# Exploring Random Walks and Brownian Motions with healthyR.ts

code
rtip
healthyrts
Author

Steven P. Sanderson II, MPH

Published

June 27, 2024

# Introduction

In the world of time series analysis, Random Walks, Brownian Motion, and Geometric Brownian Motion are fundamental concepts used in various fields, including finance, physics, and biology. Today, we’ll explore these concepts using functions from the `healthyR.ts` package.

## Random Walks

A Random Walk is a path that consists of a series of random steps. It’s a simple but powerful concept used to model seemingly unpredictable paths, such as stock prices or animal movements.

Let’s generate and plot some Random Walks using the `ts_random_walk()` function from `healthyR.ts`.

### Function Syntax

``````ts_random_walk(
.mean = 0,
.sd = 0.1,
.num_walks = 100,
.periods = 100,
.initial_value = 1000
)``````
• `.mean`: The desired mean of the random walks.
• `.sd`: The standard deviation of the random walks.
• `.num_walks`: The number of random walks you want to generate.
• `.periods`: The length of the random walk(s) you want to generate.
• `.initial_value`: The initial value where the random walks should start.

### Example

``````library(ggplot2)
library(healthyR.ts)

random_walk_data <- ts_random_walk(
.mean = 0,
.sd = 0.1,
.num_walks = 10,
.periods = 100,
.initial_value = 1000
)
``````# A tibble: 6 × 4
run     x       y cum_y
<dbl> <dbl>   <dbl> <dbl>
1     1     1  0.0725 1072.
2     1     2  0.182  1267.
3     1     3  0.110  1407.
4     1     4  0.0275 1445.
5     1     5 -0.0546 1367.
6     1     6  0.0712 1464.``````
``````random_walk_plot <- random_walk_data |>
ggplot(
mapping = aes(
x = x,
y = cum_y,
color = factor(run),
group = factor(run)
)
) +
geom_line(alpha = 0.8) +
ts_random_walk_ggplot_layers(random_walk_data)
print(random_walk_plot)``````

This code generates 10 random walks over 100 periods, starting from an initial value of 1000. The resulting plot visualizes the paths of these random walks, each represented by a different color.

## Brownian Motion

Brownian Motion, also known as Wiener Process, is a continuous-time stochastic process that is often used to model random movements in physics and finance.

### Function Syntax

``````ts_brownian_motion(
.time = 100,
.num_sims = 10,
.delta_time = 1,
.initial_value = 0,
.return_tibble = TRUE
)``````
• `.time`: Total time of the simulation.
• `.num_sims`: Total number of simulations.
• `.delta_time`: Time step size.
• `.initial_value`: Initial value of the simulation.
• `.return_tibble`: Return a tibble (TRUE) or a matrix (FALSE).

### Example

``````brownian_data <- ts_brownian_motion(
.time = 100,
.num_sims = 10,
.delta_time = 1,
.initial_value = 0,
.return_tibble = TRUE
)
``````# A tibble: 6 × 3
sim_number       t     y
<fct>        <int> <dbl>
1 sim_number 1     1     0
2 sim_number 2     1     0
3 sim_number 3     1     0
4 sim_number 4     1     0
5 sim_number 5     1     0
6 sim_number 6     1     0``````
``````brownian_plot <- ts_brownian_motion_plot(
.data = brownian_data,
.date_col = t,
.value_col = y,
.interactive = TRUE
)
brownian_plot``````

This code simulates 10 paths of Brownian Motion over 100 time units, starting from an initial value of 0. The `ts_brownian_motion_plot()` function creates a static plot of these simulations.

## Geometric Brownian Motion

Geometric Brownian Motion (GBM) is a variation of Brownian Motion where the logarithm of the variable follows a Brownian Motion. It is commonly used to model stock prices in the Black-Scholes option pricing model.

### Function Syntax

``````ts_geometric_brownian_motion(
.num_sims = 100,
.time = 25,
.mean = 0,
.sigma = 0.1,
.initial_value = 100,
.delta_time = 1/365,
.return_tibble = TRUE
)``````
• `.num_sims`: Total number of simulations.
• `.time`: Total time of the simulation.
• `.mean`: Expected return.
• `.sigma`: Volatility.
• `.initial_value`: Initial value of the simulation.
• `.delta_time`: Time step size.
• `.return_tibble`: Return a tibble (TRUE) or a matrix (FALSE).

### Example

``````gbm_data <- ts_geometric_brownian_motion(
.num_sims = 10,
.time = 25,
.mean = 0.05,
.sigma = 0.2,
.initial_value = 100
)
``````# A tibble: 6 × 3
sim_number       t     y
<fct>        <int> <dbl>
1 sim_number 1     1   100
2 sim_number 2     1   100
3 sim_number 3     1   100
4 sim_number 4     1   100
5 sim_number 5     1   100
6 sim_number 6     1   100``````
``````gbm_plot <- ts_brownian_motion_plot(
.data = gbm_data,
.date_col = t,
.value_col = y,
.interactive = TRUE
)
gbm_plot``````

This code simulates 10 paths of Geometric Brownian Motion over 25 time units with an expected return of 0.05 and volatility of 0.2. The `ts_brownian_motion_plot()` function again helps in visualizing the simulations.