Callable Bond Pricing

Copyright © Arkus Financial Services - 2014
Callable Bond Pricing
Luigi Piergallini
Date: 09/05/2014

Page 2
Callable Bond
call·a·ble bond A Callable Bond is a straight bond embedded with a call of:
European option (single call date)
Bermudan option (several call dates) The issuer can buy back from the bond holders at pre-specified prices on the pre-specified call dates.

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Why Callable Bonds
They are more attractive to borrowers
They are less attractive to lenders
Lenders
Lenders get compensated through higher coupon rates.
In order to tone down call risks with callable bonds, many issuers introduce a call protection period during which a callable bond cannot be called.
Borrowers
Callable bonds give borrowers the option to refinance when interest rates are low.

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Callable Bond Value
YIELD
PRICE
Coupon
Value of Call

Page 5
Let’s take a Bermudan callable bond:
2 year life
Call price 100
Callable until the 2nd coupon
Semi-annual coupon 4%
4%

Page 6
Underlying simulation
Simulate different paths for the underlying (interest rate)
7.09%
9.19%
11.91%
15.44%
6.81%
8.83%
11.44%
6.54%
8.48%
6.28%
0
1
3
2

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Bond Pricing
Resulting in different paths of the bond valuation
99.10046
96.7878
95.75492
96.5451
100.4394
98.716
98.3727
101.0012
99.7719
100.8344
0
1
3
2

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Callable Pricing (1)
Remember:
Call price 100
Callable till the 2nd coupon payment
Lenders
The issuer will call the bond only if the value of the bond is higher than what he needs to pay in calling it.
Borrowers
Callable bonds give borrowers the option to refinance when interest rates are low.

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Callable Pricing (2)
98.86668
96.7878
95.75492
96.5451
99.9552
98.716
98.3727
100
(101.0012>100)
99.7719
100.8344
0
1
3
2
Remember:
Call price 100
Callable till the 2nd coupon payment

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Modelling the underlying
The most well known one factor models:
Vasicek 푑푟푡=푘휃−푟푡푑푡+휎푑푊푡
Cox-Ingersoll-ROSS (CIR) 푑푟푡=푘휃−푟푡푑푡+푟푡휎푑푊푡
Ho-Lee 푑푟푡=휃푡푑푡+휎푑푊푡
Hull White (extended Vasicek) 푑푟푡=휃푡−훼푟푡푑푡+휎푑푊푡

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Cox-Ingersoll-Ross (CIR)
휃, 푘, 휎 strictly positive constants
휃 is the long term mean
푘 is the speed at which 푟푡 reverts back to the long-term mean
휎 is the local volatility of short-term interest rates
Properties
Mean reversion
For given positive 푟0, the process will never touch zero if 2푘휃≥휎2
푑푟푡=푘휃−푟푡푑푡+푟푡휎푑푊푡

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OLS Estimation of CIR parameters
The simulation of the previous equation can be illustrated as:
푟푡+1=휃푘Δ푡+1−푘Δ푡푟푡+휎푟푡Δ푡휀푡
where 휀푡~푁(0,1)
The sum square of the error (휎휀푡)2 푛−1 푖=1 must be minimised in terms of 푘 and 휃 to obtain 푘 and 휃 such that: 푘 ,휃 =argmin 푘,휃 (휎휀푡)2 푛−1 푖=1=argmin 푘,휃 푟푡+1−푟푡 푟푡 − 푘휃Δ푡 푟푡 +푘푟푡Δ푡 2 푛−1 푖=1

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Estimated Parameters
휎 = 1 푛−2 푟푡+1−푟푡 푟푡 − 휃 푟푡 +푘 푟푡 2푛−1 푖=1
The standard deviation, 휎 , of the errors is the estimated diffusion parameter:
푘 = 푛2−2푛+1+ 푟푡+1 푛−1 푖=1 1 푟푡 푛−1 푖=1−+ 푟푡 푛−1 푖=1 1 푟푡 푛−1 푖=1−푛−1 푟푡+1 푟푡 푛−1 푖=1 푛2−2푛+1− 푟푡 푛−1 푖=1 1 푟푡 푛−1 푖=1Δ푡
휃 = 푛−1 푟푡+1 푛−1 푖=1− 푟푡+1 푟푡 푛−1 푖=1 푟푡 푛−1 푖=1 푛2−2푛+1+ 푟푡+1 푛−1 푖=1 1 푟푡 푛−1 푖=1− 푟푡 푛−1 푖=1 1 푟푡 푛−1 푖=1−(푛−1) 푟푡+1 푟푡 푛−1 푖=1

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CIR simulations
Estimated Parameters (Euribor 6 months) 휎 =3.37% 푘 =0.9% 휃 =0.317%
0.00%
0.20%
0.40%
0.60%
0.80%
1.00%
1.20%
1.40%
1.60%
1.80%

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