EUROPEAN CALL OPTION APPLICATION IN INCOMPLETE MARKET-ANALYSIS AND DEVELOPMENT

Option is derivative instrument that have investmen t benefit and provide return for the writer and the holder. Option price determination is affected by risk fact or. However in Black-Scholes model option price is determined without arbitrage risk affection so it i s impossible to take return. In this study, option price formula is constructed to be more represent the con diti of financial market using incomplete market concept where financial asset, that is traded, is a ffected by arbitrage risk so it is possible for mar ket participants to take return. European call option i s defined by Esscher Transform method and option pr ice formula is determined by changing its form to linea r approximation. The result from this study is opti n price formula with linear approximation has some pr ivileges. That is easy to be applied in computation  process, more representatives in getting risk indic ation in the financial market and can predict optio n price more accurately. Linear approximation formula is ap plied in the program that can be used by option wri ter or holder and is equipped with export data feature that can be possible for further research developme nt.


INTRODUCTION
states that: "Option is defined as a contract between two parties (issuer and the holder of the option) which gives the publisher the right but not the obligation to the holder of an option to buy (call option) or sell (put option) a stock with an agreed price in the future".
Option is a hedging instrument in investing. This instrument can be used in trading strategies, for example, to avoid the collapse of stock prices, shareholders may issue a call option so that the owner of the option can exercise his right to buy shares at the price stated in the option (Soesanto and Kaudin, 2008). Therefore, to ensure the benefits of publishing option, we would require the model to calculate the estimated value of an option. The most classic models and the most popular type calculate European option pricing is the Black-Scholes models, where the yield (return) of the assets sold by the end of the period assumed to be normally distributed (Gerber and Landry, 1997). Soesanto and Kaudin (2008) state that the assumptions used in the Black-Scholes model is not realistic and its validity is questionable. This model assumes the absence of risk factors in arbitrage (Floroiu and Pelsser, 2012). It is certainly not possible in a real situation. However, the Black-Scholes option pricing model is still often used as a reference for publishers to determine the option price and the option holder to determine whether to buy or sell options issued by publisher (Andriyanto, 2009).
A condition that exists in trading stocks is grouped into two models; those are complete markets and incomplete markets. In Complete market, market players can make the exchange of goods or assets directly or indirectly in the absence of transaction costs that do not allow for arbitration even without risk arbitrage, where it is impossible in incomplete. Incomplete market is more complicated than complete market, where this model consists of uncertainty sources (Verchenko, 2010). From the perspective of theory, incomplete market is a complex learning (Staum, 2007), because market circumstances in the real world that is not affected by many factors remain that cannot be explained with certainty. Some experts have also proved that the state of Science Publications JCS the financial markets is incomplete market as described Staum (2007). The complex scope of incomplete markets does not mean that the model cannot be calculated, many experts have examined that model using assumptions, which are almost close to reality.
In determining the price of the European call option, the option price will be in accordance with the market situation in which the smaller the exercise price the greater call option price and if the greater calculated risk the smaller the call option price. So in this study, call option price will be calculated with linear approximation so that the calculation of the European call option price almost resembles the situation in the market, where the model will be derived by the method of Esscher Transform. Esscher Transform is chosen because based on the conclusions of Ruban et al. (2010) which states that the method of Esscher Transform is a method for calculating the price of an asset that has risk. Yao (2001) also states that the method of Esscher Transform provides an option price calculation model, which is efficient and consistent.
Calculation of option prices will be very useful for investors and issuers to determine the appropriate decisions in traded options. Authors designed to restrict user login system that can use this application. R language will be used to calculate the statistical formula used by the European call option pricing model. This system is expected to help stock investors who want to hedge with options traded and perform calculations efficiently.

Encountered Problems and Scopes
• Research for the European option price calculation models mostly performed on the complete market based on the Black-Scholes method and research on models of incomplete markets is still fewer • There is still no complete construction of the European call option price calculation with linear approximation in incomplete market models • Mostly the calculation formula of European call option price is very complex, so that it is difficult to calculate manually. According to Gerber and Landry (1997) it is difficult to calculate European call option price with the exact solution for incomplete markets when skewness close to 0 • The absence of a system created to facilitate the computation Will limit the scope of this research so that the discussion can be more focused and the purpose of writing can be achieved. The scopes include: • European call option price is assumed to be a stochastic process with stationary and independent increments and the risk -free interest rate is constant in accordance with stated in (Gerber and Landry, 1997) • According to Mcleod et al. (2012)

RESEARCH METHODS
Research methods are the stages that must be set before doing the research, so that research can be done directionally, clearly and efficiently. In practice, the process of research and program design depicted in the diagram Fig. 1.

Initial Assumptions
This research will be used several assumptions to limit the scope of the study. Let S(t) is a non-dividend stock price in time t. There is a stochastic process X(t), where X(t) is a process which has stationary increments and independent increments, then the final value of the option at the time 0 defined as: Call option is defined as payoff function ∏(s)≥0 with maturity date τ. At time τ, option holder will get ∏(S(τ)). In this research, it is assumed that r is constant risk-free force of interest. So the option prices at time t are defined as: * states that the expected value is taken based on equivalent martingale measure, where Equation (1) according to the observations of the stock price is: To simplify the calculations, then we will use τ = 1 and calculate the price of the option at time t = 0 so that Equation (2) can be simplified to: t) ] is a moment generating function for X(t), that is a condition where the expected value and volatility have never changed that satisfied: where M(z) = M(z,1) Equation (4 and 5): In classic model, {X(t)} is Wiener process, where for t > 0, random variable of X(t) has normal distribution with mean µt, variance σ 2 t and γ = 0. In this research it will be focused if γ ≠ 0.
To get linear approximation solution, it will be assumed that {Y(t)} is a process that has stationer and independent increment with: Science Publications

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Construct with three-parameter-family that is defined as Equation (7): With Equation (8): So moment generating function of X(t) is Equation (9):

Exact Solution
Exact solution for a European call option prices in incomplete markets will be constructed with the model developed by Gerber and Shiu (1994), using Gamma process model shifted and shifted inverse Gaussian process models. In the two models is defined X(t) and Z(t) is stochastic process that satisfy Equation (10): Then defined moment generating function of X (1) Equation (11): Price of an option is defined as the expectation of the discounted payoff, where the expectation value is taken by Esscher transforms with parameter h, where the value h = h* and z = 1 is determined, so Equation (3) is satisfied. The we will get Equation (12):

Shifted Gamma Process Exact Solution
In this model Z(t) is defined as Equation (10), assumed as a Gamma process with parameters α and β. Then the moment generating function of X (1) based on shifted Gamma process model Equation (13): So that (5) is satisfied, the value of α, β and c are given as follow: (11), (12) and (13) we will get Equation (14): So European call option with shifted Gamma process model is Equation (15): where, S(0) = non dividend stock price at time 0, K = exercise price, k = In [k/S(0)], r = constant risk free force of interest, G(.) = cumulative Gamma distribution function,

Shifted Inverse Gaussisan Process Exact Solution
In this model Z(t) is defined as Equation (10), assumed as a Inverse Gaussian process with parameter a and b. Then the moment generating function of X(1) based on Shifted Inverse Gaussian process is Equation (16): So that (5) is satisfied, then the value of a,b and c are given as follow: From (11), (12) and (16) we will get Equation (17): So that European call option with Shifted Inverse Gaussian process model is Equation (18): Science Publications

LINEAR APPROXIMATION SOLUTION
An equivalent martingale probability measure cannot be proved definitely applies in incomplete market models. But by Gerber and Shiu (1994), to get a unique answer, equivalent martingale measure will be limited to the method of Esscher transforms. By Esscher transforms with parameter h, X (t) is defined as a stochastic process with the moment generating function of X(1) is defined as follows Equation (19): Price of an option is defined as the expectation of the discounted payoff, where the expectation value is taken by Esscher transforms with parameter h, where the value h = h * is determined so that Equation (3) are satisfied, then it will be obtained Equation (20): The process X(t) is assumed with Equation (7) where k and λ as in (8). By Equation (9) and (19), cumulant generating function of X(1) on Esscher transforms with parameter h Equation (21): Substituted (6) in (22), then we will get Equation (22) Substituted k and λ from (8) in (22), then we will get Equation (23) h * will be assumed as polynomial with variable k, that is h * = a + bk +…. This form will be substituted in (21) To make the above equation to be 0, then the coefficient of each variable k will be made 0. From the constants will be obtained from the following Equation (25) By changing h within (23), then we will get expansion of cumulant generating function of X(1) based on martingale measure Equation (27) Science Publications

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Assumed that P(z) an equation with a variable that contains some parts of the Equation (29) are defined as follows Equation (30) Within (30) a will be substituted with (25), then we will get P(z) as follow Equation (31): Then (29) can be simplified and become Equation (32): From cumulant generating function in (32), we will get expansion of moment generating function of X(1) Equation (33) It can be seen in (35) where, X = σZ + µ with mean µ and variance σ 2 Equation (37): is standard normal probability density function φ(.), so that (37) can be simplified and become Equation (38): Multiplied with z, then we will get Equation (39): Within equation above, there is ze zx , that is derivative form of e zx . By using partial integration, we will get: By repeating it, it can be obtained a series as follow Equation (40) where (D) n indicate the n-th derivative. Based on (32) probability density function X(1) can be stated as follow Equation (41): Where Equation (42): Science Publications

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And Equation (43): By changing exact distribution of X(1) with the result that was gotten from linear approximation, then the formula of European call option as follow Equation (44): where, ,f x γ = σ and f 1 (x) as (41) and (42).
Based on definition of payoff function, to compute European call option prices, ∏(S(0)e x ) will be substituted by S(0)e x -K where K is exercise price. So that the formula of European call option is Equation (45): where, By using the integral calculation and probability density function of normal distribution, first term of (45) can be decomposed into Equation (46): By using the method of partial integral, the above formula can be transformed into recursive function as follows Equation (48) In this recursive function, I 0 is described as follow Equation (49): By applying the recursive function (47) , is a formula of the Black-Scholes model. Then the last term of the European call option pricing formula with linear approximation method is the rate of change (Fig. 5-6) between European call option pricing formula Black-Scholes model, where γ = 0 and the European call option pricing formula with linear approximation method, where γ ≠ 0. It is found that "the Constant Elasticity of Variance (CEV) model does not offer a correct description of equity prices", Based on that observations together with the renowned contribution of stochastic volatility formulation to option pricing, it seems to be required to have a hybrid structure of local and stochastic volatility (Choi et al., 2013).

Use Case Diagram
In Fig. 2 depicted that the admin can do some things that begin with login. After going through the login process, admin can see the data of users, changing passwords, adding to new user, changing the data of user, delete data of user, see the instructions for using the application, see the information of application and end use application with logout process. Here is the use case diagram of developed application.
In Fig. 3 depicted that the user can do some things that begin with login. After going through the login process, a new user can perform calculations, change the password, see the instructions, see the information of the application, see the glossary of terms and end use application with logout process. Based on the experiments, the application program runs accurately with GUI as shown in Fig. 4. After the user performs the calculation process, a new user could see tables and graphs calculations. Users can also export tables to Microsoft Office Excel application after the user see the results table.

Simulation
Simulations were performed in Table 1-4 using the applications that have been made. The simulation results are shown in tabular form as follows.
By using original data obtained from Microsoft Corporation (MSFT), note that the price of MSFT stock on the market on April 23, 2013 were $ 30.6 with a mean value and the volatility of the stock return data in 2010-2012 is 0.067535 and 0. Interest rates observed on April 23 th , 2013 is 0.25%. Maturity date of the call option is 1 year. Skewness value used is 0.001. Differences in the calculation of the call option price call option prices from finance.yahoo.com as follows.
A value of 0 in the model shifted inverse Gaussian suggests that this model is hard to quantify as proposed Gerber and Landry (1997).