1.Objective

The purpose of this report is to estimate the theoretical price of a European down-and-in put option with a barrier active between the moment of pricing and the option expiry, using a Monte Carlo approach and to analyze the relation between theoretical price of the option and volatility of the underlying instrument returns and time to maturity of the option.

2.Explaination of European down-and-in put option with a barrier

A European Down-and-In Put Option with a barrier active between the moment of pricing and the option expiry has the following characteristics:

Option Type:Put Option. Gives the holder the right to sell the underlying asset at a predetermined strike price at expiry.

Barrier Type:Down-and-In. The option “activates” only if the underlying instrument’s price falls below a specific barrier level during the life of the option.

Key Parameters:
Underlying Asset Price (S₀):The current price of the underlying instrument.
Strike Price (K):The price at which the option holder can sell the underlying instrument if the option is active at expiry.
Barrier Level (b): The price level below which the barrier is triggered, activating the option.
Time to maturity(t): The time until the expiry of the option.
Annualized volatility rate(σ): The price volatility of the underlying instrument. annualized risk-free rate(r):The interest rate used for discounting future cash flows.

Activation Condition:The option becomes “alive” only if the price of the underlying instrument touches or falls below the barrier b during the life of the option.

Payoff at Expiry:If the option is activated and the underlying asset price \(S_T \leq K\) at expiry:

\[ \text{Payoff} = K - S_T \] Otherwise, the payoff is zero.

This type of option is commonly used for hedging against downside risk while benefiting from cost reduction due to the barrier feature.

3.Assumptions

Assumption 1: With higher volatility, the option price increases.

Assumption 2: With longer time to maturity, the option price increases.

4. Install Libraries

knitr::opts_chunk$set(echo = TRUE, warning = FALSE, message = FALSE)
options(repos = c(CRAN = "https://cloud.r-project.org"))
install.packages("ggplot2")
library(ggplot2)
install.packages("devtools")
devtools::install_github("YunWu-ly/downInPutMC_0.1.0")
library(downInPutMC)

5.Parameter Setting

# Define parameters
S0 <- 105        # Initial stock price
K <- 110         # Strike price
r <- 0.05        # Risk-free interest rate
b <- 95          # Barrier level
nSim <- 10000    # Number of Monte Carlo simulations
sigma_vals <- seq(0.15, 0.35, by = 0.05)  # Range of volatility values
t_vals <- seq(0.5, 2, by = 0.5)           # Range of time to maturity values

6.Running Simulations

Run the Monte Carlo simulations for each combination of volatility and time to maturity and store the results.

# Store results in a data frame
results <- expand.grid(volatility = sigma_vals, time_to_maturity = t_vals)
results$option_price <- mapply(function(vol, t) {
  downInPutMC(nSim, S0, K, r, vol, t, b)  # call Monte Carlo function
}, results$volatility, results$time_to_maturity)

# Show the first few rows of the results
head(results)
##   volatility time_to_maturity option_price
## 1       0.15              0.5     4.126805
## 2       0.20              0.5     6.252793
## 3       0.25              0.5     7.947892
## 4       0.30              0.5     9.346576
## 5       0.35              0.5    10.349911
## 6       0.15              1.0     5.942222

7.Visualizing the Results

7.1 Plot: Option Price vs Time to Maturity for Different Volatilities

ggplot(results, aes(x = time_to_maturity, y = option_price, color = as.factor(volatility))) +
  geom_line(size = 1) +
  labs(title = "Down-and-In Put Option Price vs Time to Maturity",
       x = "Time to Maturity (Years)",
       y = "Option Price",
       color = "Volatility") +
  theme_minimal()

7.2 Interpretation

Interpretation 1: The option price increases with higher volatility due to greater probability of hitting the barrier.
Interpretation 2:Longer time to maturity also generally leads to higher option prices as the stock price has more time to hit the barrier.

8.Declaration

In accordance with the Honor Code, I certify that my answers here are my own work, and I did not make my solutions available to anyone else.