Derivative Securities Individual Take-Home Assignment

Problem 1: Properties of Options

Consider a four-month European call option on a dividend-paying stock. The stock price is $75, the strike price is $70, and a dividend of $1.50 is expected in three months. The risk-free interest rate is 8% per annum for all maturities.

a. What is the lower bound for the price of this call?

b. Assume that the call is currently selling for $3. Describe in detail with which strategy you can gain an arbitrage profit and how much this profit will be.

Problem 2: Properties of Options

The price of a European call that expires in six months and has a strike price of $50 is $5. The underlying stock price is $52, and a dividend of $1.00 is expected in three months. The term structure is flat, with all risk-free interest rates being 10%.

a. What is the price of a European put option on the same stock that expires in six months and has a strike price of $50?

b. Explain in detail the arbitrage opportunities if the European put price is $0.50. How much will be the arbitrage profit?

Problem 3: Binomial Trees

A stock price is currently $30. Over each of the next two three-month periods it is expected to go up by 8% or down by 10%. The risk-free interest rate is 5% per annum with continuous compounding.

a. Use a two-step binomial tree to calculate the value of a six-month European put option with a strike price of $32.
b. Use a two-step binomial tree to calculate the value of a six-month American put option with a strike price of $32.
c. Use a two-step binomial tree to calculate the value of a six-month European call option with a strike price of $32.
d. Show whether the put-call-parity holds for the European put and the European call.
e. Calculate the deltas of the European put and the European call at the different nodes of the binomial three.

Hint: You need to calculate three deltas for the call and three deltas for the put.

Problem 4: Binomial Trees

A stock price is currently $40. During each two-month period for the next four months it is expected to increase by 10% or decrease by 8%. The risk-free interest rate is 5%. Use a two-step tree to calculate the value of a derivative that pays off (max[(ST-35),0])2 where is the stock price in four months. T S

a. Use no-arbitrage arguments (you need to show how to set up the riskless portfolios at the different nodes of the binomial tree).
b. Use risk-neutral valuation.
c. Verify whether both approaches lead to the same result.
d. If the derivative is of American style, should it be exercised early?

Problem 5. Valuing Stock Options: The Black-Scholes-Merton Model

This is a Bloomberg-based exercise. Before making any calculations, please do the following:

Search for the share of Google (Bloomberg ticker: GOOGL). Download daily price data for the Google share price over the last 250 trading days.

• Go to the Options Monitor showing option contracts on the Google share (Use OMON ).
• Find the put and the call options with

o Expiration in October 2017 and
o Strike price $940

Make screenshot(s) showing the prices of these options.

Hint: You can make screenshots and email them to yourself via GRAB .

• Use LR to obtain a LIBOR value for the same day as the option price data. Choose the USD LIBOR with time horizon closest to the time-to-maturity of the options. Document with a screenshot.

Once you have this data, you can start with the calculations:

(a) Calculate with Excel the daily returns of the Google share and calculate afterwards their standard deviation over the last 250 days.

(b) Convert the daily volatility to volatility per annum.

(c) Use the Black-Scholes-Merton pricing formulas for European options and calculate the theoretical prices of a European call and a European put option with expiration in October 2017 and strike price of $940 on the Google share. Use the LIBOR rate you have downloaded as a proxy of the risk-free rate.

(d) Insert the Black-Scholes-Merton prices you just calculated in the put-call-parity. Does it hold?

(e) How would the result of (c) change if a dividend of $2 is expected in four months?

(f) Compare the Black-Scholes-Merton prices you calculated in (c) with the prices of these options given in Bloomberg (use the average of bid and ask price as the Bloomberg price). Are there any deviations between the theoretical prices you have calculated and the prices given in Bloomberg? If yes, what could be possible reasons for these deviations?

Hint: There are no dividends announced for August, September and October 2017.

Additional submission requirements for problem 5:

• Additionally to your calculations, please insert in the Word file that you will submit the screenshots you have made showing the spot price, option prices and LIBOR which you used.

• Upload the Excel file showing the calculation of the standard deviation in (a).