Assignment

1. Graphing Trading Strategies: This problem asks you to graph profit diagrams for several trading strategies. Compute the profit of the strategy at maturity with respect to different realizations of the stock price in the future and plot the profit diagram of the strategy.

a. Writing a Straddle: Sell a call with an exercise price of $40 for a price of $3.10 and sell a put with an exercise price of $40 for $5.48.

b. Buying a Strangle: Buy a call with an exercise price of $60 for a price of $2.25 and buy a put with an exercise price of $50 for $4.35.

c. Buying a bull spread: Buy a call with an exercise price of $25 for a price of $5.20 and sell a call with an exercise price of $40 for $3.10.

d. Buying a butterfly spread with calls: Buy a call with an exercise price of $25 and a price of $8.15. Sell two calls with an exercise price of $30 and a price of $6.45. Finally, buy one call with an exercise price of $35 and a price of $4.85.

e. Writing a butterfly spread with puts: Sell a put with an exercise price of $70 and a price of $1.15. Buy two puts with an exercise price of $80 and a price of $3.65. Finally, sell one put with an exercise price of $90 and a price of $6.17.

2. Building a Binomial Tree in EXCEL with Market Data: This problem walks you through the process of building a binomial tree in EXCEL and using it to price options. In addition, you will use actual market data to estimate the market parameters that you need to do this (i.e., the risk-free rate and standard deviation of returns). In this problem, we will download price data for Amzaon.com, build a binomial tree of Amazon's stock price, and finally you will price put and call options on the underlying AMZN stock using that tree.

Sources for daily stock price data and risk-free interest rates are:

• http://finance.yahoo.com/q/hp?s=AMZN+Historical+Prices

• http://www.bloomberg.com/markets/rates-bonds/government-bonds/us/

a. Download daily closing stock prices for Amazon (AMZN) for the 3rd quarter of 2015 (July-September). Compute the continuous daily return as follows:

??_t = ln(??????????_t/??????????_t-1)

Note: if you have N daily stock prices in 3 months, you will have (N-1) daily returns. Compute (and report) the standard deviation of the daily return. Compute (and report) the annualized standard deviation (based on 252 trading days in a year) as follows:

σ_?????????? = σ_??????????√252

b. Compute the UP and DOWN factors for a binomial model with weekly price changes (i.e., Δt = 1/52). Be sure to use the annualized standard deviation. Also compute the risk-neutral probability of an UP move. Let's use the 12-month Treasury rate as the risk- free rate so that everyone is doing the same thing. (We should use the weekly rate, but the entire short-end of the yield curve is so close to zero that it hardly matters which maturity we use).

c. Build a 13-week binomial tree for AMZN (this will take you from October 19 to January 15). The standard way to represent a binomial tree in EXCEL is to have each column represent a date (step). An UP move is a represented as the column directly to the right and on the same row. A DOWN move is represented as the column directly to the right and down one row. As you move across the spreadsheet the columns will grow one row longer with each step. AMZN closed at $573.15 on October 19.

d. Price a 13-week call option on AMZN with a strike price of $525. Use backward induction and risk-neutral valuation to work your way back to today's call price. In EXCEL, build a second tree. Start by filling in the last column (step 12); these are the payoffs at maturity that correspond to the stock prices in the last column in the stock- price tree you built in part c. (Hint: just use a formula like =max(price-525, 0) where price references the appropriate cell from the last column of part c.) Work back one column at time, copy and pasting the risk-neutral valuation formulas. As a benchmark, the AMZN Jan 525 call was selling for $66.40 on October 19.

e. Price a 13-week put option on AMZN with a strike price of $600. As a benchmark, the AMZN Jan 600 put was selling for $50.30 on October 19.

Hint: once you have built the trees for parts c and d; you can simply copy the tree from part d, and replace the call payoff formulas in the last column with the put payoff formulas. The tree will then use the risk-neutral valuation formulas to automatically fill in the tree and give you the put price today.

3. Build another Binomial Tree in EXCEL with Market Data: OK, now that you know how to build a tree using market data, do it again! Pick your own stock and repeat problem #2. Adjust the length of the tree to roughly match the number of weeks until the maturity date of the options that you choose to price.

Here are the restrictions on your stock and options selections.

• The stock must have options traded on it.

• We will ignore dividends for this project, but try to select stocks that don't have high dividends or are not scheduled to pay dividends prior to the maturity of the options you are pricing.

• Pick options that have at least 8 weeks to maturity so that the tree has a reasonable degree of complexity.

• Price one call and one put with the same maturity but different strike prices.

• Report the market price of those options as a benchmark.