Document Sample

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 1ST K LS , K ST Lb Ls 1ST K Ls , K ST Lb Lb 1ST K LS , K ST Lb 0 1ST 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 1ST K LS , K ST Lb Lb 1ST K LS , K ST Lb T Ls 1ST K Ls , K ST Lb 0 1ST 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 1ST 1 K 0, ST 1 K 0 K1ST 1 K K , ST 1 K 0 K 1ST 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 11 K ST 1 K K1ST 1 K 1ST 1 K K 1ST 1 K 1ST 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 11 K ST 1 K e EQ K1ST 1 K r T t r T t Ct e EQ K 1ST 1 K r T t e EQ ST 11 K ST 1 K Ke EQ 11 K ST 1 K r T t r T t e EQ K1ST 1 K r T t e EQ K 1ST 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。 - 10 -

DOCUMENT INFO

Shared By:

Categories:

Tags:
High Speed, Pricing of Options, An Introduction to The Mathematics of Financial Derivatives, Martin Baxter, Introduction to Derivatives, option pricing, martingale theory, Andrew Rennie, financial engineering, Salih Neftci

Stats:

views: | 1 |

posted: | 3/24/2011 |

language: | English |

pages: | 10 |

How are you planning on using Docstoc?
BUSINESS
PERSONAL

By registering with docstoc.com you agree to our
privacy policy and
terms of service, and to receive content and offer notifications.

Docstoc is the premier online destination to start and grow small businesses. It hosts the best quality and widest selection of professional documents (over 20 million) and resources including expert videos, articles and productivity tools to make every small business better.

Search or Browse for any specific document or resource you need for your business. Or explore our curated resources for Starting a Business, Growing a Business or for Professional Development.

Feel free to Contact Us with any questions you might have.