---
title: "Problem set #16: Binomial Monte Carlo"
author: "Milica Cudina"
date: "`r Sys.Date()`"
output: pdf_document
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
```

---

Let the **volatility** of a stock be the standard deviation of its (continuously compounded) realized return on an annual basis. Then, we can define the up and down factors in the so-called *forward binomial tree* for a **non-dividend-paying** stock as
\begin{equation}
\begin{split}
u & = e^{rh + \sigma\sqrt{h}}\\
d & = e^{rh - \sigma\sqrt{h}}\\
\end{split}
\end{equation}

Let the continuously compounded, risk-free interest rate be $0.04$. 


Consider a stock whose current price is \$100 and whose volatility is $0.25$. We will be pricing a one-year, at-the-money call option in a variety of ways here. 



## Problem #1: Analytic one period
Price the option above using a one period binomial tree. 



## Problem #2: Monte Carlo one period
Price the option above using Monte Carlo a one period binomial tree. Use $10000$ simulations. 


## Problem #3: Analytic two periods
Price the above option using a two-period binomial tree. 


## Problem #4: Monte Carlo two periods
Price the option above using Monte Carlo a two period binomial tree. Use $10000$ simulations. 


## Problem #5: Analytic one hundred periods
Price the above option using a $100-$period binomial tree. 


## Problem #6: Monte Carlo with one hundred periods
Price the option above using Monte Carlo with a hundred period binomial tree. Use $10000$ simulations.