Gold_R.R

gold_data <- read.csv("C:\\Users\\VALLARI SHARMA\\Downloads\\Gold final PS.csv")
library(ggplot2)
# Inspection of the data
str(gold_data)
summary(gold_data)
# Histogram of Gold Close Prices
ggplot(gold_data, aes(x = close)) +
  geom_histogram(binwidth = 10, fill = "blue", color = "black", alpha = 0.7) +
  labs(title = "Histogram of Gold Close Prices", x = "Gold Close Price", y = "Frequency") +
  theme_minimal()

# Density Plot
ggplot(gold_data, aes(x = close)) +
  geom_density(fill = "blue", alpha = 0.5) +
  labs(title = "Density Plot of Gold Close Prices", x = "Gold Close Price", y = "Density") +
  theme_minimal()

rowst=nrow(gold_data)
rowst



# Load the libraries
library(MASS)
library(fitdistrplus)
library(dplyr)

gold_price_subset <- gold_data %>%
  filter(close >= 1000 & close <= 1500)
subsetrows=nrow(gold_price_subset)
subsetrows

# Extract the 'close' column (Gold Close Price)
close_data <- gold_price_subset$close

# Step 1: Fit a normal distribution
normal_fit <- fitdistr(close_data, "normal")

# Step 2: Fit a log-normal distribution
lognormal_fit <- fitdistr(close_data, "lognormal")

# Step 3: Fit a t-distribution (with 5 degrees of freedom)
t_fit <- fitdistr(close_data, "t", df = 5)
# Step 4: Fit an exponential distribution
exponential_fit <- fitdistr(close_data, "exponential")

# Step 5: Fit a gamma distribution
gamma_fit <- fitdistr(close_data, "gamma")

#Step 6: Fit a Weibull distribution
weibull_fit <- fitdistr(close_data, "weibull")

#Step 7:Poisson distribution
# Convert the 'close' data to count data if necessary
close_counts <- round(data$close)

# Plot histogram of the counts to visualize the data
hist(close_counts, breaks=50, prob=TRUE, main="Histogram of Gold Close Counts", xlab="Gold Close (Counts)", col="skyblue")
# Fit a Poisson distribution
lambda <- mean(close_counts)  # λ is the mean of the counts

# Print the estimated lambda (λ)
cat("Estimated lambda for Poisson Distribution:", lambda, "\n")

# Step 8: Print summary of the fits
print("Normal Distribution Fit:")
print(normal_fit)

print("Log-Normal Distribution Fit:")
print(lognormal_fit)

print("T-Distribution Fit:")
print(t_fit)


# Step 9: Print summary of the fits
print("Normal Distribution Fit:")
print(normal_fit)

print("Log-Normal Distribution Fit:")
print(lognormal_fit)

print("T-Distribution Fit:")
print(t_fit)

print("Exponential Distribution Fit:")
print(exponential_fit)

print("Gamma Distribution Fit:")
print(gamma_fit)

print("Weibull Distribution Fit:")
print(weibull_fit)

# These results provide information about the parameters of each fitted distribution
# and their estimated standard errors. 
# This allows you to assess the uncertainty in the parameter estimates 
# and compare the goodness of fit of different distributions

# Step 10: Plot the fitted distributions
hist(close_data, breaks=50, prob=TRUE, main="Gold Close Price", xlab="Price", col="skyblue")

# Overlay the normal distribution fit
curve(dnorm(x, mean=normal_fit$estimate[1], sd=normal_fit$estimate[2]), 
      col="red", lwd=2, add=TRUE, yaxt="n")

# Overlay the log-normal distribution fit
curve(dlnorm(x, meanlog=lognormal_fit$estimate[1], sdlog=lognormal_fit$estimate[2]), 
      col="green", lwd=2, add=TRUE, yaxt="n")

# Overlay the t-distribution fit
curve(dt((x - t_fit$estimate[1]) / t_fit$estimate[2], df=5) * (1/t_fit$estimate[2]), 
      col="blue", lwd=2, add=TRUE, yaxt="n")
# Overlay the exponential distribution fit
curve(dexp(x, rate=1/exponential_fit$estimate), 
      col="purple", lwd=2, add=TRUE, yaxt="n")

# Overlay the gamma distribution fit
curve(dgamma(x, shape=gamma_fit$estimate[1], rate=gamma_fit$estimate[2]), 
      col="orange", lwd=2, add=TRUE, yaxt="n")

# Overlay the Weibull distribution fit
curve(dweibull(x, shape=weibull_fit$estimate[1], scale=weibull_fit$estimate[2]), 
      col="brown", lwd=2, add=TRUE, yaxt="n")

# Overlay Poisson distribution on the histogram
x_vals <- 0:max(close_counts)
pois_prob <- dpois(x_vals, lambda)

# Plot the Poisson distribution curve
lines(x_vals, pois_prob, col="aquamarine4", lwd=2)


# Add legend to the plot
legend("topright", legend=c("Normal", "Log-normal", "T-distribution", "Exponential", "Gamma", "Weibull","Poisson"), 
       col=c("red", "green", "blue", "purple", "orange", "brown","aquamarine4"), lwd=2)


# The graph shows a histogram of gold close prices with superimposed density plots
# for seven different probability distributions

#poison continuation

# Create a table of observed counts of each unique value in the close_counts data
observed_counts <- table(close_counts)

# Expected counts based on the Poisson distribution 
expected_counts <- dpois(as.numeric(names(observed_counts)), lambda) * length(close_counts)

# Perform chi-squared goodness-of-fit test
#Compares the observed and expected counts and calculates a chi-squared statistic.
gof_test <- chisq.test(observed_counts, p=expected_counts / sum(expected_counts))
gof_test
# Print the result
if (gof_test$p.value < 0.05) {
  cat("The data does not fit a Poisson distribution as there is significant differnce in counts.\n")
} else {
  cat("The data fits a Poisson distribution as there is significant differnce in counts.\n")
}


#Step 9: Compute AIC values for the fitted distributions
# AIC (Akaike Information Criterion) is a statistical measure used to compare the 
# relative quality of different statistical models for a given dataset.
# It balances the model's fit to the data with its complexity.

print("AIC for Normal Distribution:")
print(AIC(normal_fit))

print("AIC for Log-Normal Distribution:")
print(AIC(lognormal_fit))

print("AIC for T-Distribution:")
print(AIC(t_fit))
cat("AIC for Exponential Distribution:\n")
print(AIC(exponential_fit))

cat("AIC for Gamma Distribution:\n")
print(AIC(gamma_fit))

cat("AIC for Weibull Distribution:\n")
print(AIC(weibull_fit))

# Calculate AIC for Poisson
# Calculate log-likelihood for the Poisson model
log_likelihood_poisson <- sum(dpois(close_counts, lambda, log=TRUE))

aic_poisson <- -2 * log_likelihood_poisson + 2 * 1  # 1 parameter for Poisson (lambda)
cat("AIC for Poisson Distribution:", aic_poisson, "\n")

# Log-likelihood is a measure of how well a statistical model fits a given dataset.
# It is the natural logarithm of the likelihood function, which represents
# the probability of observing the data given the model's parameters.


#Note:
# Poisson distribution is designed for: count data 
# Gold price data is typically continuous, representing a numerical value.
# When we try to fit a Poisson distribution to continuous data, 
# the fitdistr function might encounter difficulties in finding suitable parameter
# estimates that maximize the likelihood.
# This can lead to errors or unexpected results.Hence cant do AIC(poisson_fit)


#HYPOTHESIS TESTING USING Z-TEST


# Loading necessary libraries
library(ggplot2)
library(BSDA)

data=gold_price_subset

# Converting 'time' column to Date type
data$time <- as.Date(data$time, format="%d-%m-%Y")

# Why are we dividing?
# 1.Comparison: Dividing the data into groups allows you to compare the
# average gold prices in each group. This can help you determine if there is a 
# relationship between DXY and gold prices.
# 2.Hypothesis Testing: The Z-test is used to compare the means of two groups.
# By dividing the data, you have two groups that can be compared using the Z-test.
# 3.Clarity: Dividing the data into groups can make it easier to interpret the 
# results and understand the relationship between the variables.

# Hypothesis 1: There is a negative correlation between gold prices and DXY
cor_gold_dxy <- cor(data$close, data$dxy_close, method="pearson")

# Plot the relationship between Gold price and DXY
ggplot(data, aes(x = dxy_close, y = close)) +
  geom_point() +
  geom_smooth(method = "lm", col = "blue") +
  labs(title = paste("Correlation between Gold Price and DXY:", round(cor_gold_dxy, 2)),
       x = "DXY Close", y = "Gold Close")
# Divide the data into two groups: high and low DXY
threshold_dxy <- median(data$dxy_close)
high_dxy <- subset(data, dxy_close > threshold_dxy)
low_dxy <- subset(data, dxy_close <= threshold_dxy)

# Perform the Z-test for DXY
z_test_dxy <- z.test(high_dxy$close, 
                     y = low_dxy$close, 
                     sigma.x = sd(high_dxy$close), 
                     sigma.y = sd(low_dxy$close))
# Print the result
if(z_test_dxy$p.value < 0.05) {
  cat("Null hypothesis rejected for Gold vs DXY. Conclusion: There is a significant difference in Gold prices based on DXY.\n")
} else {
  cat("Null hypothesis accepted for Gold vs DXY. Conclusion: No significant difference in Gold prices based on DXY.\n")
}


# Hypothesis 2: Positive correlation between gold and crude oil prices
cor_gold_oil <- cor(data$close, data$light_crude_oil_close, method="pearson")


# Divide the data into two groups: high and low Crude Oil prices
threshold_oil <- median(data$light_crude_oil_close)
high_oil <- subset(data, light_crude_oil_close > threshold_oil)
low_oil <- subset(data, light_crude_oil_close <= threshold_oil)

# Perform the z-test
z_test_oil <- z.test(high_oil$close, 
                     y = low_oil$close, 
                     sigma.x = sd(high_oil$close), 
                     sigma.y = sd(low_oil$close))

# Print the result
if(t_test_oil$p.value < 0.05) {
  print("Null hypothesis rejected for Gold vs Crude Oil.")
  print("Conclusion: There is a significant difference in Gold prices based on Crude Oil prices.\n")
} else {
  print("Null hypothesis accepted for Gold vs Crude Oil.")
  print(" Conclusion: No significant difference in Gold prices based on Crude Oil prices.\n")
}
# Plotting the relationship between Gold price and Crude Oil price
ggplot(data, aes(x = light_crude_oil_close, y = close)) +
  geom_point() +
  geom_smooth(method = "lm", col = "green") +
  labs(title = paste("Correlation between Gold Price and Crude Oil:", round(cor_gold_oil, 2)),
       x = "Crude Oil Close", y = "Gold Close")



# Hypothesis 3: Gold moves inversely with the S&P 500 during uncertainty
cor_gold_sp500 <- cor(data$close, data$sp500_close, method="pearson")

# Divide the data into two groups: high and low S&P 500
threshold_sp500 <- median(data$sp500_close)
high_sp500 <- subset(data, sp500_close > threshold_sp500)
low_sp500 <- subset(data, sp500_close <= threshold_sp500)

# Perform the t-test
z_test_sp500 <- z.test(high_sp500$close, 
                       y = low_sp500$close, 
                       sigma.x = sd(high_sp500$close), 
                       sigma.y = sd(low_sp500$close))

# Print the result
if(t_test_sp500$p.value < 0.05) {
  print("Null hypothesis rejected for Gold vs S&P 500.")
  print("Conclusion: There is a significant difference in Gold prices based on the S&P 500 index.\n")
} else {
  print("Null hypothesis accepted for Gold vs S&P 500.")
  print("Conclusion: No significant difference in Gold prices based on the S&P 500 index.\n")
}
# Plot the relationship between Gold price and S&P 500 index
ggplot(data, aes(x = sp500_close, y = close)) +
  geom_point() +
  geom_smooth(method = "lm", col = "red") +
  labs(title = paste("Correlation between Gold Price and S&P 500:", round(cor_gold_sp500, 2)),
       x = "S&P 500 Close", y = "Gold Close")



#Hypothesis 4: Gold is affected significantly by RSI
cor_gold_rsi <- cor(data$close, data$gold_RSI, method="pearson")

# Divide the data into two groups: high and low RSI
threshold_rsi <- median(data$gold_RSI)
high_rsi <- subset(data, gold_RSI > threshold_rsi)
low_rsi <- subset(data, gold_RSI <= threshold_rsi)

# Perform the t-test
z_test_rsi <- z.test(high_rsi$close, 
                     y = low_rsi$close, 
                     sigma.x = sd(high_rsi$close), 
                     sigma.y = sd(low_rsi$close))

# Print the result
if(z_test_rsi$p.value < 0.05) {
  cat("Null hypothesis accepted for Gold vs RSI. Conclusion: There is a significant difference in Gold prices based on RSI.\n")
} else {
  cat("Null hypothesis rejected for Gold vs RSI. Conclusion: No significant difference in Gold prices based on RSI.\n")
}
# Plotting Gold price vs RSI
ggplot(data, aes(x = gold_RSI, y = close)) +
  geom_point() +
  geom_smooth(method = "lm", col = "purple") +
  labs(title = paste("Correlation between Gold Price and RSI:", round(cor_gold_rsi, 2)),
       x = "Gold RSI", y = "Gold Close")



#Hypothesis 5: Gold is affected significantly by SMAs

# Create a condition for when the gold price is above all SMAs
above_all_smas <- subset(data, close > gold_SMA_10 & close > gold_SMA_50 & close > gold_SMA_200)

# Create a condition for when the gold price is below all SMAs
below_all_smas <- subset(data, close <= gold_SMA_10 & close <= gold_SMA_50 & close <= gold_SMA_200)

# Perform a t-test to compare gold prices above and below the SMAs
z_test_sma_combined <- z.test(above_all_smas$close, 
                              y = below_all_smas$close, 
                              sigma.x = sd(above_all_smas$close), 
                              sigma.y = sd(below_all_smas$close))

# Print the t-test result with hypothesis testing conclusion
if(t_test_sma_combined$p.value < 0.05) {
  print(" Null hypothesis accepted for Gold vs combined SMAs (10, 50, 200).")
  print("Conclusion:There is a significant difference in Gold prices when above vs below the combined SMAs.")
  
} else {
  print("Null hypothesis rejected for Gold vs combined SMAs (10, 50, 200). ")
  print("Conclusion: No significant difference in Gold prices when above vs below the combined SMAs.\n")
}

# Plotting Gold price with 10-day, 50-day, and 200-day SMAs

ggplot(data, aes(x = time)) +
  geom_line(aes(y = close, col = "Gold Price")) +
  geom_line(aes(y = gold_SMA_10, col = "SMA 10-Day")) +
  geom_line(aes(y = gold_SMA_50, col = "SMA 50-Day")) +
  geom_line(aes(y = gold_SMA_200, col = "SMA 200-Day")) +
  labs(title = "Gold Price vs Moving Averages",
       x = "Time", y = "Gold Price") +
  scale_color_manual(values = c("Gold Price" = "red", "SMA 10-Day" = "blue", 
                                "SMA 50-Day" = "green", "SMA 200-Day" = "black"))
data <- read.csv("C:\\Users\\VALLARI SHARMA\\Downloads\\Gold final PS.csv")

# Perform Pearson correlation test between gold prices and U.S. Dollar Index
cor_test_result <- cor.test(data$close, data$dxy_close, method = "pearson")

# Print the result
print(cor_test_result)

data <- read.csv("C:\\Users\\VALLARI SHARMA\\Downloads\\Gold final PS.csv")
# Calculate daily returns as percentage change in the gold closing price
data$returns <- c(NA, diff(data$close) / data$close[-nrow(data)] * 100)

# Create a condition for overbought (RSI > 70) and oversold (RSI < 30)
overbought <- data$gold_RSI > 70
oversold <- data$gold_RSI < 30

# Subset data for overbought and oversold conditions
overbought_returns <- data$returns[overbought]
oversold_returns <- data$returns[oversold]

# Average returns on all other days (when RSI is neither overbought nor oversold)
other_returns <- data$returns[!(overbought | oversold)]

# Perform t-tests to compare overbought/oversold days vs other days
t_test_overbought <- t.test(overbought_returns, other_returns, na.rm = TRUE)
t_test_oversold <- t.test(oversold_returns, other_returns, na.rm = TRUE)

# Print the results
print("T-test for Overbought Days (RSI > 70):")
print(t_test_overbought)

print("T-test for Oversold Days (RSI < 30):")
print(t_test_oversold)


# Calculate daily returns as percentage change in the gold closing price
data$returns <- c(NA, diff(data$close) / data$close[-nrow(data)] * 100)

# Define moving average crossover signal (10-day MA crosses above 50-day MA)
crossover_signal <- data$gold_SMA_10 > data$gold_SMA_50

# Subset returns based on crossover signals (buying/selling points)
crossover_returns <- data$returns[crossover_signal]
random_returns <- sample(data$returns, length(crossover_returns), replace = TRUE)  # Random trading strategy

# Perform paired t-test to compare returns of crossover vs random strategy
paired_t_test <- t.test(crossover_returns, random_returns, paired = FALSE, na.rm = TRUE)

# Print the result
print("Paired t-test for Moving Average Strategy vs Random Strategy:")
print(paired_t_test)
# Install necessary packages (if not already installed)
install.packages("zoo")
install.packages("ggplot2")

data <- read.csv("C:\\Users\\VALLARI SHARMA\\Downloads\\Gold final PS.csv")
# Check the first few rows to see if columns were added correctly
head(data)
data$rolling_mean <- rollapply(data$close, width = 30, FUN = mean, fill = NA, align = "right")
data$rolling_mean
# Filter out rows where rolling_mean is NA
valid_data <- data[!is.na(data$rolling_mean), ]

# View the first few rows of the valid data
head(valid_data)
# View the rolling mean after the first 30 rows
head(data$rolling_mean, 40)



# Convert the 'time' column to Date format (assuming it's in 'dd/mm/yyyy' format)
data$time <- as.Date(data$time, format = "%d/%m/%Y")

# Sort data by time (if not already sorted)
data <- data[order(data$time),]

# Calculate daily returns as percentage change in the gold closing price
data$returns <- c(NA, diff(data$close) / data$close[-nrow(data)] * 100)

# Calculate 30-day rolling mean of gold closing prices
data$rolling_mean <- rollapply(data$close, width = 30, FUN = mean, fill = NA, align = "right")

# Calculate 30-day rolling volatility (standard deviation of returns)
data$rolling_volatility <- rollapply(data$returns, width = 30, FUN = sd, fill = NA, align = "right")

# Plot the rolling mean of gold prices over time
ggplot(data, aes(x = time)) +
  geom_line(aes(y = rolling_mean), color = "blue", linewidth = 1) +   # Changed size to linewidth
  labs(title = "30-Day Rolling Mean of Gold Prices",
       x = "Date", y = "Rolling Mean (Gold Price)") +
  theme_minimal()
S
# Plot the rolling volatility of gold returns over time, handling missing values with na.rm = TRUE
ggplot(data, aes(x = time)) +
  geom_line(aes(y = rolling_volatility), color = "red", linewidth = 1, na.rm = TRUE) +   # Changed size to linewidth
  labs(title = "30-Day Rolling Volatility of Gold Prices",
       x = "Date", y = "Rolling Volatility (Standard Deviation of Returns)") +
  theme_minimal()

# Adding vertical lines to mark significant historical events (e.g., financial crisis of 2008)
ggplot(data, aes(x = time)) +
  geom_line(aes(y = rolling_volatility), color = "red", linewidth = 1, na.rm = TRUE) +   # Changed size to linewidth
  geom_vline(xintercept = as.Date("2008-09-15"), linetype = "dashed", color = "blue") +  # Lehman Brothers collapse
  labs(title = "30-Day Rolling Volatility of Gold Prices",
       x = "Date", y = "Rolling Volatility (Standard Deviation of Returns)") +
  annotate("text", x = as.Date("2008-09-15"), y = max(data$rolling_volatility, na.rm = TRUE), 
           label = "2008 Financial Crisis", vjust = -1) +
  theme_minimal()




# Convert the 'time' column to Date format (assuming it's in 'dd/mm/yyyy' format)
data$time <- as.Date(data$time, format = "%d/%m/%Y")

# Sort data by time (if not already sorted)
data <- data[order(data$time),]

# Check the structure of the dataset
str(data)

# Plot the gold price time series using ggplot2
ggplot(data, aes(x = time, y = close)) +
  geom_line(color = "blue", linewidth = 1) +  # Line plot for closing prices
  labs(title = "Gold Price Time Series",
       x = "Date", y = "Gold Price (Closing)") +
  theme_minimal() +  # Clean theme for the plot
  theme(axis.text.x = element_text(angle = 45, hjust = 1))  # Rotate x-axis labels for better readability



# Convert the 'time' column to Date format
data$time <- as.Date(data$time, format = "%d/%m/%Y")

# Extract 'Month' from the date to create a categorical variable for ANOVA
data$month <- format(data$time, "%m")

# Check the structure of the dataset
str(data)

# Remove any missing values if necessary
data <- na.omit(data)

# Perform one-way ANOVA to check if gold prices differ by month
anova_result <- aov(close ~ month, data = data)

# View the ANOVA table
summary(anova_result)

# If the ANOVA result is significant, perform Tukey's Honest Significant Difference (HSD) test
# This helps to understand which specific groups differ
tukey_result <- TukeyHSD(anova_result)
print(tukey_result)

# Visualize the gold prices by month to check the variation
ggplot(data, aes(x = month, y = close)) +
  geom_boxplot(fill = "skyblue") +
  labs(title = "Gold Prices by Month",
       x = "Month", y = "Gold Price (Closing)") +
  theme_minimal()




# Load necessary libraries

data <- read.csv("C:\\Users\\VALLARI SHARMA\\Downloads\\Gold final PS.csv")
# Convert 'time' column to Date format
data$time <- as.Date(data$time, format = "%d/%m/%Y")

# Extract month and year from the time variable for grouping
data$month <- month(data$time, label = TRUE)  # Label months as Jan, Feb, etc.
data$year <- year(data$time)

# Calculate monthly average of gold prices
monthly_data <- data %>%
  group_by(year, month) %>%
  summarise(monthly_mean = mean(close, na.rm = TRUE))

# Plot the monthly average gold prices across years to detect seasonality
ggplot(monthly_data, aes(x = month, y = monthly_mean, group = year, color = factor(year))) +
  geom_line() +
  labs(title = "Monthly Average of Gold Prices Across Years",
       x = "Month", y = "Average Gold Price") +
  theme_minimal()

# Calculate monthly returns from closing prices
data <- data %>%
  group_by(month) %>%
  mutate(monthly_return = (close - lag(close)) / lag(close))

# Summarize monthly returns to detect seasonality
monthly_returns <- data %>%
  group_by(month) %>%
  summarise(mean_return = mean(monthly_return, na.rm = TRUE))

# Plot monthly average returns to detect seasonality
ggplot(monthly_returns, aes(x = month, y = mean_return)) +
  geom_bar(stat = "identity", fill = "skyblue") +
  labs(title = "Average Monthly Returns of Gold Prices",
       x = "Month", y = "Average Monthly Return") +
  theme_minimal()

# Perform ANOVA test to check if there is a significant difference in returns between months
anova_test <- aov(monthly_return ~ month, data = data)
summary(anova_test)


# Calculate Historical Returns
gold_data <- gold_data %>%
  mutate(daily_return = c(NA, diff(log(close))))  # Log returns   

# Estimate Parameters for Simulation
mu <- mean(gold_data$daily_return, na.rm = TRUE)  # Expected return
sigma <- sd(gold_data$daily_return, na.rm = TRUE)  # Volatility
S0 <- last(gold_data$close)  # Last known gold price

# Run Monte Carlo Simulations
set.seed(123)  # For reproducibility
n_simulations <- 1000  # Number of simulations
n_days <- 365  # Number of trading days in a year
simulated_prices <- matrix(NA, nrow = n_days, ncol = n_simulations)

for (i in 1:n_simulations) {
  simulated_prices[1, i] <- S0  # Starting price
  for (t in 2:n_days) {
    simulated_prices[t, i] <- simulated_prices[t - 1, i] * exp(rnorm(1, mu, sigma))
  }
}

# Convert matrix to data frame for ggplot
simulated_prices_df <- as.data.frame(simulated_prices)
simulated_prices_df <- mutate(simulated_prices_df, Day = 1:n_days)
simulated_prices_long <- pivot_longer(simulated_prices_df, -Day, names_to = "Simulation", values_to = "Price")

# Calculate mean and quantiles for confidence intervals
summary_stats <- simulated_prices_long %>%
  group_by(Day) %>%
  summarize(mean_price = mean(Price),
            lower_bound = quantile(Price, 0.05),
            upper_bound = quantile(Price, 0.95))

# Create a ggplot object
p <- ggplot() +
  geom_ribbon(data = summary_stats, aes(x = Day, ymin = lower_bound, ymax = upper_bound), 
              fill = "black", alpha = 0.4) +  # Confidence interval
  geom_line(data = simulated_prices_long, aes(x = Day, y = Price, group = Simulation), 
            color = alpha("blue", 0.1), linewidth = 0.5) +  # Simulation paths
  theme_minimal(base_size = 16) +
  labs(title = "Monte Carlo Simulations of Gold Prices",
       subtitle = "Visualizing the potential future paths of gold prices with 90% confidence intervals",
       x = "Days",
       y = "Gold Price (in USD)",
       caption = "Source: Your Gold Dataset") +
  theme(panel.grid.major = element_line(color = "grey80"),
        panel.grid.minor = element_blank(),
        plot.title = element_text(hjust = 0.5, face = "bold"),
        plot.subtitle = element_text(hjust = 0.5, face = "italic"),
        plot.caption = element_text(hjust = 0, face = "italic"),
        axis.title.x = element_text(),
        axis.title.y = element_text())
# Convert ggplot to plotly for interactivity
interactive_plot <- ggplotly(p)

# Show the interactive plot
interactive_plot