inClassLab3.rmd

---
title: "inClassLab3"
author: "Marco Dodson"
date: "2024-04-16"
output: html_document
---

```{r echo=FALSE,message=FALSE,warning=TRUE,result='asis'}
##load packages
library(knitr)
###settings for R chunks
knitr::opts_chunk$set(echo=TRUE,
               message=FALSE,
               warning = TRUE,
               results='asis') #echo: whether prints the R code; message: whether shows the message outputs; warning: whether shows the warning messages; results: whether contain ## in front of the R output # nolint
```

```{r}
set.seed(123)
```


# (a) choose basic random variable z ~ U[0,20], and define the target random variable W 
## w = 50000 * (20 / 25) * exp (- Z / 25)


## (b) Monte carlo estimation with sample size n = 10^3 
```{r}
n <- 10^3
```

## (b-i) Generate a sample of Z (uniform random on 0,20 )
```{r}
Z <- runif(n, min = 0, max = 20)

```


## (b-ii) Generate a sample of W 
```{r}

W <- 50000 * (20 / 25) * exp(-Z / 25)

```

## (b-iii) estimate the fair premium 
```{r}
fair_premium_estimate <- mean(W)

```

## (b-iv) caluclate the estimation error using 99% interval 
```{r}
sigma_hat_W <- sd(W)
estimation_error <- 2.58 * sigma_hat_W / sqrt(n)

cat("Estimated fair premium (n =", n, "):", fair_premium_estimate, "\n")
cat("99% confidence interival:", fair_premium_estimate - estimation_error, ",",fair_premium_estimate + estimation_error, "\n")

```

## (c) calculate the required simulation size for an error tolerance 
```{r}
epsilon <- 5
C <- 1.1 
required_size <- ceiling((2.58 * C * sigma_hat_W / epsilon)^2)
cat("required simulation size for error tolerance of $5:", required_size, "\n")

```

## (d) compare new with old 
```{r}
cat("Previous required simulation size (exponential basic random variable):", 235433630, "\n")
cat("New required simulation size (uniform basic random variable):", required_size, "\n") 
```


# (2)

## (a) define the parameters
```{r}
S <- 50 
K <- 45
r <- .03
sigma <- .20
T <- .5 
```

## (b) monte carlo simulation 
```{r}
## generate a sample of discounted payoff with n = 10,000
n_small <- 10000
z <- rnorm(n_small)
ST_small <- S * exp((r - .5 * sigma^2) * T + sigma * sqrt(T) * z)
payoff_small <- pmax(K- ST_small, 0)
discounted_payoff_small <- exp(-r * T) * payoff_small

```

## esimtate the standard deviation of discounted payoff 
```{r}
sigma_hat <- sd(discounted_payoff_small)
```

## compute the appropriate simulation size 
```{r}
Epsilon <- .5
c <- 1.2
n_large <- ceiling((2.58 * c * sigma_hat / Epsilon)^2)

```

## generate a sample of discounted payoff with the correct simulation size 

```{r}
z <- rnorm(n_large)
ST_large <- S * exp((r - .5 * sigma^2) * T + sigma * sqrt(T) * z)
payoff_large <- pmax(K - ST_large, 0)
discounted_payoff_large <- exp(-r * T) * payoff_large

```

## estimate the true price using the mean of discounted payoff 
```{r}
put_price_estimate <- mean(discounted_payoff_large)

```

## compute 99% confidence interval 
```{r}
error <- 2.58 * sd(discounted_payoff_large) / sqrt(n_large)
lower_bound <- put_price_estimate - error 
upper_bound <- put_price_estimate + error

cat("Estimated put option price: $", round(put_price_estimate, 4), "\n")
cat("99% confidnece interval: [$", round(lower_bound, 4), ",$ ", round(upper_bound, 4),"]\n")

```


## (c) compute exact price 
```{r}
d1 <- (log(S / K) + (r + .5 * sigma^2) * T) / (sigma * sqrt(T))
d2 <- d1 - sigma * sqrt(T)
put_exact_price <- K * exp(-r * T) * pnorm(-d2) - S * pnorm(-d1)

cat("Exact put option price: $", round(put_exact_price, 4), "\n")

```

## (d) compute true error and compare with error tolerance
```{r}
true_error <- abs(put_price_estimate - put_exact_price)
cat("True Error: $", round(true_error, 4), "\n")
cat("Is the true error smaller than error tolerance?", true_error < Epsilon, "\n")

```