1. Pricing Callability. Bonds often come with early redemption options, some quite compli...

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1. Pricing Callability. Bonds often come with early redemption options, some quite compli XXXXXXXXXXcated, for the issuer (”callables”) or the investor (”putables”). Consider a callable bond with two years to maturity remaining, a 15% annual coupon, and a call schedule indicating call prices (strikes) of K0 = XXXXXXXXXXand K1 = XXXXXXXXXXin years t = 0, 1. If the issuer decides to call the bond she has to pay the call price and the coupon on the date the bond is called. Suppose that the evolution of the 1Y spot rate (in %) and the ex-coupon price1 of a 2Y non-callable bond with a 15% annual coupon is described by the lattice below with no further information on the lattice’s type available. Using a standard synthetic replication argument price the callable on an ex-coupon basis. Note that you cannot price the callable directly from the lattice because you do not know what the risk-neutral probabilities are (a safe bet would be the usual q = 1 − q = 1/2 , though).

Kushal · Accepted Answer

1. Callable Bonds – 
A. Coupon Lattice Structure.
 (
R0= 9%
Bond Price = 
Ru
=  
7
%
Coupon payments = 15
 
R
u
= 
 11
%
Coupon payments = 15
Put Price = 0.4674
Ruu
 = 13.01%
Coupon Payments = 15
Rud
= 9%
Coupon Payments =15
R
du
= 9%
Coupon Payments =15
Put Value = 2.84
Rdd
 = 5%
Coupon Payments = 15
)
B. Bond prices at each stage – 
At t=2 , the principal of 100 and 15 coupon payments will be discounted by 11% and 7% each. The probabilities given for the upside and downside scenarios are 50%. Hence, we will get the multiply the coupon payments by 50% each and add them up.
For example,
At t=2, 
For Ruu scenario 115 cashflows, and Rud scenario – 115 cashflows.
Bond price at t=1 ,
Bond price (t=1) = (115*0.5 + 115*0.5 ) / (1+ 11%)  = 103.6
This bond will be not called since its price is less than the call price of 106 at t=1.
For Rdu and Rdd scenarios,
Bond price at t=1 ,
Bond price (t=1) = (115*0.5 + 115*0.5 ) / (1+ 7%)  = 107.47 
This bond will be called since its price is more than the call price of 106 at t=1.
Hence bond price will be changed to 106 at t=1 for the downside scenario.
Bond price at t=0,
Bond price (t=0) = (106*0.5 + 103.6*0.5+15 ) / (1+ 9%)  = 110.1376
This bond will be called since its price is more than the call price of 107 at t=0.
Hence bond price will be changed to 107 at t=0. 
 (
R0= 9%
Bond Price = 
107
Ru
=  7
%
Bond price = 106 (instead of 107.47)
 
Ru
=  11
%
Bond price = 
103.6
Put Price = 0.4674
Ruu
 = 13.01%
Bond price = 100
Rud
= 9%
Bond price = 100
Rdu
= 9%
Bond price = 100
Put Value = 2.84
Rdd
 = 5%
Bond price = 100
)
C. Bond prices without the call option using the interest rate structure.
AT t=1 , the prices would be same as the 103.6 and 107.47. 
At t=0,
Bond price (t=0) = (107.47*0.5 + 103.6*0.5+15 ) / (1+ 9%)  = 120.535
Hence, the value of the call option is 
Call option = Bond price without call option – Bond price with call option = 120.535 – 107 = 13.535
D. Callable bonds have its pros and cons – 
Pros – For the issuer it makes sense embed the bonds with options, in the environment where the interest rate structure is very volatile and it is expected that the rate sin the future will fall. If the interest rates fall,

1. Pricing Callability. Bonds often come with early redemption options, some quite compli XXXXXXXXXXcated, for the issuer (”callables”) or the investor (”putables”). Consider a callable bond with two...

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