ECON – 477

Problem Set 4

Spring 2021

Due by 2:30pm CST on April 20th 2021 (submission via Canvas)

Problem 1. (6 points)

The process for the prices of a 5-year maturity zero-coupon bond and of a derivative on

the interest rate that matures in three years are decribed by the following trees. The

probablities that an analyst associates with going up and down are 60% and 40% at each

node of the tree, respectively. (NOTE: These are NOT the risk neutral probabilities.)

5-year zero coupon bond price

period =⇒ i = 0 i = 1 i = 2 i = 3 i = 4 i = 5 time (in years) =⇒ t = 0 t = 1 t = 2 t = 3 t = 4 t = 5

Z50 Z 5 1 Z

5 2 Z

5 3 Z

5 4

100

84.33

70.91 100

61.81 87.02

55.55 75.49 100

52.03 67.07 89.12

63.08 79.05 100

74.84 91.31

86.57 100

95.86

100

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Derivative

period =⇒ i = 0 i = 1 i = 2 i = 3 time (in years) =⇒ t = 0 t = 1 t = 2 t = 3

154.5

108.72

68.04 103

40.33 49.17

23.92 10.3

4.64

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1. Suppose that you hold a portfolio of 10 five-years zeros and 20 derivatives. How does

the portfolio payoff evolve over three years? Construct the tree.

2. How can you change your position in the derivative in order to make the portfolio

riskless between date t = 0 and t = 1?

3. What is the implied interest-rate tree up to t = 2?

4. What is the price of a zero-coupon bond that matures at time t = 2?

5. Suppose that at time t = 1 the interest rate is 12.48%. How many the 3-year

zero-coupon bonds do you need to hold for each unit of the derivative to obtain

a Sharpe ratio of 0.75? (Hint: Recall that the variance of a security X is σ2X =

E [ (X −E (X))2

] Problem 2. (2 points)

The following table shows some annualized, continuously compounded zero-coupon yields

and forward rates:

y(0,T) f(0,T −1,T) f(0,T −2,T) f(0,T −3,T) T = 1 4.00% ? — — T = 2 ? 6.00% ? — T = 3 6.00% ? ? ?

1. What are the missing six values, y(0,2), f(0,0,1), f(0,2,3), f(0,0,2), f(0,1,3) and

f(0,0,3)? (denoted by “?”in the table). Select values that ensure that there are no

arbitrage opportunities.

2. Suppose the six missing numbers are as you calculated in Part 1, but now f(0,1,2)

is equal to 7% rather than 6%. Are there arbitrage opportunities? If so, explain how

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you would exploit such an opportunity? Specify which bonds you would buy and

sell, the quantities, and which forward contracts you would enter and their notional

values. If not, explain briefly why the alternative forward rate is consistent with the

absence of arbitrage.

Problem 3. (2 points)

Today you observe the following term structure of swap rates.

Maturity (years) Period i ci0 0.5 1 2.6542%

1 2 2.9055%

1.5 3 3.1546%

2 4 3.3623%

2.5 5 3.6570%

3 6 3.8865%

Determine the price of a 10-year Treasury note issued 8 years ago that pays a semi-

annual coupon at 2.75% annual rate, and has face value $10,000.

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