PRICING OF CALL OPTIONS USING THE QUASI MONTE CARLO METHOD

ABSTRACT


INTRODUCTION
In the world of business and finance, investment activities are no longer a new thing.Investment activities are a form of fund allocation to get future profits [1].An alternative form of investment that is currently popular is derivative products.A derivative product refers to a type of financial instrument that heavily relies on the value of the underlying asset.The function of derivative products is to avoid investment risks and increase profits.One form of derivative product is an option.An option is a contractual agreement between the option buyer and the option seller that contains the right, not the obligation, for the option buyer to sell or buy a specific asset at a predetermined price within a specified time frame [2].
Investors can use options to prevent the impact of falling market prices on their portfolios and to speculate on stock price movements.However, a common problem with stock options trading is the losses that investors incur.One of the factors leading to losses is due to a lack of knowledge about strategies for pricing options.Therefore, determining the option price within a certain period of time is an important part of determining strategies for investing.The option price is the price that the option holder must pay to the option seller to obtain the option right [3].Various approaches exist for determining option prices, namely Monte Carlo and Quasi-Monte Carlo simulations.
Previous research indicates that when utilizing the Quasi-Monte Carlo method, the valuation of option contracts demonstrates accelerated convergence and yields a smaller standard error compared to the Monte Carlo method [4].Another research study, conducted by [5] has demonstrated that the utilization of Halton's quasi-randomized sequence within the Quasi-Monte Carlo method yields more accurate results because it produces a smaller error value, which means that the simulation in the Quasi-Monte Carlo method is closely the market's option price.In addition, recent research conducted by [6] which aims to determine the pricing of European-type stock options using the Quasi-Monte Carlo method with Halton's quasi-randomized sequence shows results in the form of the more simulations performed, the smaller the standard error value produced so that the option calculation results will be more convergent.
The Monte Carlo simulation method is a computational algorithm that requires repetition to assess risk by calculating the probability of the final result due to uncertainty involving random variables [7].The Quasi-Monte Carlo method is an improved method of the Monte Carlo simulation because it makes the Quasi-Monte Carlo method effective for use in determining option prices.The Quasi-Monte Carlo method serves as an improvement to the Monte Carlo simulations, offering a more precise approach.The Quasi-Monte Carlo is a variety of the Monte Carlo simulation that utilizes quasi-randomized sequences as a substitute for the pseudo-random sequence [8].A quasi-randomized order is a random number that has a certain pattern.The quasi-randomized sequence is divided into Halton, Van der Corput, Faure, and Sobol.
The most basic and most commonly used quasi-randomized sequences in the multidimensional form are Halton's quasi-randomized sequences [9].This research uses Halton's quasi-random sequence in Quasi Monte Carlo simulation.Then calculate the standard error value of the call option price.This method will be used to determine the price of European-type call options on Apple Inc. for the period October 1, 2021 to September 30, 2022.

Options
Options as derivative products whose value is highly dependent on the underlying asset.Options can be divided into two types, namely based on the time of exercise and based on the rights owned by the holder.According to [10], options based on the time of exercise consist of two, namely: a.An American-type This option is a contract that grants the holder to exercise the option either at the expiration date or before expiration date.

b. A European-type
This option is a contract that grants the holder to exercise the option only at the expiration date.
While options are divided into two parts according to the type of rights granted to the option holder, namely: a.A put option is an option type that provides the holder to sell an asset at a predetermined price within a specified time frame.The equation representing the payoff of a put option is as follows.
= ( −   , 0) b.A call option is an option type that provides the holder to buy an asset at a strike price within a specified time frame.The equation representing the payoff of a call option is as follows.
with   represents the call option payoff,  represents the strike price, and   denotes the stock price at maturity [11].

Stock Price Movement Model
One of the factors that affects option price trading is stock prices.Stock prices or underlying assets often occur due to demand and supply, which causes stock prices to fluctuate over time.The model for stock price movement follows Brownian motion, which assumes that the returns of stock prices follow a normal distribution.Brownian motion is a stochastic process characterized by relatively rapid changes occurring within a short timeframe [12].If   expresses the stock price at time t, then the expectation of   can be assumed as   with  denotes the constant expected stock price return.Suppose  is defined as the stock price volatility then the model for stock price movements becomes: where dW is Brownian motion.
Furthermore, Lemma Ito is used to calculate the price and time functions in the stock price movement model.By applying Lemma Ito for  = ln(  ), so the stock price movement model can be expressed in Equation ( 4) [12]: where   , ,  0 , T and  are the initial stock price at maturity, volatility, stock price, maturity time, and risk-free interest rate.Notation  in some literature is also denoted as r [13].

Normality Test
The Kolmogorov-Smirnov test is employed as a normality test to ascertain whether a specific data distribution adheres to a normal distribution.This test compares the empirical cumulative frequency (observed) with the cumulative frequency of the theoretical distribution [14].The hypothesis utilized in the Kolmogorov-Smirnov test is as below.
0 : data is normally distributed  1 : data is not normally distributed Furthermore, the largest standard deviation can be calculated using Equation ( 5): where   () is the cumulative frequency distribution and   () is the observed cumulative frequency distribution.If  <   then  0 is not rejected, which means that the data is normally distributed [14].

Return, Variance, and Volatility
Return is the result obtained from a stock investment.The return value on stocks can be positive or negative [15].If the stock return is positive, it means getting a profit (capital gain), while if the stock return is negative, it means experiencing a loss (capital loss).Stock price return can be calculated using Equation (6).
with   expresses the stock price at time t,   expresses the stock price return at time t, and  −1 expresses the stock price at time t-1 [6].
Variance is a measure of the magnitude of the spread of data that shows how much the random variable spreads to the average.The equation to calculate the variance value can be done using Equation (7).
with  2 denotes the variance,   expresses the stock return at time t, n denotes the number of stock price data, and () denotes the expected stock price return [6].The equation for calculating stock return expectations is as follows: Volatility is a measure of rapidly fluctuating stock price changes.The volatility that describes annual stock price movements in the past (historical data) can be calculated using Equation (9).
with n stating the number of days in a trading year and  2 denotes the variance of stock price returns [6].

Quasi-Monte Carlo
The option pricing method known as the Monte Carlo simulation method was first introduced by Boyle in 1977.Monte Carlo simulation is a numerical analysis method that involves random number trial samples and requires repetition, which is generally done using a computer.Random numbers are used to describe random events at any time from random variables and sequentially follow any changes that occur in the simulation process [7].
The foundation of the Monte Carlo method lies in the concept that an integral in the form of ∫ () This result can also be applied to calculate multidimensional integral forms, so Equation (10) can be written as follows.
Over time, in 1995, Paskov and Traub introduced the Quasi-Monte Carlo method as an approach to pricing options.Quasi-Monte Carlo shares similarities with Monte Carlo simulation, but it uses quasirandomized sequences as a replacement of pseudo-random sequences.Quasi-randomized sequences can generate representative samples of the probability distribution simulated in a problem.These sequences, often referred to as low discrepancy sequences are used because they provide more accurate results that can improve the performance of Monte Carlo simulation [8].The key distinction between Quasi-Monte Carlo and Monte Carlo simulation is that when the data is generated repeatedly, the sequence generated by quasirandomized will be the same because they already have their patterns, while the sequence from the Monte Carlo simulation will produce different values for each repetition [17].
The Quasi-Monte Carlo method uses a quasi-randomized numbers   to approximate the function f as the average of the functions evaluated on the set of points  1 ,  2 , … ,   .
In the Monte Carlo simulation,   is a pseudo-random sequence, while in the Quasi-Monte Carlo method,   is a quasi-randomized sequence [16].
The Quasi-Monte Carlo method requires a uniform quasi-randomized sequence so that the results given are more accurate.Several quasi-randomized sequences can be used, including Halton, Faure, Van der Corput, and Sobol.

Haltons's Quasi-Randomized Sequence
Halton's quasi-randomized sequence is the most basic low discrepancy sequence in multidimensional form.Halton's quasi-randomized sequence is a d-dimensional sequence on the unit hypercube [0,1]  [5].For example, given an integer  ≥ 1, Halton's quasi-randomized sequence can be written as follows: where n is the 1st, 2nd, ..., Mth terms of the sequence,   () is a constant in the sequence i that is nonnegative to satisfy the value of n, b is a base of prime numbers, and I is the smallest integer from | ln() Furthermore, the value of   () that has been obtained will be entered into Equation (15).
The d-dimensional Halton's quasi-random sequence can be written as follows: () = (  (1),   (2), … ,   ()) The first dimension of Halton's quasi-randomized sequence is Van der Corput's quasi-randomized sequence of base 2 and the second dimension of Halton's quasi-randomized sequence is Van der Corput's quasi-randomized sequence of base 3.The d-dimension of Halton's quasi-randomized sequence is Van der Corput's quasi-randomized sequence using the d-th prime as the base [18].
For example, the following quasi-random Halton series of dimension one, dimension two, and dimension three for the first term to the sixth term are given in Table 1 [11].
The steps taken to determine the European-type call options price using the Quasi-Monte Carlo method with Halton's quasi-randomized sequence are as follows.
a. Calculating the stock price return value (  ) using Equation (6).
b. Normality test on stock price returns.c.Calculating the value of stock price volatility () using Equation (9).d.Determining parameters obtained from historical data in the form of risk-free interest rate (r), initial stock price ( 0 ), strike price (K), maturity time (T), and volatility ().
e. Generating quasi-random Halton numbers and simulating stock prices (  ) by substituting the parameters.
f. Determine the payoff value of the call option using Equation (2).
g. Estimate the call option price by averaging the call option payoff values.

RESULTS AND DISCUSSION
The data utilized in this research is the daily closing stock prices of Apple Inc. (AAPL) obtained through https://finance.yahoo.com.The analyzed data is from October 1, 2021, to September 30, 2022, totaling 252.The first step is calculating the stock price return using Equation ( 6).Based on the data used, 251 stock price returns were obtained.

Normality Test of Stock Price Return
The stock price return obtained is then tested for normality assumptions first.The normality test is used to identify whether stock returns are normally distributed or not using Kolmogorov-Smirnov (KS).The Kolmogorov-Smirnov test hypothesis used is as follows.
H0 : the returns of Apple Inc.'s stock price are normally distributed H1 : the returns of Apple Inc.'s stock price are not normally distributed Using R software, the results are shown in Table 2.In this research, the significance level or α value is 5%.Based on Table 2, p-value more than the value of α, so H0 is not rejected.This means that the stock price return of Apple Inc. is normally distributed.

Volatility
After conducting the normality test, the subsequent stage involves computing the expected return, variance, and volatility obtained based on the daily stock return value using Equation (7), Equation (8), and Equation (9).
From the return that has been obtained, investors also need an expected return to get an idea of the benefits of investing in Apple inc.shares.The expected return is the rate of return expected by investors from an investment by averaging the return value.The expected return in this research is -0,0001.To find out how far the data spread from the expected return value, it is necessary to know the variance value.Higher values of variance indicate more risk in investment returns, while lower values of variance indicate that returns tend to be more stable.The value of variance in this research is 0,0004.Volatility describes the annual movement of stock prices in the past.The higher the stock price volatility value, the higher the option price.The volatility value in this research is 0,320.The volatility value obtained is the standard deviation value of the annual stock price returns of Apple Inc. for the period October 1, 2021, to September 30, 2022.

European-Type Call Options Price Calculation
Some of the parameters required to calculate the European-type call options price using Quasi-Monte Carlo are as follows.

Risk-free interest rate (r)
The risk-free interest rate used is the reference interest rate published by the US central bank of 4% or 0.04.The higher the interest rate, the higher the option price.

Initial stock price (𝑆 0 )
The price of the stock underlying the option at the time of calculation.The initial stock price in this period is $138,20.

Volatility (𝜎)
A measure of how much the price of the underlying stock fluctuates.Higher volatility tends to increase the price of the option.Annual stock price volatility is 0,320.

Maturity time (T)
The time remaining until the option expires.The longer the time remaining, the higher the value of the option.The maturity time used is one year.

Strike price (K)
The pre-determined price for the purchase of shares under the option.The strike price used is $50.
Next, Halton's quasi-randomized sequence is generated, and the stock price is simulated using Equation (4) by substituting the predetermined parameters.Then proceed to determine the payoff value of the call option using Equation (2).The next step is to estimate the price of the European-type call options by averaging the call option payoff values.The European-type call options price using Quasi-Monte Carlo with Halton's quasi-randomized sequence is given in Table 3.The results of the call option pricing using the Quasi-Monte Carlo simulation method are summarized in Table 3.As the number of simulations (M) performed increases, a convergence trend is observed, which is indicated by a decrease in the standard error.The European-type call option price for Apple Inc. stock determined at the 1,000,000th simulation is $90,163.This convergence shows that the calculated option price stabilizes with a smaller and smaller standard error, reaching 0.045 in the last simulation.This means that if the call option is purchased at a price greater than the Quasi-Monte Carlo calculated price, then investors should consider it before buying because the call option is overpriced.However, if the call option is purchased at a price smaller than Quasi-Monte Carlo calculated price, the investors are recommended to buy because the call option is underpriced.