GEOMETRIC BROWNIAN MOTION IN ANALYZING SEASONALITY OF GOLD DERIVATIVE PRICES

Complex financial markets, influenced by complex and interconnected factors, require proper decision making. Gold derivatives, as an increasingly popular trading instrument, have experienced significant growth. However, with high profit potential also comes significant risk. Market analysis, including technical analysis and leveraging seasonality, can be an important tool in reducing risk and making smart decisions. This research aims to assess the effectiveness of the Geometric Brownian Motion (GBM) model in predicting Gold Derivative prices through the application of the Mean Absolute Percentage Error (MAPE) test.In this study, the Brownian Motion Geometric Method and Simple Moving Average are combined to analyze the seasonality of gold derivative prices to provide a view of price movement patterns. The results showed that the Brownian Motion Geometric Method was effective in predicting the price of gold derivatives, with a low error rate. In addition, seasonality analysis reveals monthly price movement patterns that can be a guide for traders and investors. This research provides valuable insights for decision making in gold derivatives trading in dynamic and complex financial markets.


INTRODUCTION
In the financial market, there is a complex system that is influenced by various factors that are interconnected and not linear. Therefore, understanding when and how to make decisions in trading this market is very significant (Naranjo et al., 2018). Currently, many investors and traders are interested in derivative trading instruments. This instrument patterns formed using the Simple Moving Average (SMA) method. The results of the combination test between these mathematical theories will be used as a reference in determining the trend seasonality of gold derivative price movements. The seasonality of price movement trends can be used as a description and reference for traders and investors in conducting technical and fundamental analysis.
This research aims to assess the effectiveness of the Geometric Brownian Motion (GBM) model in predicting Gold Derivative prices through the application of the Mean Absolute Percentage Error (MAPE) test. The obtained test results were used to determine the Seasonality trend, offering traders and investors valuable insights for conducting technical and fundamental analyses.

Return
Return asset or is the rate of return or profit generated from an investment in a financial asset. Return for a period( ) with is defined as the logarithmic comparison of the asset price and the asset price in the previous period. Mathematically, it is formulated as follows ( = 1,2,3, … , ( ) ( ) (Ruppert & Matteson, 2011):

Normal Distribution
The normal distribution is the level of data distribution that collects around the mean (mean) and spreads to the right and left with a certain standard deviation. The mean determines the location of the center of the curve, while the standard deviation controls how wide or narrow the data spreads around the mean. A random variable with the opportunity density function can be expressed as a normal distribution with parameters and if it satisfies the following equation ( ( ) (Navidi, 2006): for , , and . Mathematically, it is written as .−∞ < < ∞ − ∞ < < ∞ > 0~( , 2 )

Log-Normal Distribution
The normal distribution is generally not suitable for data with high skewness or outliers. The log-normal distribution, which is a subset of the normal distribution, is often a good choice for such data sets (Navidi, 2006): The relationship between normal and log-normal distribution can be stated as follows: -If~( , 2 ), then the random variable is log-normally distributed with parameters and . = 2 -If is log-normal distribution with parameters and 2 , then the random variable = ln is normally distributed ( , 2 ). The probability density function of a log-normal random variable with parameters and is:

Kolmogorov-Smirnov Normality Test
Eduvest -Journal of Universal Studies Volume 3, Number 8, August, 2023 1561 http://eduvest.greenvest.co.id The Kolmogorov-Smirnov normality test is a statistical method used to test whether a data sample comes from a normal distribution or not. This test was developed by mathematicians named Andrey Kolmogorov and Nikolai Smirnov, whose aim is to determine the degree to which data fits a normal distribution (Siegel et al., 1997). If a function with a cumulative distribution is normally distributed with empirical cumulative probability , then p Testing the normality of the data using the Kolmogorov-Sminorv test procedure is as follows (Daniel, 1989): Determine the hypothesis 0 And : 1 0 : = (Sample data comes from normal distribution). 1 : ≠ (Sample data is not from a normal distribution).

2.
Set a significance level of 3.

Stochastic Process
The stochastic process {W,t.,tεT} consists of a group of random variables, where each W(t) is a random variable. The set T is the set of indices that represent time. If T can be calculated, then stochastic processes are categorized as discrete-time processes. In this case, T can be expressed as the set T={,t-1.,,t-2.,...,t-n.} where ,t-i. is a quantifiable value of time, such as a non-negative integer. If T consists of uncountable sets, then stochastic processes are categorized as continuous-time processes, suppose T is an infinite set [0.∞), then W(t) for t≥0 is a random variable representing continuous-time stochastic processes (Ross, 2014).
Theorem ̂ In general, in finance with a continuous time model, it is often assumed to be an It,o. Suppose there is a continuous function ( , ) which depends on the variables and , where is an ̂ process. which satisfies the stochastic differential equation ( Equation (2.9) is referred to as a formula ̂ Theorems ̂ become important tools in financial analysis and modeling involving stochastic processes and continuous time processes, and make it possible to calculate the expected values of process-dependent functions and Brownian Motion.

Geometric Brownian Motion
Geometric Brownian Motionor also known as the Wiener process, is a stochastic process that has a continuous nature. Brownian motion is formed from the symmetrical random walk equation ( ) by finding the limits of the random walk distribution. A stochastic process , ≥ 0} is called Geometric Brownian Motion if it fulfills the following three conditions (Dmouj, 2006): i.
(0) = 0, with a probability of 1. ii. Each change is normally distributed, which means it has a mean of 0 and a standard deviation of 1.
In general, the GBM model is expressed through equation (2.8) as follows:

Mean Absolute Percentage Error (MAPE)
Suppose there are two data sets with periods For = 1,2,3, … , that is actual ( )and predictive data, each of which consists of observations, then ( ) Mean Absolute Percentage Error (MAPE) with the following equation (Maricar, 2019):

The level of prediction accuracy scale can be concluded based on the following summary
The use of the Simple Moving Average to read trends can be seen in the following table. Source: Achelis, S.B

Seasonality
Seasonality in the financial market refers to the tendency found in the foreign exchange (forex) market where there is a pattern that repeats over a certain period of time. These patterns can be related to changes in trading volume, volatility, or the direction of price movements over a certain period of time in a year (Girardin & Namin, 2019). Seasonality refers to certain times of the year marked by changes, which may impact economic, political or business aspects. According to Fountas et al (2016), all physical activity, including activity in the forex and commodity fields, is influenced by seasonality.

RESEARCH METHOD
The data used in this study is historical data from the closing price of gold derivatives for a period of 10 years, from December 4 2012 to December 30 2022. The data source for this research was obtained from historical data on derivative gold prices on the website.investing. com. This study uses 4 research technical stages, namely as follows:

Application of the Geometric Brownian Motion Model
The technical steps taken at this stage are as follows: 1) Data grouping. At this stage, historical data on gold derivative prices are grouped based on the number of periods. 2) Determine the in-sample data. Data in sample is determined as much data, which will then be used to build the model. 3) Determine the data out sample. Data out sample is determined as much , used to validate the model. 4) Calculating the value of return on assets with data in sample. 5) Performing the data normality test in sample return on assets using the Kolmogorov-Smirnov test.
6) Calculating the expected value of the mean asset price parameter (), variance (̂2), and volatility (̂) from the in sample return data obtained 7) Modeling and predicting the price of gold derivatives using the Geometric Brownian Motion method. 8) Validating the model using out sample data and calculating the error value of gold derivative prediction prices using the Mean Absolute Percentage Error (MAPE) method.
9) Analyze the results of error calculations using Mean Absolute Percentage Error (MAPE). 10) Repeat the steps in points 4-7 to predict the price of gold derivatives in the 2023 period.

Trend Pattern Analysis Using the Simple Moving Average Method
The technical steps taken at this stage are as follows: 1) Group data. At this stage the data is grouped based on a period of 1 month. 2) Calculating the moving average value using equation (2.18).
3) Analyze trend patterns based on Table 2.3.

Seasonality analysis
The technical steps taken at this stage are as follows: 1) Grouping data from trend pattern analysis.
2) Analyze the probability of a trend pattern repeating.
3) Analyze the probability of the 2023 gold price prediction trend pattern.

Price prediction using Geometric Brownian Motion
To get the GBM model solution, assume a function ( , ) = ln . Using the theorem̂in equation (2.9), then the following equation is obtained: : Predict the price of gold derivatives at a point in time − 1 : Drift values : Value volatility In-sample data is used to build a gold price model, while out-sample data is used to validate the model. There are no standard rules regarding the comparison of the amount of in-sample and out-sample data used for predictive research, this is adjusted to the character and objectives of the research. Like the research conducted by Moniaga (2011) using a comparison of 70% in-sample data and 30% out-sample data, besides that another study conducted by Sari, et al (2020) used a comparison of 50% in-sample data and 50% out-ofsample data. out sample.
At this stage a prediction of gold derivative prices for a period of 1 year will be made using the Geometric Brownian Motion model to see its feasibility in predicting gold derivative prices, which are presented in the following table: 2019-2021 2022 3:1 Grouping is based on the number of periods, with the aim of seeing the amount of influence the data has on the prediction accuracy of research conducted with the assumption that the data is normally distributed. Data N1 uses an in sample and out sample ratio of 1:1, namely 2021 for in sample and out sample in 2022. Data group N2 uses a 2:1 ratio with 2 years of in sample data, namely 2020-2021. The N3 data group uses a data group of 3 years, namely 2019 to 2021. the normality test of the in sample return data on derivative gold prices using the Kolmogorov-Smirnov is as follows: Hypothesis: H0 : Data in sample returns are normally distributed. H1 : Data in sample return is not normally distributed. Significance Level :α = 5% The calculation results are obtained as presented in the following table: The estimated mean return, variance, and volatility will be the parameters in constructing the Geometric Brownian Motion model for derivative gold prices. Based on Equation (2.12) with the model parameters of mean return, variance, and volatility in Table 4.3, the derivative gold price model with Geometric Brownian Motion is as follows: data in sample and out sample, where the N1 data group shows better accuracy with a comparison of the actual amount of data in the sample and out sample by 1: 1 compared to N2 and N3 with a larger comparison in the data in the sample.
Based on the MAPE test analysis that was carried out in the previous step, the prediction of gold derivative prices in 2023 within a period of 1 year with as many as 260 predictive data will use historical data in 2022. Starting from January 3, 2022 to December 30, 2022.
The return value is tested for normality with H0 accepted because the value of ,Dmax.<,D-t. with α = 0.05, it can be concluded that the data in sample return is normally distributed and can be continued for estimation of model parameters.
Parameter estimation is obtained with a return expectation value (μ) of -0.00014016, return volatility (σ 2 ) of 0.0082134, and a variance value (σ) of 0.00006746.  At this stage, the calculation of the average moving value is carried out using the Simple Moving Average method. This calculation is carried out with the aim of smoothing out the fluctuations in the movement of the trend within a range of monthly periods with a value of 20. After getting the results of the calculation of the moving average, at this stage an analysis of the trend movement pattern is carried out. Drawing conclusions from the analysis of trend movement patterns is based on references in Table 2.3. which is depicted in the plot as follows.

Figure 3. Trend movement pattern in January 2013
From the movement in January 2013 it can be seen that the price of gold derivatives moved up and down crossing the SMA (20) line. So it can be concluded that the price movement pattern in January 2013 was Ranging.

Figure 4. Trend movement pattern for January 2020
From the movement in January 2020, it can be seen that the price of gold derivatives is moving above the SMA (20) line. So it can be concluded that the type of pattern for price movement in January 2020 is a trend movement.

Figure 5. January 2021 trend movement pattern
From the movement in January 2021, it can be seen that the price of gold derivatives has moved to cross the SMA (20) line from top to bottom. So it can be concluded that the type of price movement pattern in January 2021 is a Reversal movement.

Personality analysis
Seasonality analysis is carried out to see seasonal movements, or patterns of trend movements that occur repeatedly. From the results of the analysis of the trend movement pattern in the previous step, the probability of a repetition of the movement pattern for each monthly period within 10 years is determined, and the results are summarized in the following table: Based on the table, in 2023 there is a possibility of gold derivative price movements following trends, reversals and ranging with different probabilities for each month. This probability is based on probability data for the seasonality of gold derivative price movements over the previous 10 years.
The results of the seasonality analysis show that January has the best level of probability and prediction for making transactions with a probability of predicting the trend pattern movement of 70%. Meanwhile, February, March and April show the lowest probability of predicting the movement of the gold derivative pattern with a probability of 10%.

CONCLUSION
Based on the problems raised in this study, it can be concluded that by grouping 3 variations in the comparison of the number of n In Sample and Out Sample, a Derivative Gold Price Brownian Motion Geometry model is obtained with a prediction error value calculated using the Mean Absolute Percentage Error (MAPE) each of 3.12%, 5.99% and 6.36%. It indicates that the prediction accuracy of gold derivative prices using the Geometric Brownian Motion model is very good. So it can be concluded that the Geometric Brownian Motion method can be used to predict the price of gold derivatives. Of the three variations of the ratio of n In Sample and Out Sample, the N1 variation shows better accuracy with a ratio of 1:1 compared to the other 2 variations.
The results of the seasonality analysis obtained the probability of repetition of each pattern of monthly gold derivative price movements for 10 years. It is indicated that the price movement pattern of gold derivatives that occurs in the prediction for 2023 will be correlated according to the probability level of the results of the analysis.The results of the seasonality analysis show that January has the best level of probability and prediction for making transactions with a probability of predicting the trend pattern movement of 70%. Meanwhile, February, March and April show the lowest probability of predicting the movement of the gold derivative pattern with a probability of 10%.