Stock Option Model

Authors@R: person(“Annice”, “Najafi”, email = “annicenajafi27@gmail.com”)

Summer 2024

This is an R implementation of the stock option model from Chapter 1, section 3 (“A Stock-Option Model”) of the “Introduction to Stochastic Dynamic Programming” book by Sheldon Ross.

An option for a stock is the ability to buy a stock for a pre-determined price. That means you can pay some money to have the right to buy a specific share of a company for a pre-determined price even if the stock price increases later on.

First let’s start by loading relevant libraries

#Load libraries
library(stats)
library(latex2exp)
library(ggplot2)

Define the maximum expected return function

# Define the probability density function
f <- function(x) {
  dnorm(x, mean = 0, sd = 1)
}

#Define V_n
#' Recursive function returns the maximal expected return
#'
#' @param n number of days
#' @param s current price of the stock
#' @param c past price that we buy the stock at
#' @return maximal expected return value
#' @export
V <- function(n, s, c) {
  
  if (n == 0) {
    return(max((s-c), 0))
  }
  
  
  #Remember F(x) is f(x)dx so we can multiply by the pdf and integrate with
  #respect to x.
  hold.integrand <- function(x) {V(n - 1, s + x, c) * f(x)}
  
  expected_future_value <- integrate(hold.integrand, -Inf, Inf)$value
  
  # Two possible scenarios here
  # Case I) immediately sell the option
  # Case II) wait and see what happens, hoping for better future value
  return(max(s - c, expected_future_value))
}

Let’s test

# Parameters
initial_stock_price <- 10  # Initial stock price
c <- 8  # stock purchase price
n <- 3  # Number of days

result <- V(n, initial_stock_price, c)

Proving Lemma 3.1

s_values <- seq(0, 100, by = 5)

V_n_s <- c()
for(i in 1:length(s_values)) {
  s <- s_values[i]
  V_n_s_value <- V(n, s, c)
  V_n_s <- c(V_n_s , V_n_s_value)
}

hold.df <- data.frame(s = s_values, V_n_s = (V_n_s_value - s_values))

ggplot(hold.df, aes(s, V_n_s))+
  geom_line(size=2, color = "#538392")+
  labs(x="s", y=TeX("$V_n(s) - s$"))+
  ggtitle(TeX("$V_n(s) - s$ is decreasing in s"))+
  theme(
    panel.grid.major = element_blank(),
    panel.grid.minor = element_blank(),
    panel.background = element_blank(),
    axis.line = element_line(colour = "black"),
    plot.title = element_text(hjust = 0.5, size=20),
    axis.text = element_text(size = 15),
    text = element_text(size=18))+labs(tag="A")

## Warning: Using `size` aesthetic for lines was deprecated in ggplot2 3.4.0.
## ℹ Please use `linewidth` instead.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.

We just proved Lemma 3.1 using R!

Annice
Summer 2024

References:

Ross, S. M. (2014). Introduction to stochastic dynamic programming. Academic press.