---
title: 'Project #2'
author: "Milica Cudina"
date: "`r Sys.Date()`"
output:
  html_document:
    df_print: paged
urlcolor: blue
---
<!-- The author of this template is Dr. Gordan Zitkovic.-->
<!-- The code chunk below contains some settings that will  -->
<!-- make your R code look better in the output pdf file.  -->
<!-- If you are curious about what each option below does, -->
<!-- go to https://yihui.org/knitr/options/ -->
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```{r echo=FALSE, warning=FALSE, include=FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  fig.align="center",
  fig.pos="t",
  strip.white = TRUE
)
```
<!-- ... down to here. -->

---

# Forwards and futures. 

## Problem #1 (25 points)
### The Dow index and Dow futures
Please, in no particular order read the Wikipedia entry about *futures contracts* and listen to the fun and educational episode of *Planet money* about credit risk in cafeteria futures. Both are linked below:

[Wikipedia](https://en.wikipedia.org/wiki/Futures_contract): Futures

[Planet money](https://www.npr.org/sections/money/2016/01/06/462149561/episode-15-delicious-cake-futures): Delicious cake futures

**(10 points)** Provide the definitions of the forward contract and the futures contract. Then, highlight at least twosimilarities and at least two differences between the two. 

The files "dow.csv" and "dow-futures.csv" contain index prices and the futures prices for delivery on a particular date in the future. 

**(5 points)**
Import the data into a data frame (or two?). Draw the time-plot of the evolution of the closing index prices and the futures prices on the same coordinate system. You do **not** need to put the calendar days on the horizontal axis, but you **do** need to label your axes and give your time-plot a title indicating the dates. Make sure that you plot the two trajectories in **different** colors indicating in the text which color corresponds to which company. 

Here is a link to a straightforward tutorial which will help you accomplish the above neatly: 

[Timeline plotting](https://statisticsglobe.com/draw-multiple-time-series-in-same-plot-in-r) 

**(10 points)**
What do you notice when you compare the timeplots? Which of the **formulae** for calculating futures prices you encountered on the Wikipedia entry come in handy?

# Call price properties

## Problem #2 (25 points)
### Apple calls
The file "apple-calls.csv" contains call prices observed on February 14th for European call options on the stocks of Apple Inc with expiry on May 19th and with varying strike prices. 

**(5 points)**
Import the data into a data frame. Create a line plot of the mid price (defined as the midpoint between the bid and the ask price) as it depends on the strike. 

**(5 points)**
What do you notice about the monotonicity and convexity of the above curve? 

**(15 points)**
**Prove** that the same conclusions regarding the monotonicity and convexity of the call prices with respect to the strike must hold in general **under the no arbitrage assumption**. 

# Put-call parity

## Problem #3 (25 points)
**(5 points)**
Using put-call parity, find the formula for the interest rate if all other ingredients are known. 

**(5 points)** 
Based on the above, what can you say about the interest rates $r$ you expect to obtain for varying values of the strike price? Bear in mind the **standing assumptions** we made in the beginning of the semester. 

**(10 points)**
Based on the data set "apple-parity.csv", calculate the continuously compounded, risk-free interest rate for all the strike prices given. Be careful about how you use the *bid* and *ask/offer* prices of the options. Set $T=0.25$. Plot the values of the interest rate you obtain as they depend on the strike prices. 

**(5 points)**
What do you notice about the above plot? Does it or does it not agree with your prediction from the second question in the problem? Substantiate your answer. 

## Problem #4 (25 points)
To what do you attribute the discrepancies you observed in the previous problem? Think about it a bit and note your ideas. Then, consult the paper by *Brenner* and *Galai* uploaded into Canvas (under *Files* in the folder *articles*). Provide at least **four** substantiated reasons. 

*Note that this question is worth a quarter of your score for this project! A few lines of your musings will give you hardly any credit at all.*