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					                            The Key Role Penalty Played




                                    C. Y. Huang*


Department of Finance, National Sun Yat-sen University, 70 Lien-Hai Rd., Kaohsiung,
       Taiwan (R.O.C). E-mail: zajihuang@gmail.com Tel: 886-937256585


                                  Derek, M. H. Chiang


Department of Money and Banking, National Chengchi University, 64, Sec.2, ZhiNan
            Rd., Wenshan District, Taipei City 11605,Taiwan (R.O.C).


                                      K.T. Wang


Department of Finance, National Sun Yat-sen University, 70 Lien-Hai Rd., Kaohsiung,
                                 Taiwan (R.O.C).


                                      Abstract

     In breach of contract, one has to compensate another for penalty, but the dispute
is easily formed contemporaneously, especially the proportion of penalty. We
consider that contractors must pay for breaking a contract. Here we take an
appropriate option pricing model to examine what is the role penalty played in a
fair-trade and factors influenced on. By way of pricing, we find the penalty is a kind
of premium to execute the right of breach of contract for both contractors. We also
suggest that is important to set a fair proportion of penalty in a contract, because the
consumers usually belong to the disadvantaged minority. In other words, from the
viewpoint of fair dealing, the criteria use to set the proportion penalty of sellers
should be more strictly.


Keywords: Penalty; Breach of Contract; Fair Trades.


*
    Corresponding author.

                                          -1-
JEL code: D11, D18


                                         Introduction

The buyer and seller are both selfish in a trading, for this reason, the contractors will
add penalty when designing a contract to avoid someone will breach it. The penalty
when breaching the contract often results in dispute, however we consider that the
penalty can help the agreement to be fair-traded which the both returns in the date of
contract are same. Herewith we focus on the “rationality and capability of penalty” in
light of simple option pricing model to explain.


Option holder has a right, not an obligation, to excise, and option writer must comply
with an agreement responsibly. The asymmetry of right and obligation result in holder
must pay a premium to compensate the writer. There is analogy between the option
and a trading, the buyer will actively buy good or service in bloom market, but the
seller would like to violate a contract by selling them with a higher or better market
price, be more than exercise price in contract, rather than the strike price that initially
designed. On the contrary, the seller will energetically sell good or service to another
in bear market, however another will breach to buy in market using market price. In
view of the above, it seems as if we consider that the buyer and seller have call and
put respectively.


To price the penalty, firstly we see the final payoffs of both buyer and seller. And then
we use the risk neutral pricing method, originated by Harrison and Kreps (1979) and
Harrison and Pliska (1981), to find the discounted expected return on the date of the
contract. Finally, we will analyze the influence of the parameters in our model and
show their economic intuitions.


This study is organized as follows. Methodology is described in section II. Section III
presents the economic intuitions in our model. Finally, the last section is conclusion.


                                    Analytic Model

In the risk neutral world, S denotes the asset or service price which follows
Geometric Brownian Motion (GBM) with drift and diffusion term proportional to S ,
that is,
(1)            dSt  rSt dt   St dWt
Where


                                             -2-
r : the risk-free interest rate,
 : the volatility of asset or service price S ,
W : the Standard Brownian Motion.

We assume that market is perfect with no tax, risk-free rate and etc., strategy
decisions of both buyer and seller are independent and the option is European type.
Accordingly, whatever buyer executes or breach the contract, the final payoff is
(2)              CT   ST  K 1ST  K  LS , K  ST  Lb    Ls 1ST  K  Ls , K ST  Lb 
                           Lb 1ST  K  LS , K  ST  Lb    0 1ST  K  LS , K ST  Lb 


Where
K : the exercise price in the contract,
ST : the market price in maturity date,
Lb : the penalty of buyer given he breached of contract,
Ls : the penalty of seller given he breached of contract,
T : the maturity date of contract,

1 : a indicator function equals to 1 if condition holds, otherwise is 0.
  



The first part of right hand side shows that buyer can get the profit                                  ST  K    if both

have a mind to execute. Here we see that the contract will be carry out depends on
whether seller feel like to sell even if buyer want to buy. The second part illustrates
when seller does not feel like to meet their agreement, the buyer only receives the

penalty Ls rather than  ST  K  . The third part interprets the buyer pay Lb to the

seller if he decides to breach the contract but seller wants to sell. Finally, the final part
shows that they terminate the contract voluntarily. Similarly, we have the final payoff
of the seller as (3),
(3)              P   K  ST 1ST  K  LS , K  ST  Lb    Lb 1ST  K  LS , K  ST  Lb 
                  T

                        Ls 1ST  K  Ls , K  ST  Lb    0 1ST  K  LS , K  ST  Lb 


Often, the penalty was set to be proportional of strike price and then we can define the
Lb and Ls in the contract as follow,


                                                         Lb   K



                                                             -3-
                                                                            Ls   K


Where
 : the penalty ratio of buyer,
 : the penalty ratio of seller.
Plugging the Lb and Ls into term of (2) and (3), we have
(4)                                  CT   ST  K 1ST 1   K 0, ST 1  K 0   K1ST 1   K   K , ST 1  K 0
                                                  K 1ST 1   K 0, ST 1  K 0
                                                 PT
Here we observe the difference between CT and PT is barely the sign, so we only
use this model to analyze the buyer as below and the analysis of seller just
contrariwise with a negative sign.


Under assumption of independent, the (4) can be rewritten as
(5)                                  CT   ST  K 11  K  ST 1   K    K1ST 1   K 1ST 1  K  
                                                                                                                        
                                            K 1ST 1  K 1ST 1   K 


In (5), we see that (5) will degenerate to only buyer (seller) has option if   
(    ), and both of them have obligation to execute the contract if  ,    .


By Harrison and Kreps and Harrison and Pilska, we can get the price1 on date of
contract, t , is
(6)                                  Ct   St N  d1   1    Ke  r T t  N  d 2     Ke  r T t 
                                                                                           

                                                         
                                               St N d1  1    Ke r T t  N d 2 
                                                                                                
                                              Pt


Where,

d1 
        ln         St
                 1  K       r    T  t  ,
                                    1
                                    2
                                        2


                              T t


d2  d1   T  t ,


d1 
        ln         St
                 1   K      r    T  t  ,
                                    1
                                    2
                                        2


                              T t




1
    The proof shows in the appendix.

                                                                                -4-
d2  d1   T  t ,

N  is the cumulated standard normal distribution.


                        Analytic Results and Economic Intuition

By (6), we see that Pt will be positive if Ct is negative, besides Ct is positive if
 Pt is negative. Herewith buyer and seller will not have equally nonnegative excess
return contemporaneously rather than zero. In the other words, with proper  and  ,
the buyer and seller in the contract both will be on the equal footing, Ct  Pt  0 , on
date of contract.

To catch on the influence of  and  as follow, we can take partial derivative of
(6) with respect to  and  respectively, and then we have
                   Ct
                         K   1  N  d 2  
                             r T t
(7)
                   
                       0
                   Ct
(8)
                   
                        Ke   N d 2
                            r T t
                                          
                       0
In (7) and (8), we see that the discounted expected return of buyer will be increasing
if  is lower and  is higher. This suggests that the discounted expected value of
contract will be higher when less limitations of contract for contractor’s advantage,
and it consists with the economic intuition.


Taking partial derivative of (6) with respect to  , we can obtain the influence of
volatility on (6) given other variables remain constant,
                   Ct Ke r T t  
                                       1    e 2 d2  1    e 2 d2 
                                                   1 2                1 2
(9)                   
                        2                                              
                                0                      0

                         0
         Ct
Here         is negative2, and be different form the case3 of    . This shows that
         




2
    From (9), we have

                                                     -5-
the seller deprives the buyer of deserved profit by paying  K to compensate the
buyer, and then acquire the higher price in spot market on time T .


Continuously, if we feel like to know the effect of asset return, we can take partial
derivative of (6) with respect to r ,
                                                                                            
                 Ct
(10)
                  r
                       T  t  Ke
                                     r T t  
                                                                                 
                                                  N  d 2   N d 2   N  d 2    N d 2 
                                                                                                                   
                                 0             
                                                           0                    ?          
                                                                                             
                                                 Ct
In (10), we see that the likelihood of
                                                  r
                                                       0 will increase if  N  d 2    N d 2 .                       
This show N d 2        and 1  N  d2  will raise if r increase, that is to say seller, not

buyer, enjoy the profit of asset price goes up. This result is quite a few different from
   that buyer can get the profit of asset price arise.

Finally, we can take partial derivative of (6) with respect to T  t  , by the way we can
know the change in the value of Ct changes as time to maturity,




                                 
               1    e  2 d2 
                             1 2


1    e 
          1 d2
              2
           2
                         1d2   0
               1    e 2 2 
               1
                           1   
                 Ct
and then we have       0.
                  
3
  In case of    , we can get the C t is
                            Ct  St N  d1    Ke                       1    Ke                  N  d2 
                                                           r T  t                     r T  t 


Therefore we have
                                       Ct
                                            1    T  t Ke r T t  n  d 2 
                                       
                                           0
                          Ct
                                T  t  Ke  r T t   1    (T  t ) Ke  r T t  N  d 2 
                           r
                                              0                                     0

                                0
                           Ct       1     r T t 
                                                           n  d 2  rKe    a  N  d 2  
                                                                          r T t
                                             Ke
                         T  t  2 T  t
                                                                                                   0
                                                     0

                                   0

                                                                -6-
                                                                                   
                     Ct                          N  d 2    N d 2   rKe  r T t 
(11)                          N  d2   N d2
                   T  t   
                                                
                                                                               
                                                                                    
                                        0                        ?                
                                 1     K  r T t   1   e 12 d22  1   e 12 d22 
                             
                                                          
                                                                                          
                                2 2 T  t 
                                                            0

(11) shows similar result with (10), it shows the seller takes the benefit from the buyer
as whom waits till the maturity date comes. In sum, we believe in our model the right
holding with buyer almost fall away, instead seller get the most profit. This is a
possible reason why the consumer always be considered as an inferiority.
Consequently, to eliminate this asymmetry between buyer and seller for a fair trade,
we can setup the  and  as


                                    
(12)                  , r, T  t     : Pt  EQ  ST   K  0
                           ?
                                 ?   

                                    
(13)                  , r , T  t     : Pt  K  EQ  ST   0
                           ?
                                 ?   

That is why we think that  and  is a significant and adjustable role to make
buyer and seller be on an equal footing.


                                         Conclusion

We take a simple and easy option pricing model to explain the rationality of penalty
designed in the contract, which is a kind of the expense while anyone breaks a
contract. In the other word, the saboteur pay penalty to compensate another. Moreover,
the penalty ratio can reduce the asymmetry between buyer and seller on date of
contract. Continuously, we see the seller is always a superiority of a contract and that
is the reason we consider the buyer to be inferiority. Finally, we consider that we must
control rigorously penalty of the seller.


                                          Reference

Baxter, M. and A. Rennie, 1996. Financial Calculus: A Introduction to Derivative
       Pricing. (Cambridge University Press).


Black, F. and M. Scholes, 1973. The Pricing of Options and Corporate Liabilities,


                                                -7-
    Journal of Political Economy 81, 637-654.
Harrison, J. M. and D. Kreps, 1979. Martingales and Arbitrage in Multiperiod
    Securities Markets, Journal of Economic Theory 20, 381-408.


Harrison, J. M. and S. Pliska, 1981. Martingales and Stochastic Integrals in the
    Theory of Continuous Trading, Stochastic Processes and Their Applications, 11,
    215-260.


Hull, J. C., 2006, Options, Futures, and Other Derivatives, 6th edition. (Prentice Hall).


Appendix

     The proof of (6) is


                   EQ  ST  K 11  K  ST 1   K    e   EQ   K1ST 1   K  
          r T  t                                                    r T t
Ct  e
                                                                                                   
                     EQ   K 1ST 1  K  
         r T  t 
    e
                                                      
                   EQ  ST 11  K  ST 1   K    Ke   EQ 11  K  ST 1   K  
       r T  t                                              r T t
   e
                                                                                                 
                     EQ   K1ST 1   K  
         r T  t 
    e
                                               
                          EQ   K 1ST 1  K  
            r T  t 
     e
                                                      

                                                                                
     St N  d1   Ke  r T t  N  d 2     St N d1  Ke  r T t  N d 2 
                                                                                                
       Ke
                    r T  t 
                                    
                                  N d 2   Ke
                                                            r T  t 
                                                                          N  d 2 

     St N  d1   Ke
     
                                      r T  t 
                                                                                
                                                    N  d 2     St N d1  Ke   N d 2 
                                                                                 r T t
                                                                                                 
                                    
       Ke  r T t  N d 2   Ke  r T t  1  N  d 2  

     St N  d1   1    Ke
     
                                                      r T  t 
                                                                                      
                                                                    N  d 2     St N d1  1    Ke
                                                                                                             r T  t 
                                                                                                                             
                                                                                                                           N d2 
                                                                                                                                
                    r T  t 
      Ke


Q.E.D。



     The proofs of (9)、(10) and (11) show as below, respectively,




                                                                                 -8-
Ct                                                             

     St n  d1 
                  
                       d1   1    Ke r T t  n  d 2 
                                                                
                                                                    d1   T  t                                  
                                                                 
      St n d1      
                   
                         d1  1    Ke  r T t  n d 2
                                                                 
                                                                     d1   T  t                                   
       1    T  t Ke r T t  n  d 2   1    T  t Ke  r T t  n d 2                     
                                                            1                                                          1
       1    T  t Ke                                                      1    T  t Ke
                                              r T  t              1 d2                          r T  t                  1 d2
                                                                          2                                                          2
                                                                 e     2
                                                                                                                            e     2

                                                            2                                                         2
                r T t 
          Ke                 1    e 2 d2  1    e 2 d2 
                                                   1   2                      1   2
                                                                
                2                                               


Q.E.D。



Ct                                                                                                    
     St n  d1   d1   1   T  t  Ke  r T t  N  d 2   1    Ke  r T t  n  d 2   d 2 
 r              r                                                                                     r
                  
      St n d1      
                  r
                         d1  1   T  t  Ke   N d 2
                                                         r T t
                                                                                          
                                         
      1    Ke   n d 2
                       r T t

                                        r
                                                   
                                             d 2   T  t  Ke  
                                                                     r T t




     T  t  Ke
                    r T  t 
                                1    1  N  d 2    1    N d 2   
                                                                                               
       T  t  Ke
                               r T  t 
                                                                 
                                              N  d 2   N d 2   N  d 2                N  d 
                                                                                                        2




Q.E.D。




                                                                                  -9-
   Ct                                                                                       
 T  t 
            St n  d1 
                          T  t 
                                     d1   1    rKe  r T t  N  d 2   St n d1
                                                                                           T  t 
                                                                                                     d1                     
                                                              
              1    Ke  r T t  n  d 2 
                                                           T  t 
                                                                      d 2   1    rKe r T t  N d 2            
              1    Ke  r T t  n d 2           T t   d   arKe
                                                                           2
                                                                                                   r T  t 




                                                              1  a  
            1  a  rK                         N  d2                         n  d2 
                                   r T  t                                   r T  t 
                                                                           K
                                                               2 T t
                                                                1      r T t 
              1    rK  r T t  N d 2                
                                                                 2 T t
                                                                          K            n d2         
              arKe  r T t 

                                   N  d2   N d2   
                                                         1      r T  t 
            rKe
                    r T  t 
                                                         
                                                     2 T t
                                                                  K             1    n  d 2   1    n d 2 
                                                                                                                                  
              rKe         N  d 2    N d 2 
                      r T  t 
                                                                  
                                                  1    
            rKe    N  d 2   N d 2  
                 r T t
                                             2 2 T  t  K   1    e 2 2  1    e 2 2 
                                                               r T t
                                                                       
                                                                       
                                                                              1d2             1d2
                                                                                                    
                                                                                                    
                                                                                             
              rKe  r T t   N  d 2    N d 2 
                                                                  
Q.E.D。




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