16. Given the following information, prepare the HL bond price tree for three dates, t = 0; 1; 2. Th

16. Given the following information, prepare the HL bond price tree
for three dates, t = 0; 1; 2. The parameter =12 , and = 0:8.

T Zero-Coupon Rate

1
0.06

2
0.07

3
0.08

17. Bond pricing in the Vasicek
(1977) model: assume an interest rate process

dr = k( r) dt +
dB

where base parameter levels are r = k = = = 0:1, T = 1, and dB is
a standard Brownian motion. Assume also that the market price of risk = 0. In
each of the following three cases, compute the bond price for each value of the
given parameter, holding the other parameters at their base levels.

(a)
k = f0:1; 0:2; 0:4g.

(b)
= f0:05; 0:10; 0:15g.

(c)
= f0:05; 0:10; 0:20g.

For
each of the three cases, explain the direction in which the bond price changes.
That is, provide an economic explanation for why the bond price increases or
decreases with the given parameter, holding the other parameters at their base
levels.

18.
(Extending the model) In this problem we extend the Vasicek model
to allow the mean rate to become stochastic. Think of a situation in which the
Federal Reserve makes mi-nor adjustments to short-term market rates to manage
the temperature of the economy. The model comprises the following two
equations:

dr =
k( r) dt + dB

d = dB

The Brownian motion dB is the same for both the interest rate r
and its mean level . Answer the following questions:

Sundaram
& Das: Derivatives – Problems and Solutions . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 445

(a)
Given the bond price function P (r; ; T ), write down the process
for dP using Ito’s lemma. T denotes the time to maturity. t may be used to
denote current time.

(b)
Suppose the market price of risk is zero for both stochastic
variables r and . Then the bond’s instantaneous return will be given by E(dP )
= rP dt. Using this identity, derive the pde for the price of the discount
bond, stating clearly the boundary condition for the bond price.

(c) Guess a functional form for the solution of the pde. Use the guess
to derive a closed-form expression for the price of the bond.

(d) Will bond prices be higher or lower in this model versus a model
in which = 0, where the mean rate is constant?

19.
Write a function in Octave for
the Cox, Ingersoll, and Ross (CIR 1985) model and price the bond when the
values are r = k = = = = 0:10, and T = 5 years.

20.
In the CIR model, compute the yield curve from 1 to 10 years when
r = k = = =

= 0:10.

21.
Find a set of parameters in the
CIR model such that the yield curve from 1 to 10 years is of upward-sloping
shape.

 

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