callable bond pricing

Copyright © Arkus Financial Services - 2014 Callable Bond Pricing

Luigi Piergallini Date: 09/05/2014

Callable Bond Pricing


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.


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.


Callable Bond Value

YIELD

PR

ICE

Coupon

Va

lue

of

Ca

ll


Let’s take a Bermudan callable bond:

2 year life

Call price 100

Callable until the 2nd coupon

Semi-annual coupon 4% 4%


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


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


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.


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


Modelling the underlying

The most well known one factor models:

Vasicek 𝑑𝑟𝑡 = 𝑘 𝜃 − 𝑟𝑡 𝑑𝑡 + 𝜎𝑑𝑊𝑡

Cox-Ingersoll-ROSS (CIR) 𝑑𝑟𝑡 = 𝑘 𝜃 − 𝑟𝑡 𝑑𝑡 + 𝑟𝑡𝜎𝑑𝑊𝑡

Ho-Lee 𝑑𝑟𝑡 = 𝜃𝑡𝑑𝑡 + 𝜎𝑑𝑊𝑡

Hull White (extended Vasicek) 𝑑𝑟𝑡 = 𝜃𝑡 − 𝛼𝑟𝑡 𝑑𝑡 + 𝜎𝑑𝑊𝑡


Cox-Ingersoll-Ross (CIR)

𝜃, 𝑘, 𝜎 strictly positive constants

𝜃 is the long term mean

𝑘 is the speed at which 𝑟𝑡 reverts back to the long-term mean

𝜎 is the local volatility of short-term interest rates

Properties

Mean reversion

For given positive 𝑟0, the process will never touch zero if 2𝑘𝜃 ≥ 𝜎2

𝑑𝑟𝑡 = 𝑘 𝜃 − 𝑟𝑡 𝑑𝑡 + 𝑟𝑡𝜎𝑑𝑊𝑡


OLS Estimation of CIR parameters

The simulation of the previous equation can be illustrated as:

𝑟𝑡+1 = 𝜃𝑘∆𝑡 + 1 − 𝑘∆𝑡 𝑟𝑡 + 𝜎 𝑟𝑡∆𝑡𝜀𝑡 where 𝜀𝑡~𝑁(0,1)

The sum square of the error (𝜎𝜀𝑡)2 𝑛−1

𝑖=1 must be minimised in terms of 𝑘 and 𝜃 to obtain 𝑘 and 𝜃 such that:

𝑘 ,𝜃 = argmin𝑘,𝜃

(𝜎𝜀𝑡)2

𝑛−1

𝑖=1

= argmin𝑘,𝜃

𝑟𝑡+1 − 𝑟𝑡

𝑟𝑡−

𝑘𝜃∆𝑡

𝑟𝑡+ 𝑘 𝑟𝑡∆𝑡

2

𝑛−1

𝑖=1


Estimated Parameters

𝜎 =1

𝑛 − 2

𝑟𝑡+1 − 𝑟𝑡𝑟𝑡

−𝜃

𝑟𝑡+ 𝑘 𝑟𝑡

2𝑛−1

𝑖=1

The standard deviation, 𝜎 , of the errors is the estimated diffusion parameter:

𝑘 =𝑛2 − 2𝑛 + 1 + 𝑟𝑡+1

𝑛−1𝑖=1

1𝑟𝑡

𝑛−1𝑖=1 − + 𝑟𝑡

𝑛−1𝑖=1

1𝑟𝑡

𝑛−1𝑖=1 − 𝑛 − 1

𝑟𝑡+1𝑟𝑡

𝑛−1𝑖=1

𝑛2 − 2𝑛 + 1 − 𝑟𝑡𝑛−1𝑖=1

1𝑟𝑡

𝑛−1𝑖=1 ∆𝑡

𝜃 =𝑛 − 1 𝑟𝑡+1

𝑛−1𝑖=1 −

𝑟𝑡+1𝑟𝑡

𝑛−1𝑖=1 𝑟𝑡

𝑛−1𝑖=1

𝑛2 − 2𝑛 + 1 + 𝑟𝑡+1𝑛−1𝑖=1

1𝑟𝑡

𝑛−1𝑖=1 − 𝑟𝑡

𝑛−1𝑖=1

1𝑟𝑡

𝑛−1𝑖=1 − (𝑛 − 1)

𝑟𝑡+1𝑟𝑡

𝑛−1𝑖=1


CIR simulations

Estimated Parameters (Euribor 6 months)

𝜎 = 3.37%

𝑘 = 0.9%

𝜃 = 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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