HW: 2
Course: M339W/M389W - Financial Math for Actuaries
Page: 1 of 4
University of Texas at Austin
HW Assignment # 2
Black-Derman-Toy.
2.1. Black-Derman-Toy. Provide a complete solution to the following problem(s):
Problem 2.1. (4 points) Yield volatility
Consider the following values of interest rates from an incomplete Black-Derman-Toy interest rate tree for
the effective annual interest rates.
r0 = 0.10,
ru = 0.15,
ruu = 0.19, ruuu = 0.27
rd = 0.13,
rud = 0.15, ruud = 0.20.
Calculate the yield volatility in year one of a zero-coupon bond with maturity two years from today.
Solution: First, we realize that much of the provided BDT tree is not useful at all. In fact, the yield
volatility we are looking for is equal to the volatility of effective interest rates in the first step. So, our answer
is
1
0.15
σ1 = ln
= 0.07155.
2
0.13
Problem 2.2. (16 points) Source: Problems 24.12 and 24.13 from the McDonald textbook.
Here is an incomplete Black-Derman-Toy interest rate tree with effective annual interest rates at each node.
0.13843
0.10362
rud
0.08000
0.07676
0.08170
(i) (2 pts) Calculate rud .
(ii) (10 pts) What is the 3-year zero coupon bond price per $100 at maturity implied by this tree?
Assume that the bond is issued at time 0.
(iii) (4 pts) What volatilities of annual effective interest rates were used to construct the above tree?
Instructor: Milica Čudina
HW: 2
Course: M339W/M389W - Financial Math for Actuaries
Solution:
(i)
2
rud
= ruu × rdd = 0.08170 × 0.13843 ≈ 0.01131.
So, rud ≈ 0.10635.
(ii)
P (0, 3) =
1
(1.08170 × 1.07676 × 1.08)−1
4
+ (1.10635 × 1.07676 × 1.08)−1
+ (1.10635 × 1.10362 × 1.08)−1
+ (1.13843 × 1.10362 × 1.08)−1
1
= 0.794969563 + 0.777257266 + 0.758340311 + 0.736970918 ≈ 0.76572.
4
(iii)
1
ru
ln( ) ≈ 0.15.
2
rd
ruu
1
σ2 = ln(
) ≈ 0.13.
2
rdu
σ1 =
Instructor: Milica Čudina
Page: 2 of 4
HW: 2
Course: M339W/M389W - Financial Math for Actuaries
Page: 3 of 4
Problem 2.3. (16 points) Source: Problems 24.12 and 24.13 from the McDonald textbook.
Here is an incomplete Black-Derman-Toy interest rate tree with effective annual interest rates at each node.
ruu
0.09908
0.10689
0.08000
0.08112
0.08749
(i) (2 pts) Calculate ruu .
(ii) (10 pts) What is the 3-year zero coupon bond price per $100 at maturity implied by this tree?
Assume that the bond is issued at time 0.
(iii) (4 pts) What volatilities of annual effective interest rates were used to construct the above tree?
Solution:
(i)
2
ruu = rud
/rdd = 0.106892 /0.08749 = 0.130592.
(ii)
B(0, 3) =
1
(1.08749 × 1.08112 × 1.08)−1
4
+ (1.10689 × 1.08112 × 1.08)−1
+ (1.10689 × 1.09908 × 1.08)−1
+ (1.130592 × 1.09908 × 1.08)−1
i
1
= 0.787548 + 0.773745 + 0.761101 + 0.745145 ≈ 0.766885.
4
So, per $100, the price is 76.6885.
(iii)
1
ru
ln( ) ≈ 0.10.
2
rd
1
ruu
σ2 = ln(
) ≈ 0.10.
2
rdu
σ1 =
Instructor: Milica Čudina
HW: 2
Course: M339W/M389W - Financial Math for Actuaries
Page: 4 of 4
Problem 2.4. (14 points) Consider the following Black-Derman-Toy interest-rate tree modeling the future
evolution of annual effective interest rates. The period-length is one year.
0.06
0.055
0.05
0.05
rd
rdd
(i) (2 points) Calculate the interest rate denoted in the tree by rdd .
(ii) (2 points) Assume that the volatility of the effective interest rates for the second year equals σ1 = 0.1.
Calculate rd in the above tree.
(iii) (10 points) Consider a two-year interest rate cap with the notional amount of $1,000 and an annual
effective cap rate of KR = 0.05. Calculate the price of this interest rate cap.
Solution:
(i) According to the Black-Derman-Toy model, we have
0.06
0.05
=
0.05
rdd
(ii) Again, in the BDT tree, we have
ru = rd e2σ1
⇒
⇒
rdd =
0.052
= 0.0417.
0.06
rd = 0.055e−2·0.1 = 0.045.
(iii) This is a two year cap, so there is only one node at which payment can take place: the “up”-node.
The price is
1
1
1
×
×
× (0.055 − 0.05) × 1000 = 2.26.
2 1.05 1.055
Instructor: Milica Čudina