Option pricing is one of the main topics in finance, closely related to the calculation of integrals. Traditional option pricing theory includes the Black-Scholes model, binomial tree, and the finite difference method. Most of these methods can be valued analytically.

As the path of the underlying asset becomes more complex and dynamic, it might be better to use numerical methods such as Monte Carlo simulations to remove the assumptions of the analytical method. Simulation methods have proven to be valuable and flexible computational tools. Monte Carlo simulation, in particular, performs well in a variety of environments due to its ability to cope with uncertainty. However, one of the drawbacks of this method is that the standard error of the estimate increases with the number of simulation runs. As an alternative, Boyle (1976) and Larsson (2020) found that reducing variance is an effective way to reduce errors and obtain more accurate results.

Data-driven models also draw attention to pricing financial derivatives. Studies using machine learning have shown that future stock prices can be predicted with near-accuracy based on past stock prices. Chowdhury et al. (2020) concluded that data mining results are similar to those of the Black-Scholes equation. Furthermore, Ivaşcu (2021) confirms that decision tree algorithms such as Random Forest, XGBoost, and LightGBM are effective for option valuation. More recently, Ke and Yang (2019) found that deep learning models such as LSTM outperform Black-Scholes.

In this paper, we predict the price of KOSPI200 options and show the results of investment simulations. We use traditional methods including the Black-Scholes model and binomial tree, and simulation methods such as Monte Carlo and importance sampling. We also apply machine learning and deep learning models. Besides, we want to see how well these methods work for different timeframes. By examining these models closely, we aim to offer valuable insights that can help investors and financial professionals make better decisions in the dynamic world of finance.

In Section 2, we describes the concept of options. We briefly explain the concepts of option positions, call and put options, and their payoffs. Section 3 describes various option pricing methods including the traditional option pricing models such as the Black-Scholes model, and the binomial tree, and also the non-traditional option pricing methods. Simulation methods, such as Monte Carlo simulation, antithetic sampling, control variables, and importance sampling are used to improve the limitations of traditional methods. We also briefly discuss machine learning and deep learning methods which use historical data to predict option prices. Section 4 shows the results of our experiments. We compare the actual and predicted prices of KOSPI200 options. We also demonstrate an investment simulation that shows the expected return rate of option pricing methods. Finally, Section 5 summarizes and concludes our paper.

An option is a type of derivative, a contract that exchanges the right to buy or sell at a predetermined price within a predetermined period of time. A person who buys an option is called an option buyer, and a person who sells an option is called an option seller. The right to purchase a specific product at a set price on the option expiration date is called a call option, and the right to sell it is called a put option. Options have value because they are a type of complex transaction, and are affected by the price of the underlying asset, the period until maturity, market volatility, risk-free interest rate, and the exercise price of the option.

The type of option is primarily defined by the date on which the option is exercised. Most options are European or American options. European options are contracts that can be exercised only at expiration, while American options can be exercised at any time before. Options that are calculated similarly to these two options are called “vanilla options.” In contrast to regular vanilla options, “exotic options” have relatively complex and unusual features. Exotic options may include products developed for specific markets, and representative examples include Asian options and binary options.

There are three terms that describe where an option is traded in relation to the price of the underlying asset. These terms are called “at the money,” “in the money,” and “out of the money.” If there is intrinsic value by comparing the exercise price of the option and the price of the underlying asset, it is called an in the money state. If there is no intrinsic value, it is called an out of the money state. When the spot price and futures price are the same, the state is at the money (Larsson, 2020).

If S_t is the underlying stock price and K is the strike price, the call option can be defined as follows:

• In the money status : S_T > K

• At the money status : S_T = K

• Out of the money status : S_T < K

On the other hand, the put option is as follows:

• In the money status : S_T < K

• At the money status : S_T = K

• Out of the money status : S_T > K

In options trading, “long position” and “short position” are terms that refer to holding and selling an option contract, respectively. Long position refers to purchasing and holding a specific type of option contract. Options held by the buyer can only be exercised during a certain period of time in the future, and if there is profit, profit can be earned by exercising the option contract. Short position refers to selling a specific type of option contract. Sellers make a profit by writing option contracts and selling them to other investors. In other words, a long position is when the stock price is expected to rise, and a short position is when the stock price is expected to fall and the person is trying to make a profit.

The position of the call option can be expressed as:

• the payoff from a long position in a call option : max(S_T − K, 0)

• the payoff from a short position in a call option : min(K − S_T, 0)

The Put option is expressed as:

• the payoff from a long position in a put option : max(K − S_T, 0)

• the payoff from a short position in a put option : min(S_T − K, 0)

Determining option prices is one of the most important aspects of derivatives trading. Among them, the most traditional and famous methods are the Black-Scholes model and binomial tree model. The Black-Scholes model was designed by Fisher Black and Myron Scholes in 1973. With this model, they presented a closed-form mathematical formula for measuring the price of European call options. However, even after the Black-Scholes model was introduced, limitations were revealed in the market, making it difficult to apply this model in some special situations. In particular, because American options can be exercised at any time, a model that considers discrete times was needed. Accordingly, the binomial tree model developed by Cox-Ross-Rubinstein appeared in 1979. This model considers stock price movements at discrete time steps and is useful when evaluating more complex options.

3.1. Black-Scholes model

The Black-Scholes model calculates the theoretical price of an option by considering the stock price, exercise price, interest rate, investment period, and volatility of the underlying asset. At the time of its introduction, this model was recognized as a simple and effective tool for calculating accurate prices. However, since the assumption about the random movement of stock prices is made using a stochastic process called geometric brownian motion, there are limitations in assuming that this assumption always accurately represents stock movements in the real world. The model also makes several assumptions: Arbitrage opportunities do not exist in the market, there are no taxes, and the risk-free interest rate is constant and volatile.

Basically, the Black-Scholes model shows how to determine the price of an option contract using a simple formula. The calculation formulas for European call option c and put option p are as follows (Chowdhury et al., 2020).

where, S is the stock price; K is the exercise price; r is the risk free interest rate; T is the time to expiration; σ : volatility of stock price; q is the dividends; N(x) is the cumulative normal distribution function (MacBeth and Merville, 1979).

3.2. Binomial tree model

The binomial tree model is a model that considers option prices at discrete time steps. This model calculates the price of an option by dividing it into several time stages until the expiration date of the option and creating a binomial tree in which the stock price rises or falls at each stage (Rubinstein, 1994). Each node represents the stock price at a specific point in time and the option price at that point in time. The model calculates option value by considering possible future price movements at each node of the tree, which includes calculating the intrinsic value of the option and the expected value of the option in the future. The binomial tree model is recognized to converge to the Black-Scholes model as the number of time steps increases. This will be confirmed by the experimental part using real KOSPI 200 options data.

Consider the binomial tree model of the first step. This model starts at time t = 0 when the underlying asset price is S_t. The price may rise by u or fall by d at each step. At this time, if the probability of rising is p, the probability of falling is 1 − p. Therefore, the price at time t = 1 is as follows (Dar and Anuradha, 2018).

As a result, according to Benninga and Wiener (1997), if the number of price increases during t period is j, the stock value at time t becomes

3.3. Monte-Carlo method and variance reduction techniques

In the experimental part, we predict the price of KOSPI200 options and calculate the profit/loss of an investment simulation. Accordingly, we find out which method is the most predictive and which is the most profitable. The price prediction is performed for three cases: Weekly, monthly, and 50 days to maturity referring to Martinkute - Kauliene et al. (2013).

The methods used to predict option prices and calculate profit/loss include traditional methods, and models in Monte Carlo simulations with variance reduction techniques. We also make a comprehensive comparison using machine learning methods such as XGBoost, random forest, and multilayer perceptron.

The machine learning methods include inputs such as S (stock price), K (strike price), r (risk-free rate), T (time to expiration), σ (volatility), and q (dividend rate), similar to other methods for fair comparison. However, machine learning methods necessitate training data, unlike traditional methods where input values can be directly inserted into a predefined formula. Consequently, in a machine learning method for predicting option prices, data from up to one day prior is employed as the training data.

We use XGBoost (Chen and Guestrin, 2016) without parameter tuning and for random forest (Breiman, 2001), we set the number of estimators to 500, maximum depth to 8, and maximum features to 5 to optimize performance. The MLP (Haykin, 1994) model was built based on Culkin and Das (2017). The first layer consisted of 120 nodes, and the model consisted of a total of 7 hidden layers. The output layer comprised a single node with a linear activation function to predict option prices. For training the MLP, we employed the mean squared error (MSE) loss function and RMSProp optimizer. We trained the model using a batch size of 32 and iterated over 50 epochs to optimize performance.

4.1. KOSPI 200 data description

The KOSPI 200 index includes 200 stocks that best represent the market capitalization, liquidity, and industry composition of the KRX (Ahn et al., 2008). The KOSPI 200 option, introduced in July 1997, is an option based on the KOSPI 200 index as an underlying asset. Secondary derivatives exist such as the KOSPI 200 Weekly option, which has a one-week expiry, and the Mini KOSPI 200 option, which enables small investments by reducing the contract size by 1/5. The KOSPI 200 option is a European option that can only be exercised at expiration, and the expiration date is fixed to the second Thursday of every month. The trading history of KOSPI 200 options, including S (stock price), K (strike price), T (time to expiration), σ (implied volatility), and q (dividend rate) was obtained from data provided by KRX market data system. We collected the data by crawling the option transaction history. The r (risk-free rate) corresponds to the KOFR (Korea overnight financing repo rate). The price of the KOSPI 200 option is expressed in premium points. The multiplier is 250,000(KRW), which means that the actual price equals the quoted price multiplied by 250,000(KRW).

We use one year of data from January to December 2023. However, we need earlier data to apply the machine learning method in predicting Jan 2023 option price. Therefore, we use data from November 2022 for our analysis.

Table 1 presents a summary of call and put options within the dataset. The analysis reveals a higher transaction volume for put options as opposed to call options. Furthermore, it is observed that put options exhibit lower maximum and quantile values compared to their call counterparts. This can also be seen in the histogram in Figure 1, where the number of options close to 0 is much higher for puts than for calls.

4.2. Results

This paper demonstrates a comprehensive examination of various option pricing methods ranging from traditional models like the Black-Scholes and binomial tree to advanced techniques involving machine learning and simulation methods. The primary objective was to discern the most effective methods for predicting KOSPI200 option prices and to evaluate the profitability of these predictions through investment simulations.

Our findings reveal that traditional models such as the Black-Scholes and the binomial tree models have shown superior performance in predicting option prices. This allows us to make reasonable inferences about using these models in actual option pricing.

Machine learning models, especially random forest and XGBoost, fell short in accuracy when predicting option prices compared to traditional methods, but performed well in maximizing returns on investment. This outcome was remarkable in January 2023, leveraging the difference between predicted and actual prices to find undervalued options, generating significant profits.

This study suggests that while traditional models are efficient in terms of prediction accuracy, machine learning methods can provide investors with a competitive edge by finding opportunities that traditional models might overlook. The adaptability of these methods to various market conditions and their ability to handle complex, dynamic scenarios prove invaluable in the constantly evolving finance fields.

It’s important to note that these results are specific to a particular option and time period, so generalization is not advisable. Future research could explore a more thorough study with a wider variety and longer period of options data. By continuing to refine these models and strategies, the financial community can better explore the intricacies of market dynamics, enhance investment strategies, and ultimately achieve more robust financial outcomes.