The paper analyses coupon bonds linked to variable interest rate in a contingent claim approach such that it can be decomposed in elementary options on interest rate and options to default. It is considered the case of continuous arithmetic average of interest rate in a simple capitalization to value the variable coupon paid by the bonds at maturity. The paper determines the expected interest rate on the bonds and the risk spread due to the default risk.Contingent claim, Asian option, Stochastic continuous process

Giandomenico, Rossano

English

Research Papers in Economics

 
Graduate at University of Bologna (Italy)                    
e-mail: rossano1976@libero.it 
1
Valuing Coupon Bonds Linked 
to Variable Interest Rate 
 
 
 
Rossano Giandomenico                                          
 
 
 
Abstract: The paper analyses coupon bonds linked to variable interest rate in a contingent claim approach such that it 
can  be  decomposed  in  elementary  options  on  interest  rate  and  options  to  default.  It  is  considered  the  case  of 
continuous arithmetic average of interest rate in a simple capitalization to value the variable coupon paid by the 
bonds at maturity. The paper determines the expected interest rate on the bonds and the risk spread due to the default 
risk. 
 
 
 
The model and its assumptions 
 
Traditionally a coupon bond linked to variable interest rate is seen by the management as a bond 
plain vanilla. Indeed, it can be decomposed in elementary options on interest rate such that it is 
a  structured  product.  We  will  show  how  a  coupon  linked  to  variable  interest  rate  is  a 
combination of Call option and Put option. The value of the coupon is computed as follows: 
 
                                                                                        T 
RT  =  [1/(T − t)]  ∫ rt dt 
                                                                                      t 
 
BT  =  RT (T − t) Face Value 
 
 
Where  RT  denotes the continuous arithmetic average of interest rate  rt and  BT  denotes the 
coupon paid by the bonds at maturity. Now we see how to decompose the pay-off of the bonds: 
 
 
RT   =  rt  +  max [ RT  − rt , 0 ]  −  max [ rt − RT , 0 ] 
 
 
The third and the fourth term are respectively the pay-off of a Call option on interest rate and a 
Put option on interest rate. Now we denote the expectation of the pay-off as follows: 
 
C(T)  =  E{max [ RT  − rt , 0 ]} 
 
P(T)  =  E{max [ rt  − RT , 0]} 
 
Along Zhang(1998) and Brigo,Mercurio(2006), we have the following: 
 
C(T)  =  F(rt) k q N[d2]  −  rt N[d1] 
 
P(T)  =  rt N[-d1]  −  F(rt) k q N[-d2] 
 
  
  2
Where: 
N[…]  denotes the cumulative normal distribution 
 
 
ln[kF(rt)/ rt]  −  ½ s
2
F (T – t)/2 
                                d1   = 
sFÖ(T – t)/3 
 
 
d2  =  d1  + sFÖ(T – t)/3 
 
 
k  = 1 + (1/24)( – ½ s
2
F)
2 (T – t)
2  + (1/576) ( – ½ s
2
F)
4 (T – t)
4 
 
q  =  exp[– s
2
F (T – t)/12]   
 
 
F(rt) denotes the future on the interest rate  rt 
 
The dynamic of the future  F(rt)  is given by the following stochastic continuous process: 
 
 
dF(rt) / F(rt)  =   sF  dWr 
 
 
sF    denotes the instantaneous volatility of the future F(rt)   
 
dWr  denotes a standard Wiener process capturing the interest rate risk 
 
Now the price of the options at initial time is given by: 
 
C(t)  =  C(T) D(t,T)   
 
P(t)  =  P(T) D(t,T)   
 
We use the price of a default free zero coupon bond D(t,T) with the same maturity of the coupon 
bonds as discount factor. Hence, the expected interest rate  R
e on the coupon is given by: 
 
R
e  =  rt  +  C(T)  –  P(T)  +  Risk Spread  
 
Thus, the expected value of the coupon  B
e is: 
  
B
e  =  R
e (T − t) Face Value 
 
Whereas, the expected total value of the debts  L* is given by: 
 
L* = (Face Value  +  B
e) Number Bonds 
 
Now, along Black-Scholes(1973) and Merton(1974), we can value the debts by using the option 
to default. We can see how to apply the default option by using the pay-off. Therefore, we have 
the following final pay-off for the coupon bonds, where we denote with Lt the initial value of 
the debt and with  At  the initial value of assets: 
 
LT  =  min [ L*, AT ] 
 
  
  3
Where: 
LT  –  L*  =  min [ AT  – L* , 0 ] 
 
 
At this point, we can note that this is the same final pay-off of a short position on a Put option  
written on the company value  At  and with exercise price  L*. Thus, we have: 
 
LT  –  L*  =   - max [ L* - AT , 0 ] 
 
This result is intuitive; it reflects the option of stakeholders to walk away if things go wrong. 
Straightforward, the initial value of the debt is: 
 
Lt  =  L* D(t,T)  -  P(At , L*, T – t) 
 
We denote with P(…) the value of European Put option on the firm’s underlying asset At , 
maturing at time  T  and with exercise price  L*. 
Now we assume that the dynamic of assets is given by the following stochastic continuous 
process: 
dAt  / At  =  µt dt  +  sA dWA 
 
 
µt   denotes the drift of the process 
 
dWA  denotes a standard Wiener process capturing the assets-risk  
 
sA     denotes the instantaneous volatility of assets 
 
 
Thus, we have the following price for the Put option by using  D(t,T) as numeraire: 
 
 
P(At , L*, T – t)   =   D(t,T) L* N[h2]  –  At N[h1] 
 
Where: 
N[…]  denotes the cumulative normal distribution 
 
 
ln(L*D(t,T) / At )  +  ½ s
2
(t,T) (T – t) 
                          h2    = 
s(t,T)Ö(T – t) 
 
 
h1   =   h2  –  s(t,T)Ö(T – t) 
 
 
                                                                  T 
s
2
(t,T)   =   [1 / (T − t)]  ∫  sA
2  +  sD
2  –  2 r(A,D) sA(t) sD
 dt
 
                                                                 t 
 
 
sD denotes the instantaneous volatility of the discount factor 
 
                             r(A,D)   represents the correlation between the total value of assets                         
                                            and the discount factor 
  
  4
Conclusion 
 
We have showed how coupon bonds can be decomposed in elementary options on interest rate 
and  options  to  default.  Obviously,  if  the  interest  rate  paid  by  the  coupon  is  not  computed 
continuously the options on interest rate become European options. We have to note that for 
sake of simplicity we assumed that the coupon will be paid just at maturity of the bonds but it is 
possible to extend the model to multiple coupon paid during the life of the bonds. In fact, it can 
be seen as multiple debts model developed in Giandomenico(2006), the basic idea is that the 
coupons are seen as debts maturing before the time of maturity, straightforward, the default 
options become an option on option. We have to observe that the expected interest rate of the 
coupon is determined essentially by the future rate. Thus, the formula permits to calibrate the 
model to the market expectations. Indeed, some problems can arise in the use of the model; for 
instance, it is possible to have a future rate just for short maturity but in this case we can use the 
interest rate swap. Moreover, it is possible to don’t have a future rate on the interest-rate; in this 
case, along Heath-Jarrow-Morton(1992) it is possible to extract the forward rate implied by the 
yield curve straightforward from the interest rate tem structure. In fact, they take the observed 
yield curve as initial condition for the forward rate curve, they assume that the forward rate 
curve  reflects  the  expectation  of  the  market  on  the  future  interest-rates  such  that  to  avoid 
arbitrage opportunity it determines the yield curve. We definitely conclude by noting that the 
model can be extended to price various kinds of coupon bonds from default free bonds, where 
essentially we don’t have the options to default, to structured bonds. 
 
 
 
 
References 
 
 
Bjòrk, T. :  Arbitrage Theory in Continuous Time  
 Oxford University Press, 1998 
 
Black, F., Scholes, M. : The Pricing of Options and Corporate Liabilities  
 Journal of Political Economy, pag. 637-659, 1973 
 
Brigo, D.,Mercurio, F. : Interest Rate Model  
 Springer Finance, 2006 
 
Giandomenico, R. :    Asset Liability Management in Insurance Company 
Social Science Research Network, 2006 
 
Heath, D., Jarrow, R., Morton. A. :   Bond Pricing and the Term Structure of Interest Rates: a New 
Methodology for Contingent Claims Valuation 
Econometrica, 60(1), pag. 77-105, 1992 
 
Merton, R. :  On the Pricing of Corporate Debt: The Risk Structure of Interest Rate  
 The Journal of Finance, Vol. 29, pag. 449-470, 1974 
 
Zhang, P. :  Exotic Options  
  World Scientific, 1998 

Valuing Coupon Bond Linked to Variable Interest Rate

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