# Final Exam Practice: fixed income -- answers

A. Binomial Option Pricing 30 points

Assume that the interest rate starts at 4% and in each period and either increases by 2% or decreases by 2% (from 4% up to 6% or down to 2% would be the first move). The risk-neutral probabilities of ups and downs are all 1/2.

1. What is the price now of a discount bond with face of \$100 maturing one year from now?
```
This is a one-period discount bond with face of \$100 and an interest rate
equal to the initial rate of 4%.  Therefore, the price is

100
---- ~ \$96.15
1.04

```
2. What is the price now of a discount bond with face of \$100 maturing two years from now?
```
interest tree:

8%
/
6%
/    \
4%        4%
\    /
2%
\
0%

discount bond price:

100
/
94.34
/       \
92.49           100
\       /
98.04
\
100

```
3. What is the price today of a two-year collar with a cap price of 5% and a floor price of 3%? The underlying notional is \$1,000.
```
cash flows:

30
/
10
/     \
0           0
\     /
-10
\
-30

price (pre-cash flow)

30
/
24.151
/         \
-0.267               0
\         /
-24.706
\
-30

calculations:

10 + (30 + 0)/2/1.06 = 24.151
-10 + (0 - 30)/2/1.02 = -24.706

(24.151 - 24.706)/2/1.04 = -.267

```
B. Binomial Option Pricing 30 points

Assume that the interest rate starts at 6% and in each period and either increases by 2% or decreases by 2% (from 6% up to 8% or down to 4%). The risk-neutral probabilities of ups and downs are 1/2.

1. What is the price now of a discount bond with face of \$100 maturing one year from now?
```
\$94.3 = 100/1.06

```
2. What is the price now of a discount bond with face of \$100 maturing two years from now?
```
quoted spot rates:

10%
/
8%
/    \
6%         6%
\    /
4%
\
2%

discount bond prices

100
/
92.6
/      \
\$89.0          100
\      /
96.2
\
100

92.6 = 100/1.08, 96.2 = 100/1.04, 89.0 = 0.5 * (92.6 + 96.2)/1.06

```
3. What is the price today of an interest rate floor with a strike of 7% and two periods to maturity? The underlying notional is \$100 (so the cash flow is 2 if the rate is 5%).
```
cash flows

0
/
0
/   \
0       1
\   /
3
\
5

Values

0
/
0.46
/      \
\$2.99          1
\      /
5.88
\
5

```
0.46 = 0.5 * 1/1.08, 5.88 = 3 + 0.5*(1 + 5)/1.04, 2.99 = 0.5*(0.46 + 5.88)/1.06