Reading 56: Valuing Bonds with Embedded Options-LOS c 习题精 div, framework, following, valuation

土豆妮

Session 14: Fixed Income: Valuation Concepts
Reading 56: Valuing Bonds with Embedded Options

LOS c: Illustrate the backward induction valuation methodology within the binomial interest rate tree framework.

 

 

 

Using the following interest rate tree of semiannual interest rates what is the value of an option free bond that has one year remaining to maturity and has a 5% semiannual coupon rate?

        7.30%

6.20%

        5.90%

 

A) 98.98.

B) 97.53.

C) 98.67.

---

 

The option-free bond price tree is as follows:

		100.00
A → 98.89
98.67		100.00
	99.56
		100.00

As an example, the price at node A is obtained as follows:

Price_A = (prob × (P_up + (coupon / 2)) + prob × (P_down + (coupon / 2)) / (1 + (rate / 2)) = (0.5 × (100 + 2.5) + 0.5 × (100 + 2.5) / (1 + (0.0730 / 2)) = 98.89. The bond values at the other nodes are obtained in the same way.

The calculation for node 0 or time 0 is> >

0.5[(98.89 + 2.5) / (1+ 0.062 / 2) + (99.56 + 2.5) / (1 + 0.062 / 2)] => >

0.5(98.3414 + 98.9913) = 98.6663> >

土豆妮

Using the following interest rate tree of semiannual interest rates what is the value of an option free semiannual bond that has one year remaining to maturity and has a 6% coupon rate?

         6.53%
6.30%
         5.67%

A) 98.52.

B) 99.81.

C) 97.53.

---

The option-free bond price tree is as follows:

		100.00
A ==> 99.74
99.81		100.00
	100.16
		100.00

As an example, the price at node A is obtained as follows:

Price_A = (prob × (P_up + coupon/2) + prob × (P_down + coupon/2))/(1 + rate/2) = (0.5 × (100 + 3) + 0.5 × (100 + 3))/(1 + 0.0653/2) = 99.74. The bond values at the other nodes are obtained in the same way.

The calculation for node 0 or time 0 is

0.5[(99.74 + 3)/(1+ 0.063/2) + (100.16 + 3)/(1 + 0.063/2)] =

0.5 (99.60252 + 100.00969) = 99.80611

土豆妮

For a putable bond, callable bond, or putable/callable bond, the nodal-decision process within the backward induction methodology of the interest rate tree framework requires that at each node the possible values will:

A) include the face value of the bond.

B) be, in number, two plus the number of embedded options.

C) not be higher than the call price or lower than the put price.

---

At each node, there will only be two values. At each node, the analyst must determine if the initially calculated values will be below the put price or above the call price. If a calculated value falls below the put price: V_i,U = the put price. Likewise, if a calculated value falls above the call price, then V_i,L = the call price. Thus the put and call price are lower and upper limits, respectively, of the bond’s value at a node.

土豆妮

A bond with a 10% annual coupon will mature in two years at par value. The current one-year spot rate is 8.5%. For the second year, the yield volatility model forecasts that the one-year rate will be either 8% or 9%. Using a binomial interest rate tree, what is the current price?

A) 102.659.

B) 101.837.

C) 103.572.

---

The tree will have three nodal periods: 0, 1, and 2. The goal is to find the value at node 0. We know the value in nodal period 2: V₂=100. In nodal period 1, there will be two possible prices:

V_1,U=[(100+10)/1.09+(100+10)/1.09]/2= 100.917

V_1,L=[(100+10)/1.08+(100+10)/1.08]/2= 101.852

Thus

V₀=[(100.917+10)/1.085+(101.852+10)/1.085]/2= 102.659

土豆妮

A bond with a 12% annual coupon will mature in two years at par value. The current one-year spot rate is 14%. For the second year, the yield volatility model forecasts a lower bound of 12% for the one-year rate and a standard deviation of 10%. In a binomial interest rate tree describing this situation, what are the forecasted values for the bond in the first nodal period?

V1,U: upper rate value

V1,L: lower rate value

A) 97.680 101.125

B) 94.676 97.664

C) 97.683 100.000

---

The value of the bond for the lower rate is easy; since that forecasted rate is the coupon rate: V_1,L = 100. The value for the upper rate will be determined by the lower rate and the standard deviation: i_1,U = i_1,L × (e^{2 × s}) = 0.12 × (e^0.20) = 0.14657. Thus, V_1,U = (112 / 1.14657) = 97.683.

土豆妮

Which of the following is a correct statement concerning the backward induction technique used within the binomial interest rate tree framework? From the maturity date of a bond:

A) the corresponding interest rates and interest rate probabilities are used to discount the value of the bond.

B) the corresponding interest rates are weighted by the bond's duration to discount the value of the bond.

C) a deterministic interest rate path is used to discount the value of the bond.

---

For a bond that has N compounding periods, the current value of the bond is determined by computing the bond’s possible values at period N and working “backwards” to the present. The value at any given node is the probability-weighted average of the discounted values of the next period’s nodal values.

土豆妮

Why is the backward induction methodology used to value a bond rather than a forward induction scheme?

A) Future interest rate changes are difficult to forecast.

B) The price of the bond is known at maturity.

C) The convexity of a bond changes over time.

---

The objective is to value a bond's current price while the bond price at maturity is known. Therefore, price at maturity is used as a starting point, and we work backward to the current value.

土豆妮

With respect to interest rate models, backward induction refers to determining:

A) the current value of a bond based on possible final values of the bond.

B) convexity from duration.

C) one portion of the yield curve from another portion.

---

Backward induction refers to the process of valuing a bond using a binomial interest rate tree. For a bond that has N compounding periods, the current value of the bond is determined by computing the bond’s possible values at period N and working "backwards."

土豆妮

For an option-free bond where the coupons and maturity value are known and assuming constant interest rate volatility, which of the following sets of information will allow an analyst to construct the entire tree? The:

A) beginning interest rate at the root only.

B) lowest interest rate in each nodal period.

C) interest rate at the root and in the final nodal period.

---

Given the lowest interest rate in each nodal period, the interest rates at the other nodes can be calculated. The interest rate at any node above the lowest is larger than the one below it by a factor of e^{2 × σ}. Neither of the other sets of information are sufficient for constructing the tree.

土豆妮

A binomial model or any other model that uses the backward induction method cannot be used to value a mortgage-backed security (MBS) because:

A) the cash flows for the MBS are dependent upon the path that interest rates follow.

B) the prepayments occur linearly over the life of an interest rate trend (either up or down).

C) the cash flows for an MBS only depend on the current rate, not the path that rates have followed.

---

A binomial model or any other model that uses the backward induction method cannot be used to value an MBS because the cash flows for the MBS are dependent upon the path that interest rates have followed.