Reading 56: Valuing Bonds with Embedded Options-LOS d 习题精 following, div, center, interest

土豆妮

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

LOS d: Compute the value of a callable bond from an interest rate tree.

 

 

Using the following tree of semiannual interest rates what is the value of a 5% callable bond that has one year remaining to maturity, a call price of 99 and pays coupons semiannually?

        7.76%
6.20%
        5.45%

A) 99.01.

B) 97.17.

C) 98.29.

---

 

The callable bond price tree is as follows:

		100.00
A → 98.67
98.29		100.00
	99.00
		100.00

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

Price_A = min[(prob × (P_up + (coupon / 2)) + prob × (P_down + (coupon/2)) / (1 + (rate / 2)), call price] = min[(0.5 × (100 + 2.5) + 0.5 × (100 + 2.5)) / (1 + (0.0776 / 2)), 99} = 98.67. The bond values at the other nodes are obtained in the same way.

土豆妮

Using the following tree of semiannual interest rates what is the value of a callable bond that has one year remaining to maturity, a call price of 99 and a 5% coupon rate that pays semiannually?

	7.59%
6.35%
	5.33%

A) 98.26.

B) 99.21.

C) 98.65.

---

The callable bond price tree is as follows:

		100.00
	98.75
98.26		100.00
	99.00
		100.00

The formula for the price at each node is:

Price = min{(prob × (P_up + coupon/2) + prob × (P_down + coupon/2)) / (1 + rate/2), call price}.

Up Node at t = 0.5: min{(0.5 × (100 + 2.5) + 0.5 × (100 + 2.5)) / (1 + 0.0759/2), 99} = 98.75.

Down Node at t = 0.5: min{(0.5 × (100 + 2.5) + 0.5 × (100 + 2.5)) / (1 + 0.0533/2), 99} = 99.00.

Node at t = 0.0: min{(0.5 × (98.75 + 2.5) + 0.5 × (99 + 2.5)) / (1 + 0.0635/2), 99} = 98.26.

土豆妮

A callable bond with an 8.2% annual coupon will mature in two years at par value. The current one-year spot rate is 7.9%. For the second year, the yield-volatility model forecasts that the one-year rate will be either 6.8% or 7.6%. The call price is 101. Using a binomial interest rate tree, what is the current price?

A) 101.000.

B) 100.558.

C) 100.279.

---

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 for all the nodes in nodal period 2: V₂=100. In nodal period 1, there will be two possible prices:

V_1,U =[(100+8.2)/1.076+(100+8.2)/1.076]/2 = 100.558

V_1,L =[(100+8.2)/1.068+(100+8.2)/1.068]/2= 101.311

Since V_1,L is greater than the call price, the call price is entered into the formula below:

V₀=[(100.558+8.2)/1.079)+(101+8.2)/1.079)]/2 = 101.000.

---

土豆妮

Which of the following is the appropriate "nodal decision" within the backward induction methodology of the interest tree framework for a callable bond?

A) Min(par value, discounted value).

B) Max(call price, discounted value).

C) Min(call price, discounted value).

---

When valuing a callable bond using the backward induction methodology, the relevant cash flow to use at each nodal period is the coupon to be received during that nodal period plus the computed value or the call price, whichever is less.

土豆妮

Eric Rome works in the back office at Finance Solutions, a limited liability firm that specializes in designing basic and sophisticated financial securities. Most of their clients are commercial and investment banks, and the detection, and control of interest rate risk is Financial Solution’s competitive advantage.

One of their clients is looking to design a fairly straightforward security: a callable bond. The bond pays interest annually over a two-year life, has a 7% coupon payment, and has a par value of $100. The bond is callable in one year at par ($100).

Rome uses a binomial tree approach to value the callable bond. He’s already determined, using a similar approach, that the value of the option-free counterpart is $102.196. This price came from discounting cash flows at on-the-run rates for the issuer. Those discount rates are given below:

Using the binomial tree model, what is the value of the callable bond?

A) $101.735.

B) $102.196.

C) $95.521.

---

The price of the callable bond is $101.735.

---

What is the value of the call option embedded in this bond?

A) $6.675.

B) $0.461.

C) $12.924.

---

Given in the problem is the value of the bond’s option-free counterpart: $102.40. From Part A we’ve determined the price of the callable bond to be $101.735. From the relationship: 

V_call = V_noncallable – V_callable 

We can determine that the value of the call option is $102.196 – $101.735 = $0.461.

---

Which of the following steps that Rome would go through in calculating the effective duration of this callable bond is incorrect?

A) Add the option-adjusted spread (OAS) to each of the spot rates in the interest rate tree to get a "modified" tree.

B) Impose a small parallel shift to the interest rates used in the problem by an amount equal to +?.

C) Given the assumptions about benchmark interest rates, interest rate volatility, and a call and/or put rule, calculate the OAS for the issue, using the binomial model.

---

Calculating effective duration for bonds with embedded options is a complicated undertaking because you must calculate values of V₊ and V_–. Given the information in the problem, this requires following seven steps:

Step 1: Given the assumptions about benchmark interest rates, interest rate volatility, and a call and/or put rule, calculate the OAS for the issue, using the binomial model.

Step 2: Impose a small parallel shift to the interest rates used in the problem by an amount equal to +D_i.

Step 3: Build a new binomial tree using the new yield curve.

Step 4: Add the OAS to each of the 1-year forward rates in the interest rate tree to get a “modified” tree. (We assume that the OAS does not change when the interest rates change.)

Step 5: Compute the new value for V₊ using this modified interest rate tree.

Step 6: Repeat steps 2 through 5 using a parallel shift of -D_I to obtain the value for V_–.

Step 7: Use the formula duration = (V_– + V₊) / 2V₀(DI).