terms of the

exercise date or the underlying asset types. So,

European options do not allow for exercise before

maturity and the exercise date is defined in the

option contract. American options, on the other

hand, allow for exercise at any point of time before

maturity and are more attractive for trading.

Considering some fundamental asset types, options

can be either stock options, stock index options,

future options, or product options. Many more

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option types exist, but in this work, we focus on the

two most known, the American and European ones.

2.2 Option Pricing Methods

The key idea behind option pricing is that options

are traded in exchanges in mature markets, or

specially organized exchanges in less developed

markets. So, typically, at the initiation of an option

contract, the buyer pays the option price (premium)

to the option seller (writer). The premium defines

the maximum profit the seller can receive from the

transaction. Consequently, fair, and accurate option

pricing is key for the efficiency of option markets.

Following finance theory, the determinants of

option price are the following:

• The current value of the underlying asset.

• The value variance of the underlying asset or

volatility.

• The dividends of the underlying asset.

• The strike price of the option.

• The expiration date of the option or time to

maturity.

• The risk-free interest rate during the option life.

Based on the above, a variety of pricing methods

and variations have been introduced to price options

accurately. The Black-Scholes model is the

predecessor of all and since its introduction in 1973

remains the most influential, [3]. It offers an

analytical method to estimate the theoretical

arbitrage-free price of an option provided that some

market parameters are known. Another widely used

model is the Binomial which was introduced in

1978 and follows a discrete-time approach, [4].

Except for those two popular methods, many

variations and novel approaches have been

introduced, as the domain is very active and the

stakes in the finance industry are very high.

However, despite the introduction of more

sophisticated methods, the traditional ones seem to

outperform some comparative studies for American

options, where analytical solutions cannot be

generated, [5]. In the following, we review some

representative works of artificial neural networks

approaches in pricing, that use the Black-Scholes

model as baseline.

2.3 Black-Scholes Model

The Black-Scholes model introduced by Fisher

Black and Myron Scholes in 1973 is considered one

of the most influential models in finance, [3]. It

assumes that stock prices follow a random walk

move and, for a market to be efficient, stock prices

should not follow a pattern that could be predicted.

If this assumption does not hold, stock future prices

can be predicted and there could be financial gain.

Since its introduction, there have been many

variations, but in its initial version there is no

dividend until the option maturity date, no

transaction fees are charged, and the risk-free rate

and volatility are known constants.

The model is parametric and its famous formula

for the arbitrage-free price of an option can be used

to price options as a function of current stock or

underlying asset price, option strike price, option

time to maturity or expiration, risk-free rate and

volatility of the underlying stock return. The

formula for call option price is the following:

=󰇛1󰇜󰇡

󰇢(2) (1)

with

1=

++2

2



(2)

2=

+2

2

 (3)

The formula for the put option price is:

= 󰇡

󰇢󰇛2󰇜 󰇛1󰇜 (4)

where:

• C: the price of the call option

• P: the price of the put option

• S: the current price of the stock or underlying

asset

• N(d): the cumulative normal probability density

function

• K: the strike price of the option

• σ: the volatility of the stock or underlying asset

• T: the time to expiration of the option

• rf: the risk-free interest rate

Several adjustments have been proposed to the

initial model to account for limitations that do not

hold for all options. The model is widely used and

referenced in almost every option pricing related

work, and interested readers can find a decent

review of the work of Hull among others, [6].

2.4 Artificial Neural Networks

Artificial neural networks were initiated back in

1943 by the work of McCulloch and Pitts, where the

idea was to use mathematical formulation on the

concept of a biological neuron to be able to execute

computations mimicking brain neurons'

functionality. In the past decades, there has been

exponential research developments, driven mainly

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by big data and computing power evolution in the

past decade. Applications were so successful, that

we can find numerous domains which utilize the

power of artificial neural networks and deep

learning models. A groundbreaking work that set the

ground for further developments was the universal

approximation theorem, which proves that an

artificial neural network can approximate any

continuous function in a closed interval based on

input variables. The importance of the theorem is

quite high as, based on it, we can use an artificial

neural network, even with one hidden layer, to

model any complex non-linear relationship, [9].

This is a key benefit of artificial neural network

models, as most real-world problems cannot be

modeled and solved analytically.

A domain where complexity and non-linearity

are combined with real-time transactions and

stochastic processes, and where mathematical

modeling is not feasible, is derivatives markets.

Under some abstractions and limitations, we can

model analytically, but again not all problems can

be solved. Option pricing is an example of a key

problem for the financial industry, that can partially

be modeled by parametric methods and solved

analytically. Black-Scholes variations and Monte

Carlo simulation are the key parametric traditional

methods. However, following the advent of data and

artificial neural networks, researchers in the nineties

proposed alternative nonparametric approaches

based on machine learning. They were among the

first researchers to study a data-driven approach and

propose the utilization of artificial neural networks

for option pricing, [10]. Those initial approaches

opened a new research direction for financial

derivatives pricing using machine learning methods.

Some early works reported a quite high level of

accuracy, [11], [12], followed by recent works with

rich analysis and benchmarking, [7], [8]. As the

number of works in machine learning-based option

pricing is increasing, not all researchers agree to

positive results. Controversy on whether artificial

neural networks outperform compared to traditional

methods and in what settings is still under research.

Usually, works were based on plain neural network

architectures and limited data, so reported weak

results for neural networks compared to Black-

Scholes, something expected as neural networks

require large training datasets. Also, some

researchers claimed that results differ if we examine

options in the money out of the money, or other

factors. However, an increasing corpus of papers

agree on the high level of neural network accuracy

for option pricing compared to traditional models.

It is important to mention though that in all

works the Black-Scholes model is still used as a

benchmark to test for errors in pricing. So, despite

the promising performance of artificial neural

networks, theoretical models are still dominant.

However, the research direction is towards

developing neural network architectures that can

offer increased accuracy.

3 Data and Methods

This work aims to explore the accuracy level of

neural network architecture for call option pricing

estimation using empirical data for testing. The

approach we followed is to:

• use a multilayer perceptron network architecture,

• train it using artificial data generated from the

Black-Scholes formula, which is considered a

benchmark method for all option pricing

methods, and next,

• test the network in real call options market data

for a portfolio of thirty-five stocks randomly

selected from the S&P 100.

The key research question that we explore is how

well an artificial network performs in real market

data when trained with synthetic data. This work

builds on some related works, [13], [14], however it

examines real testing data, instead of artificially

generated. Our approach examines the case that

testing data, that a neural network is going to use for

prediction, does not follow the same distribution as

the training data. In similar works, we see that the

performance of multilayer perceptron networks in

option pricing is quite high, using artificial data for

both training and testing, something that is

expected, in general. So, we learn the Black-Scholes

with artificial data and test the accuracy in real data.

The approach we followed comprises the phases

below:

• Generate artificial call options data using the Black-

Scholes formula for a range of realistic values.

• Define a multilayer perceptron model with initial

parameters.

• Train the model with the artificial dataset.

• Validate the model with a subset of the artificial

dataset.

• Collect real market data for call options for thirty-

five randomly selected S&P100 stocks.

• Test the model with the real market dataset.

• Evaluate the model using the real market data as the

benchmark.

The entire work for the data, both the generation

of artificial data and collection of market data, was

executed by specific modules developed in Python

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3.11, [15]. The multilayer perceptron was

implemented in Python, using the Keras library and

Tensorflow as the computational engine. For the

computations, a typical desktop computer was used

with Intel Core i5 at 2.90 GHz and 8GB RAM.

3.1 Training Dataset

The training dataset was generated with artificial

data using the Black-Scholes formula for a range of

values, replicating many call options. Even though

in real trading call option prices deviate from the

Black-Scholes formula, it is a baseline method to

calculate option prices formally. For this work, we

generated around 6.5 million call option prices in

total, with the process taking 19 minutes of CPU

time. For the generation, we used stock price values

ranging between 10 and 200 USD, with strike prices

as a multiple of stock prices to avoid extreme

values. So, strike prices range between 10 and 300

USD. The volatility was selected between 10% and

60% with a step of 5%, and the risk-free rate was

ranging between 1% and 2%. Finally, the time to

maturity was selected between 0.1 and 1 year. In

Table 1 the values for the training set are

summarized.

Table 1. Parameters for the 6.5m call option prices

artificial dataset.

Parameter

Range

Strike price (K)

0-290 (USD)

Dividend rate (q)

0%

Volatility (a)

10%–60%

Stock price (S)

10–200 (USD)

Maturity (T)

0.1 to 1 year

Risk-free rate (r)

1%–3%

The distributions of the generated call prices

and the strike prices are depicted below for

reference (Figure 1, Figure 2). The stock prices

follow a uniform distribution, while the strike prices

and the call prices are right-skewed. Even if the

dataset is artificial, as soon as the objective is to

learn the Black-Scholes formula, the call price is

generated in the dataset by the actual Black-Scholes

formula, and this is used in the training phase from

the artificial neural network to learn the formula.

Fig. 1: Artificially generated strike price

distribution

Fig. 2: Artificially generated call prices distribution

3.2 Testing Dataset

For the testing phase, we utilized data originated

from publicly available market data for thirty-five

randomly selected S&P 100 stocks. As market data

do not strictly follow the theoretical calculations,

and on the other hand include some extreme values

that are not met in practical trading, we performed

several adjustments. In total, 3,500 records were

collected. The decisions taken for data preparation

are the following:

• The stocks selected are a random subset of S&P

100 stocks, to include more diverse data, instead

of picking based on some criterion, like revenues

or market capitalization.

• Stock prices refer to the closing price of the

previous day. We compared the closing prices

with the ask and the average of the bid and ask,

and we did not see deviations, so we kept the

previous closing price.

• Dividend was collected from market data as

provided (forward dividend and yield).

• For the implied volatility we used the market-

provided volatility, as the mean value over the

last 30-day period, derived from the average of

the put and call implied volatilities for options

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with the relevant expiration date, based on

market data.

• We focused on call options, but the same

analysis can be applied to put options.

• We selected in-the-money call options.

• We filtered the data, excluding not realistic and

non-representative observations from the data to

obtain more meaningful results.

Some further filtering was applied, as followed

in other works, [13], [14], to exclude nonrealistic

and non-representative observations. Some very

high option prices were excluded to avoid large

deviations between theoretical and observed option

prices. The distributions of the call and strike prices

for the dataset are depicted below (Figure 3, Figure

4, Figure 5). The stock prices vary while the strike

prices and the call prices are right-skewed, in a

similar way to the artificial dataset. Also, we

calculated the theoretical call option prices using the

Black-Scholes formula to use them as a benchmark.

In general, it seems that even if the market data can

be considered they depart from the theory, the

distributions are not substantially different from the

artificial data.

Fig. 3: Market collected call prices distribution

Fig. 4: Market collected strike prices distribution

Fig. 5: Actual call prices versus theoretical BS

calculated

3.3 Artificial Neural Network

For the artificial neural network, we selected the

architecture of a multilayer perceptron (MLP), as a

commonly used approach for such works and it also

fits well in finance settings.

The model was trained using the entire set of

parameters as input features:

• strike price,

• stock price,

• risk-free interest rate,

• time to maturity, and

• volatility,

• and the call price as the output.

Following some similar approach, the input

variables were normalized, [10]. After some

experimentation, we used a network with one input

layer, three hidden layers of 120 neurons each, and

one output layer for the call option price output. The

first and the third hidden layers utilize the Elu

activation function and the second the Relu

activation function. The model was trained using the

artificial dataset of 6.5m instances, split into 80%

subset used as training sample and the remaining

20% used as validation sample. The test was

performed on the real market dataset and not the

synthetic one, where the performance of the model

was evaluated. The model hyperparameters were

tuned to a 25% dropout rate, a rule-of-thumb

dropout rate to prevent overfitting. The number of

epochs used for training was 100 and the batch size

(the number of samples processed before updating

the model) was set to 64. Finally, the loss function

was optimized using mean square error (MSE).

4 Results and Discussion

The key research question in this work is to explore

the accuracy of an artificial neural network in

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estimating option prices on real market data, having

been trained with artificial data originating from the

Black-Scholes formula. The approach we followed

was to test the model on a random set of S&P 100

stocks and their market-based option data after we

had trained it and tuned the hyperparameters with

synthetic data. For both training and testing we

utilized Python tailor-made libraries along with the

Tensorflow module, [15]. The set of tuned

parameters was exported into a file that was used in

all testing scenarios using the market dataset. We

focused on in-the-money call options, but the same

approach can fit at out-of-the-money options.

The results from the testing phase of the model

are presented in Table 2, and the normalized

predicted call prices against the actual ones are

depicted in Figure 6. As we can see, the Root mean

square error is equal to as low as 06560. Also, from

the histogram (Figure 7) of the differences between

actual and predicted call prices, we can see that the

error is very small in general.

Table 2. Testing Error Results with Market Data

Mean Squared Error:

0.004304196575715178

Root Mean Squared Error:

0.0656063760294316

Mean Absolute Error:

0.04031098310438521

Mean Percent Error:

0.22251207590972308

Fig. 6: Predicted call prices against actual ones

Fig. 7: Histogram of difference between predicted

and actual prices

To check model accuracy, we performed

additional testing with the market dataset, but we

replaced call option price, as the output variable,

with the value calculated from the Black-Scholes

formula. Even if the market price is not identical to

the calculated one, as shown previously, it can be

used as a benchmark.

So, for the in-the-money call options, the results

are presented in Table 3, and the normalized

predicted call prices against the real ones are

depicted in Figure 8. As we can see, the Root mean

square error is as low as 0.0284 that is comparable

to the market dataset, and from the histogram

(Figure 9) of the differences between actual and

predicted call prices, we can also see that the error

follows the same pattern as in the network.

Table 3. Testing Error Results with BS prices

Mean Squared Error:

0.0008069991156531251

Root Mean Squared Error:

0.02840772985743713

Mean Absolute Error:

0.02018331088989721

Mean Percent Error:

0.06679347660794391

Fig. 8: Predicted call prices against actual ones

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Fig. 9: Histogram of the difference between

predicted and actual prices

So, overall, we can claim that the model, even

in a preliminary setting, is comparable to the Black-

Scholes formula for calculating option prices at

market data and can evaluate options in acceptable

accuracy. Provided that market prices are not strictly

derived from the Black-Scholes formula, it is

reasonable to assert that the level of error is within

reasonable limits. Some additional experiments can

be performed including put options and combining

variations of volatility estimations for both in and

out-of-the-money options. Also, the trained model

can be benchmarked to various alternative machine

learning models in terms of accuracy and

computational performance.

5 Conclusion

In this work, we explored the accuracy of an

artificial neural network on call option pricing using

real market data for testing and artificial option

pricing data for training. We used a multilayer

perceptron model, a large synthetic dataset for

training, generated from the Black-Scholes formula,

and a real market dataset, comprising thirty fine

randomly selected S&P 100 stocks. As the baseline

for pricing errors and estimations in the study, we

used the Black-Sholes model. From the results, we

can see that a multilayer perceptron is capable of

learning the BS function accurately using synthetic

data and estimating prices for options with a high

level of accuracy, competitive with the Black-

Scholes formula.

Other relevant works using artificial neural

networks conclude in similar results, however, this

work adds the experimentation of using actual

market data. Provided that the model is not static,

but it can be retrained using additional data,

including mixed artificial and actual data, its

accuracy can be increased, and it can become more

valuable for practitioners, who might select machine

learning paradigms for option pricing in various

assets and markets. Some limitations in this work

include the training sample, the specific network

architecture, and the limited focus on S&P 100

stock options. As artificial neural networks are data-

driven, developing appropriate training datasets is

critical for their performance, so there is a need for

diverse training datasets. Also, in this work, we did

not proceed to feature engineering or advanced

sampling for the training, something that can be

examined further in subsequent works. In addition,

alternative network architectures can be tested or

further experimentation with hyperparameters can

be performed, and focus can be expanded to

additional assets. In the future, we plan to develop

training processes using market data from a variety

of sources.

Despite the limitations, it is evident that

machine learning models can be used by

practitioners as main or alternative methods for

option pricing, however, it is necessary to build

appropriate user-friendly software solutions to

deploy similar machine learning models on web

environments or mobile phone settings. This work,

and any future contributions, aim to the

development of this fast evolving area.

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Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The author developed the present research in its

entirety.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

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DOI: 10.37394/232032.2024.2.2

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