# Valuation of a Convertible Bond in an Arbitrage Free Framework

Recall that according to the arbitrage-free framework, the value of a bond with an embedded option is equal to the sum of the arbitrage-free values of its parts.

The arbitrage-free approach can be used to value convertible callable or putable bonds. Each component of the bond, including straight bond, call option of the stock, and call and/or put option on the bond) can be valued separately.

$$\text{Value}_{\text{convertible bond}} = \text{Value}_{\text{straight bond}}+\text{Value}_{\text{call option on the issuer’s stock}}$$

\begin{align*} & \text{Value of convertible bond} \\ & = \text{Value of straight bond} + \text{Value of call option on the issuer’s stock} \end{align*}

\begin{align*}\text{Value of convertible bond} & =\text{Value of straight bond}\\&+\text{Value of call option on the issuer’s stock}\end{align*}

\begin{align*}\text{Value of collable convertible bond} & =\text{Value of straight bond}\\&+\text{Value of option on the issuer’s stock}\\&-\text{Value of issuer’s call option }\end{align*}

\begin{align*}\text{Value of collable putable convertible bond } & =\text{Value of straight bond}\\&+\text{Value of call option on the issuer’s stock}\\&-\text{Value of issuer’s call option}\\&+\text{Value of investors put option}\end{align*}

## Question

The value of a callable putable convertible bond is most accurately expressed as:

1. Value of straight bond + Value of call option on the issuer’s stock.
2. Value of straight bond + Value of call option on the issuer’s stock +Value of issuer call option – Value of investor put option.
3. Value of straight bond + Value of call option on stock – Value of issuer call option + Value of investor put option.

#### Solution

\begin{align*} & \text{Value of callable putable convertible bond} \\ & = \text{Value of straight bond} \\ & + \text{Value of call option on stock} \\ & – \text{Value of issuer call option} \\ & + \text{Value of investor put option} \end{align*}

Reading 30: Valuation and Analysis of Bonds with Embedded Options

LOS 30 (p) Describe how a convertible bond is valued in an arbitrage-free framework.

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