Question

1. Assume the following interest rate tree :

			2.27%
		1.69%
	1.22%		1.69%
0.85%		1.25%
	0.90%		1.25%
		0.93%
			0.92%
Current 6-month rate	6 months	12 months	18 months

1. Calculate value of the (option-free) bond using binomial model.
2. Assume the bond is callable on 1 Jun 2018 and 1 Dec 2018 at $102.00, calculate the value of the bond.
3. Calculate value of the call option and comment on who bears the cost of the call option.

Answer 1

					0.0227	100
					97.78039
			0.0169
			97.25503
	0.0122				0.0169	100
	96.83609				100.0169
0.0085			0.0125
		98.77995
96.62283
	0.009				0.0125	100
	98.05217				100.0125
			0.0093
			99.08932
					0.0092	100
					100.0092

Ans. A. The value of option free model using binomial tree is 96.622823 with no coupon.

Ans b. The call option is out of the money as the price of the bond at 1 June 2018 and 1 Dec 2018 does not exceed 102 i.e. the call price.

So the value of the call option is 0 and the bond will trade as if its an option free bond and the value of such a callable bond = 96.62283

Empirically it being observed that price of a option free bond is more than price of a callable bond.

Ans c. value of call option = value of option free bond – value of callable bond

= 96.62283 – 96.62283

=0

The option seller bears the cost of the call option as he has the option to reduce the upside potential of the bond, which in turn makes the bond less attractive

So the investor would like to pay less than a option free bond, so the bond issuer shall bear the cost of call option.

PLEASE LIKE THE ANSWER IF YOU FIND IT HELPFUL OR YOU CAN COMMENT IF YOU NEED CLARITY / EXPLANATION ON ANY POINT.

Thanks.

For the calculating value at each node we use backward induction
In this example we start the backward induction technique from the 2.27% node at time period 3
The bond will mature after 1 year so value at that node is
PV of 100 discounted at 2.27% for 1 year :
Value = 100/1.0227 = 97.780
Similarly value at 1.69% , 1.25% & 0.92% can be calculated at time period 3
Now for calculating Value at 1.69% at time period 2 we need to do average of values arrived at 2.27% and 1.69% and then we will calculate Present value of the same.
=((97.804+100.017)/2)/1.0169
= 97.25503
Similarly on these lines the rest of the tree via backward induction can be solved.