# MATH133A California Irvine Derivative Market Problems Paper

Upon graduating from UCI, you start a job in finance earning \$72,000 per year and decides to buy a home. You can afford to pay no more than \$3,000 per month for the monthly payment on 30-year fixed rate home loan(mortgage) and the current conventional 30-year fixed mortgage interest rate you qualify to borrow at is 5%

1. What is the maximum size home loan you can afford?
2. Suppose you borrow the maximum loan amount found in part(a). How much total interest do you pay in the first 2 month?

Suppose black rock stock is trading at \$400 and pays a discrete dividend of \$3.00 to its sharebolders once per quarter, the first dividend of \$3 to be paid in 0.25 years from today, the second dividend of \$3 to be paid in 0.5 years from today, the third dividend of \$3 to be paid in 0.75 years from today, and the fourth dividend of \$3 to be paid in 1 year from today. The continuously compound risk-free interest rate that you can lend or borrow at is 8%. In this problem, you are to use continuous compounding of interest in all computations.

• A)What is the fair value price for a one-year prepaid forward contract on BLK?
• B)Suppose the market prices for a normal(not prepaid) 1-year forward contract on Blackrock are as follows:
• a)Bid price for BLK forward contract:\$520.95
• b)Ask price for BLK contract: \$530.95

Does an arbitrager opportunity exist? If so, specify the exact components of the portfolio you need to capture the arbitrage, and the exact arbitrage profit.

3. Your are a bond portfolio manager at Blackrock and the investment committee has asked you to buy a bond with price B1, and sell short a certain quantity N of a second bond with price B2

Bond with price B1 is a 1-year zero coupon bond with a yield-to-maturity’s of 1%

Bond with price B2 is a 2-year zero coupon bond with a yield-to-maturity of 2%

The resulting portfolio value is II = B1-N*B2

• A)How would you choose N to optimally hedge the interest rate exposure of the portfolio II and thus minimize its sensitivity to interest rate changes? Find a numerical value for N.
• B)Using the value of N that you found in part (a), what is your portfolio’s profit or loss if both of the yield-to-maturities of bond B1 and B2 suddenly decrease by 1%? Round your numerical answer to the nearest 4th-decimal place.

4. Consider this table of risk-free zero coupon bond (ZCB) prices:

 Quarter 1 2 3 4 5 6 7 8 ZCB 0.9804 0.9612 0.9423 0.9238 0.9057 0.888 0.8706 0.8535
• A)Given the zero coupon bond prices in the table above, what is the 8th quarter interest rate swap price?
• B)Suppose there is an abrupt stock market crash and resulting flight to quality which investors sell Rosalyn assets such as stocks and aggressively buy risk-free bond. Consequently all of the zero coupon bond prices in the table suddenly increase by 0.01. What os the new 8th quarter interest rate swap price?

5. An investor enters into a 2-year swap agreement to purchase crude oil at \$51.25 per barrel. Soon after the swap is created, forward prices rise and the new 2-year swap price is \$61.50. If interest rate are 1% and 2% on 1st and 2nd year zero coupon government bonds, respectively, what is the gain or loss to be made from unwrapping the original swap agreement?

6. An iron butterfly, also known as the iron fly, is the name of an advanced, neutral-outlook options trading strategy that involves buying and holding four different option at three different strike prices. Suppose you construct an iron butterfly position on Blackrock stock as follows, with Blackrock currently trading at \$400. All options have an expiration date of one year from today and the constant a>0.

1. Long one out-of-the-money put option with premium \$5 and strike price of 400-a

2. Short one at-the-money put with premium \$35

3.Short one at-the-money call with premium \$45

4.Long one out-of-the-money call with with premium \$15 and strike price of 400+a

A) Draw the profit diagrams at expiration for the Blackrock iron butterfly, with the constant a as a free parameter, assuming the risk-free interest rate is zero.

B) For what range of values for a does an arbitrage opportunity exist.

Suppose the risk-free rate is zero and the market price for Blackrock stock option that expire in one year are as follow

 Strike Price \$350 \$450 Call Premium \$25 \$75 Put Premium \$15 \$60
• A)Specify the component and present cost of a long synthetic one-year forward agreement on Blackrock stock using only the option with strike price \$350 in the table. The position should have a payoff at expiration that os identical to a one-year forward agreement with a forward agreement with a forward price of \$350
• B)Using any of the options in the table, does an arbitrage opportunity exist? If so, specify the exact components of the portfolio you need to capture the arbitrage, and the exact arbitrage profit.

8. Consider an investment that is long ten S&P 500 index futures contracts at a price \$2700.00. The initial margin requirement is \$36.000 per contract and the maintenance margin is \$30.000 per contract. The risk-free rate is zero.

A) What is the notional value of your long exposure to the S&P 500 index?

B) Upon opening the position you deposit, the minimum initial margin required in your future trading account. At what S&P trading account. At what S&P 500 index future price will a margin call occur?

9. For this problem, please circle True or false for each statement concerning derivative market.

1. True or false. Consider 1-year European put and call option on Blackrock stock with the same strike price. If the strike price is equal to the fair value price for the 1-year forward price on Blackrock stock, the put premium and call premium must be equal, otherwise an arbitrage opportunity exists.
2. True or False. The maximum loss of a short put at expiration is unbounded.
3. True or False. The slope coefficient computed by regressing a particular stock’s historical returns on the S&P 500 index returns is call beta, the stock’s beta with respect to the S&P index, beta is important in finding variance minimizing hedged portfolios.
4. Bond price move in the opposite direction as bond yields.
5. The maximum loos of a short stock position is equal to the price at which you enter the short position.
6. The premium of a call option increase as its strike price increase.
7. A swap provides a means for replacing a stream of uncertain and charitable payment with a fixed, non-variable payment stream that is certain.
8. The zero cost collar options trading strategy involves buying a call option and selling a put option on the same stock with the same expiration date, same premium, and higher stroke price than the call option.
9. Suppose your go long 10 Eurodollar futures contracts at a price of 97. When the future contract settle at expiration, 3-month LIBOR is 1%. Ignoring commissions and margin interest, your position results in a \$50,000 loss.
10. The current U.S. dollar / Chinese yuan currency spot rate is \$0/130 per yuan. The fair value price for the U.S. dollar / Chinese yuan exchange rate for a 2-year forward contract is \$0.1408. If the U.S. dollar denominated annual interest rate is 6.0%, the Chinese yuan-denominated annual interest rate must be 3%.

Fei opened a 100-share long position in Blackrock stock using the Robinhood iPhone app when Blackrock had a bid price of \$390 and an offer price of \$400. A few minutes later he added another 100-share long position in Blackrock stock when the bid price was \$440 and the offer price was \$450. He sold all 200 shares at the end of the day when Blackrock stock had a bid price of \$475 and offer price of \$485. The Robinhood iPhone app features zero trading commissions.

Mingkang was also trading Blackrock stock on the same day, using the interactive breakers a platform that was used in the Math 133A trading competition, his first 100 shares were bought when Blackrock had a bid price of \$394.99 and an offer price of \$395.00, and his second 100-share were bought when Blackrock stock had a bid price of \$444.99 and an offer price of \$445.00. He sold all 200 shares at the end of the day when Blackrock stock had a bod price of \$480.00 and offer price of \$480.01. Interactive Brokers charge a \$0.005 per share commission.

Who I earned a higher net trading profit, Fei or Mingkang? Make sure you carefully account fort trading profit and cost.