Florida Atlantic University Dr. Hongwei Long

Midterm Exam (Financial Mathematics 1—STA 6446), February 21, 2017

Name:

  1. The price of a stock is $36 and the price of a three-month call option on the stock with a strike price of $36 is $3.60. Suppose a trader has $3,600 to invest and is trying to choose between buying 1,000 options and 100 shares of stock. How high does the stock price have to rise for an investment in options to lead to the same profit as an investment in the stock?

Solution. Let the desired stock price be S. Then, we must have

100 (S-36) =1000 (S-36)-1000 (3.60),

which yields S=40. So the stock price has to rise to $40 in order to lead to the same profit for an investment in options and an investment in stock.

  1. On July 1, 2012, a Japanese company enters into a forward contract to buy $1 million with yen on January 1, 2013. On September 1, 2012, it enters into a forward contract to sell $1 million on January 1, 2013. Describe the profit or loss the company will make as a function of the forward exchange rates on July 1, 2012 and on September 1, 2012. Let be the spot exchange rate (all the exchange rates are measured as yen per dollar) on January 1, 2013.

Solution.Note that and are the forward exchange rates for the contracts entered into July 1, 2012 and September 1, 2012, and is the spot rate on January 1, 2013. (All exchange rates are measured as yen per dollar). The payoff from the first contract is million yen and the payoff from the second contract is million yen. The total payoff is therefore million yen.

  1. A fund manager has a portfolio worth $10 million with a beta of 0.95. The manager is concerned about the performance of the market over the next two months and plans to use three-month futures contracts on the S&P 500 to hedge the risk. The current level of the index is 1150, one contract is on 250 times the index, the risk-free rate is 5% per annum, and the dividend yield on the index is 2% per annum. The current 3 month futures price is 1159.

(a)What position should the fund manager take to eliminate all exposure to the market over the next two months?

(b)Calculate the effect of your strategy on the fund manager’s returns if the level of the market in two months is 1,200. Assume that the futures price is 0.30% higher than the index level at this time.

Solution.

a)The number of contracts the fund manager should short is

Rounding to the nearest whole number, 33 contracts should be shorted.

b)The following computation shows that the impact of the strategy. If the index in two months is 1,200, the futures price is 1200×1.0030. The gain on the short futures position is therefore

The return on the index is =0.333% in the form of dividend and in the form of capital gains.. The total return on the index is 4.681%. The risk-free rate is 0.833% per two months. By the CAPM, we have

The expected return on the portfolio=+β=0.833%+(0.95)(4.681%-0.833%)=4.4886%. The gain on the portfolio is 0.044886(10,000,000)=$448860. When this is combined with the gain on the futures the total gain is $80910.

  1. The cash prices of six-month and one-year Treasury bills are 94.0 and 89.0. A 1.5-year bond that will pay coupons of $4 every six months currently sells for $94.84. A two-year bond that will pay coupons of $5 every six months currently sells for $97.12. Calculate the six-month, one-year, 1.5-year, and two-year zero rates.

Bond principal ($) / Time to maturity (years) / Annual coupon* ($) / Bond price ($)
100 / 0.50 / 0 / 94.00
100 / 1.00 / 0 / 89.00
100 / 1.50 / 8 / 94.84
100 / 2.00 / 10 / 97.12

*Half the stated coupon is assumed to be paid every six months.

Solution.

The 6-month Treasury bill provides a return of in six months. This is per annum with semiannual compounding or per annum with continuous compounding. The 12-month rate is with annual compounding or with continuous compounding.

For the 1 year bond we must have

where is the 1 year zero rate. It follows that

or 11.5%. For the 2-year bond we must have

where is the 2-year zero rate. It follows that

or 11.3%.

  1. Suppose that zero interest rates with continuous compounding are as follows:

Maturity(years) / Rate (% per annum)
1 / 2.0
2 / 3.0
3 / 3.7
4 / 4.2
5 / 4.5

(a) Calculate forward interest rates for the second, third, fourth, and fifth years.

(b)What is the 5-year par yield? Assume that a coupon is paid at the end of each year.

(c)Value an FRA where you will pay 5% (compounded annually) for the third year on $1 million.

Solution.
(a) The forward rates with continuous compounding are as follows:

Year 2: (3.0%*2-2%*1)/1=4.0%

Year 3: (3.7%*3-3.0%*2)/1=5.1%

Year 4: (4.2%*4-3.7%*3)/1=5.7%

Year 5: (4.5%*5-4.2%*4)/1= 5.7%

(b ) We have m=1, d=exp(-0.045*5)=0.79852, and

A=exp(-0.02*1)+exp(-0.03*2)+exp(-0.037*3)+exp(-0,042*4)+exp(-0.045*5)=4.46076.

So, the par yield (coupon rate) is

c=(100-100d)m/A=[100-100*(0.79852)]*1/(4.46076)=4.52.

(c )The forward rate is 5.1% with continuous compounding or with annual compounding. The 3-year interest rate is 3.7% with continuous compounding. From equation (4.10), the value of the FRA is therefore

or $2,078.85.

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