Deck 26: Modeling Term Structure Movements

Question 1

Suppose that the one-year and two-year zero-coupon rates are 6% and 7%, respectively (assume continuous compounding). After one year, let the one-year zero-coupon rate move down to 4% or up to 9%. What must be the probability of the up move for the rates to be arbitrage-free?
(a) 0.20
(b) 0.25
(c) 0.50
(d) 0.60

Answer 1

A

Question 2

"Equilibrium" models of the term-structure

A) Are general equilibrium models of all securities in the economy.
B) Are models which match observed term structure curves perfectly.
C) Include such models as Vasicek (1977) and Cox, Ingersoll, and Ross (185).
D) Are models which ensure that "disequilibrium" phenomena, such as negative interest rates, cannot occur.

Answer 2

Include such models as Vasicek (1977) and Cox, Ingersoll, and Ross (185).

Question 3

A $100 face value one-year risk-free discount bond is priced at $95. After one year, the two-year bond will be worth either $91 or $97. What (rounded to the nearest dollar) is the highest possible price of the two-year bond that is arbitrage-free?

A) $91
B) $92
C) $95
D) $97

Answer 3

$92

Question 4

Which of the following statements is implied by the existence of no-arbitrage in a risk-neutral pricing framework?

A) All bonds will have the same expected return irrespective of maturity.
B) Expected long-term bond rates of return will be higher than expected short-term bond rates of return.
C) Each period, bonds of all maturities will have the same expected rate of return.
D) The probability of rates moving up versus moving down is fifty-fifty.

Question 5

A $100 face value one-year risk-free discount bond is priced at $95. The two-year discount bond is priced at $90. After one year, the two-year bond will take one of three possible prices with defined probabilities. Which of the following sets of prices is acceptable from a no-arbitrage standpoint?

A) $\{ 99.74,94.74,89.74 \}$ with equal probabilities.
B) $\{ 99.74,94.74,89.74 \}$ with probabilities $\{ 0.25,0.50,0.25 \}$ .
C) Both (a) and (b).
D) The problem is not well-defined because there is no unique martingale probability measure for the two-year bond.

Question 6

A $100 face value one-year risk-free discount bond is priced at $95. The two-year discount bond is priced at $90. After one year, the two-year bond will take one of three equiprobable prices, spaced $5 apart. The middle value of these possible prices is

A) $90.25
B) $94.75
C) $95.00
D) $100.00

Question 7

In the Black-Scholes framework, return volatility is assumed to be constant over the life of the option. This is not theoretically appropriate for pricing options on (default-risk-free) bonds because

A) Empirically, interest rates are known to change over time, so bond volatility will change over time.
B) Even if interest rates are constant over time, the duration of the bond will change over time, and duration is a measure of volatility.
C) Constant return volatility on a zero-coupon bond is possible only if the bond price does not change over time.
D) The bond price at maturity is known for certain, so volatility must go down as maturity approaches.

Question 8

The term "no-arbitrage" class of term-structure models refers to

A) Models which focus on bond prices directly rather than interest rates.
B) Models which work under the martingale measure directly rather than under the actual or "statistical" measure.
C) Models whose parameters never have to be re-estimated since no-arbitrage ensures that they cannot change from day to day.
D) Models which are capable of matching the observed term-structure perfectly.

Question 9

A $100 face value one-year risk-free discount bond is priced at $95. The two-year discount bond is priced at $90. After one year, the two-year bond will be worth either $91 or $97. The probability of this bond moving to a price of $97 is

A) 0.37
B) 0.50
C) 0.58
D) 0.62

Question 10

Which of the following is not sufficient for a pricing tree for risky bonds to be free of arbitrage?

A) The existence of a risk-neutral pricing probability measure.
B) The existence of a general equilibrium in the asset markets.
C) All normalized (discounted) assets are martingales.
D) On the tree, the gross risk-free one-period return is straddled by the return when rates move up and when rates move down.

Question 11

In the Black-Scholes formula, interest rates are assumed to be constant. This is not appropriate for pricing options on bonds primarily because

A) The value of a bond is constant if interest rates are constant.
B) Constant rates would mean no volatility in bond prices and no option value.
C) Payoffs would be discounted at a constant rate.
D) None of the above.

Question 12

"No-arbitrage" models of the interest rate differ from "equilibrium" models of the interest rate in that

A) They have a larger number of free parameters enabling them to fit the yield curve exactly.
B) They do not admit arbitrage whereas an equilibrium model may admit arbitrage under some conditions.
C) Equilibrium models were derived in the academic literature whereas whereas no-arbitrage models were developed mainly by practitioners.
D) They allow for the possibility that the market is in disequilibrium

Question 13

If we use the Black-Scholes model for bond options, then we assume that bond prices are lognormal, as the underlying asset in the Black-Scholes model is assumed to have a lognormal distribution. Which of the following is not a consequence of this assumption?

A) Bond prices are non-negative.
B) Interest rates are non-negative.
C) Bond prices are positively skewed.
D) Interest rates are not skewed.

Question 14

Suppose that the one-year and two-year zero-coupon rates are 6% and 7%, respectively (assume continuous compounding). After one year, let the one-year zero-coupon rates move down to $r _ { d }$ or up to $r _ { u } = 1.2 r _ { d }$ , with equal probability. The rate $r _ { u }$ that is arbitrage-free under these conditions is

A) 5.92%
B) 6.63%
C) 7.27%
D) 8.73%