Filter by Categories
Accounting
Banking

Finance




Pricing a Convertible Bond Using The Black-Scholes Model: an Example


In order to price a convertible bond using the Black-Scholes model, the following two steps would need to be taken:

  • calculate the investment value of the bond.
  • make adjustments to the investment value to account for the effect of early conversion on the maturity of the bond and the term of the warrant and also any potential losses in accrued interest due to a forced conversion.

Suppose a convertible bond has the following features:

Maturity date 25 January 2017
Par value $1,000
Issue size $150 million
Term 5 years
Call protection 5 years
Coupon 6%
Coupon payment frequency 6 months
Issue price 100%
Redemption price 100%
Underlying stock price at issue $75
Conversion price $100
Conversion premium at launch 25%
Denomination per bond $1,000
Conversion ratio (=denomination per bond/conversion price) 10
5-year risk-free rate 5.5%
Credit spread 3%
Existing shares outstanding 10 million
Shares created by the convertible 1 million
Dilution (=shares created/existing shares outstanding) 10% (=1/10)
Stock price volatility 35%
Dividends nil

Calculating the investment value of the bond requires determining the periodical coupon: C= 6%x1000/2 = 30, and the periodical discount rate: YTM= (risk-free rate + credit spread)/2 = (5.5% + 3%)/2 = 8.5%/2= 4.25%. Now we plug in these figures into the following formula:

Investment Value

It follows that:

Investment Value Example

The following table enlists all cash flows generated by the bond over its life:

Period (t) Cash flow Present value cash flow
0.5 $30 $28.77
1.0 $30 $27.60
1.5 $30 $26.47
2.0 $30 $25.39
2.5 $30 $24.36
3.0 $30 $23.37
3.5 $30 $22.41
4.0 $30 $21.50
4.5 $30 $20.62
5.0 $1030 $679.31
Total $899.80

Adjusted exercise price= investment value/conversion ratio = $899.80/10= $89.98

To calculate the value of a call warrant, we first figure out d1 and d2:

d1 Example
d2 Example

d1= -0.2726, d2= -0.2726 – 0.7826 = – 1.0552

value of a call warrant = N(-0.2726) x $75 – N(- 1.0552) x $89.98. e-0.055×5 = $19.48

value of the call component in each bond = value of the call warrant x conversion ratio = $19.48 x 10= $194.8

value of the warrant = [1/(1+λ] x value of the call component =  (1/1.1) x $194.8 = $177.09

Therefore, the theoretical value of the convertible bond is:

  • convertible bond value = investment value + value of the embedded call = $899.80 + $194.8 = $1,094.6
  • Taking into account the effect of dilution, convertible bond value = $899.80 + $177.09 = $1,076.9

This convertible bond is worth $1,094.6 before dilution, and $1,076.9 after dilution.



Tutorials
This section contains quite a vast collection of easy-to-understand explanatory manuals, practical guides, and best practices how-tos covering the main themes of this ...
Watch on Youtube
Remember to read our privacy policy before submission of your comments or any suggestions. Please keep comments relevant, respectful, and as much concise as possible. By commenting you are required to follow our community guidelines.

Comments


  • Etienne d'Ouala
    April 15, 2022 at 3:57 am

    Great post. Are you sure your d1 is correct ? should not it be + vol^2 /2 ?

Leave Your Comment

Your email address will not be published.*