# Valuing Bonds with Embedded Options

The steps for valuing a bond with an embedded option in the presence of interest rate volatility are as follows:

Step 1: Generate an interest rate tree using the yield curve and interest rate volatility assumptions.

Step 2: Determine whether the embedded option will be exercised at each node.

Step 3: Calculate the present value of the bond using the backward induction method.

Recall that backward induction involves starting from maturity and working backward from right to left to determine the bond’s value at each node.

#### Example: Valuation of Bonds with Embedded Options Assuming Interest Rate Volatility

The following interest rate tree has been calibrated, assuming a 15% interest rate volatility and the applicable benchmark yield curve.

We can determine the value of a three-year default-free 5% annual coupon bond, both callable and putable at par one year and two years from today.

#### Callable Bond

Formula:

$$V=0.5\left[\frac{V_u+C}{1+i}+\frac{V_d+C}{1+i}\right]$$

Where

V = Value of the bond at each node.

C = Coupon payment.

The bond value at each node must meet the call rule (call the bond if the price exceeds 100) for the option to be exercised. \begin{align*} V_{2,uu} &=\frac{105}{1.065}=98.59 \\ V_{2,ud} &=\frac{105}{1.055}=99.53 \\ V_{2,dd} &=\frac{105}{1.035}=101.45 \\ V_{2,dd} &= 101.45 > 100. \end{align*} The bond will thus be called at100. To continue with the valuation, the bond’s value at this node will be taken as 100. \begin{align*} V_{1,u} &=0.5\left[\frac{98.59+5}{1.05}+\frac{99.53+5}{1.05}\right]=99.10 \\ V_{1,d} &=0.5\left[\frac{99.53+5}{1.025}+\frac{100+5}{1.025}\right]=102.21 \end{align*} $$V_{1,d} > 100,$$ implying that the bond is called at100.

$$V_0 =0.5\left[\frac{99.10+5}{1.015}+\frac{100+5}{1.015}\right]=103.00$$

This can be shown in the following tree:

#### Putable Bond

The value of the bond at each node is calculated using the formula:

$$V=0.5\left[\frac{V_u+C}{1+i}+\frac{V_d+C}{1+i}\right]$$

Further, the embedded put option will be exercised at each node of the tree if the value of the bond is less than 100. This is known as the put rule. \begin{align*} V_{2,uu} &=\frac{105}{1.065}=98.59 \\ V_{2,ud} &=\frac{105}{1.055}=99.53 \\ V_{2,dd} &=\frac{105}{1.035}=101.45 \end{align*} Notice that both $$V_{2,uu}$$ and $$V_{2,ud}$$ are less than100. In this case, the bond will be put at $100. To continue with the valuation, the values of the bond at these nodes will be taken as$100.

\begin{align*} v_{1,u} & =0.5\left[\frac{100+5}{1.05}+\frac{100+5}{1.05}\right]=100\\ v_{1,d} & =0.5\left[\frac{100+5}{1.025}+\frac{101.45+5}{1.025}\right]=103.12 \\ v_0 &=0.5\left[\frac{100+5}{1.015}+\frac{103.12+5}{1.015}\right]=105.00 \end{align*}

This can be shown in the following tree:

## Question

Given the following interest rate tree:

The value of a two-year,6% annual coupon bond, with a par value of $100 and callable at par at the end of year 1 is closest to: 1.$101.30.
2. $101.53. 3.$102.64.

#### Solution

The value of the bond at each node is calculated using the formula:

$$V=0.5\left[\frac{V_u+C}{1+i}+\frac{V_d+C}{1+i}\right]$$

The call rule will then be used to establish if the embedded call option will be exercised at each node.

\begin{align*} v_{1,u} &=0.5\left[\frac{100+6}{1.085}+\frac{100+6}{1.085}\right]=97.696 \\ v_{1,d} &=0.5\left[\frac{100+6}{1.055}+\frac{100+6}{1.055}\right]=100.474 \end{align*}

$$v_{1,d}>100$$. Therefore, the bond is called at \$100.

$$v_0=0.5\left[\frac{97.696+6}{1.035}+\frac{100+6}{1.035}\right]=101.30$$

Reading 30: Valuation and Analysis of Bonds with Embedded Options

LOS 30 (f) Calculate the value of a callable or putable bond from an interest rate tree.

Shop CFA® Exam Prep

Offered by AnalystPrep

Featured Shop FRM® Exam Prep Learn with Us

Subscribe to our newsletter and keep up with the latest and greatest tips for success

Daniel Glyn
2021-03-24
I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way!
michael walshe
2021-03-18
Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.
Nyka Smith
2021-02-18
Every concept is very well explained by Nilay Arun. kudos to you man!