senior advanced writer

FIN 410

Fall 2015

HW #2

1) Real Options Part 1

The Milwaukee Brewers (a professional baseball team) recently announced a new promotion which

allowed fans to pay $1,000 for a pack of 10 “Timeless Tickets”, each of which was good for admission to

one game at any time in the team’s future (including playoff games). See:

http://espn.go.com/mlb/story/_/id/12167650/milwaukee-brewers-offer-less-tickets. For this question,

think about a “Timeless Ticket” as a financial option.

a) Is the “Timeless Ticket” a call or a put option? Is it American or European? Are the Brewers buying or writing the option? Explain why. (1 point)

b) What is the underlying on the option? What is the contract size? What is the strike? What is the premium per unit? What is the maturity? (1 point)

c) For a stock option with this strike price, what should the option premium be equal to? Why? What should the option delta be? Why? (2 points)

d) For a stock option of this type (i.e., call or put), as the time to maturity approaches infinity, what should the option premium converge to? Why? What should the option theta converge to as the

time to maturity approaches infinity? Why? (2 points)

e) The Brewers are a “small-market” team, meaning that they usually have less money available to spend on players, scouts, executives, and so on when compared to “big-market” teams like the

New York Yankees. With this in mind, can you explain, in terms of option contracts, why a

“small-market” team like the Brewers might be especially motivated to offer the “Timeless

Tickets” promotion? (1 point)

2) Real Options Part 2

Now imagine that the Brewers structured the promotion in (1) slightly differently, and offered “Timeless

Tickets” that granted the buyer the right, but not the obligation, to buy a ticket to any one regular-season

game in the team’s future for $100. We’re going to figure out a fair price for this option in this question.

To start, download the Excel data set “Brewers Tickets Sim 410” from Angel. This data set contains

(simulated) data for the price of an average ticket to each regular-season Milwaukee Brewers game from

2006 to 2015. (There are 162 games in a Major League Baseball season.)

After you do the calculations below, you’ll upload this data set to Angel as part of your HW2

submission.

In order to calculate an option price, we’ll need the following inputs (ignore dividends):

𝑆0: Let’s say that an average ticket for the next Brewers game (the first game of 2016) costs $70. 𝐾: $100, from the question description above. 𝑇: The most common option maturity is five years, so let’s say the life of the option is five years (1825 days) instead of infinitely long.

𝑟: Look up the current five-year Treasury yield (our proxy for the risk-free rate) as of 10/27/15 from this table: (http://www.treasury.gov/resource-center/data-chart-center/interest-

rates/Pages/TextView.aspx?data=yield)

a) What value are you using for the risk-free rate? _____________ (1 point)

𝜎: To calculate our realized volatility measure, take the following steps:

-Generate the series of log returns in Excel in the yellow box. Each date’s return should be given by

ln⁡(𝑆𝑡/𝑆𝑡−1), where 𝑆𝑡 is the ticket price for each day 𝑡. (Don’t calculate returns for Game 1, the first day of each season.) (1 point)

-Calculate one-, five-, and ten-year realized volatilities of log returns using STDEV.S in the green box.

The “one-year” realized volatility means the volatility from last year up to now, and so on. (1 point)

-Next, calculate one-, five-, and ten-year realized percentage volatilities (also in the green box). The

formula for percentage volatility is:

𝜎 = 100 ∗ 𝑠𝑡𝑑𝑒𝑣(ln(𝑟𝑒𝑡𝑢𝑟𝑛𝑠)) ∗ √(1/Δ)

Where 1/Δ is equal to the number of days in the year, which is 162 in our case. These are the 𝜎 values you’ll use in calculating the option price. (1 point)

b) In class, we talked about how different investors might choose different time horizons when calculating realized volatility. For calculating the option value of the “Timeless Ticket”, you’ll

have to choose a horizon, too. Explain which volatility horizon you picked, and why. There is no

“right” or “wrong” horizon to choose – I’m more interested in your reasoning. (1 point)

-Now that we have all the inputs, we can calculate the price of the option. Use an option price calculator

web site. Here are two that you can try:

http://www.math.columbia.edu/~smirnov/options13.html (Use 100 tree nodes)

https://www.m-x.ca/marc_options_calc_en.php

c) What is the premium that the Brewers should charge for the “Timeless Ticket” in this question?

List the value for 𝜎 that you used, and remember to pay attention to the option type (call or put) and style (American or European). (1 point.)

3) Binomial Trees

Given the model inputs below, find the price of an American call, American put, European call, and

European put by using binomial trees, either in Excel or by hand.

𝑆0: $50 𝐾: $52 𝑟: 4% APR 𝜎: 0.25 𝑇: 3 months Δ: 1/12 (one month)

a) Show your calculations for the values of 𝑢, 𝑑, and 𝑝. (1 point)

b) Attach a copy of your trees (whether you used Excel or drew by hand), with the option prices clearly indicated. (4 points)

c) Show that put-call parity holds for your European call and put prices. (1 point)

d) Show that your calculated price for the American options is greater than or equal to your calculated price for the European options (calls and puts). (1 point)