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Black-Scholes model is important to calcurate option premium in stock market. And variety of jump diffusion model as time-series of stock price are studied. In this paper, we propose new jumps diffusion model with corelational companies in order to culculate option pricing in a stock market. This model express corelation of companies as directed graph structure have a weight of corelation coeffect among companies. And calcurate option premiums together. Then we exhibit monte- carlo algorithm of proposed model. After simulate this model comparing with standard jump model and chang parameters, we discuss effectiveness of it.

In the early 1970's, Black-Scholes model(Black and Scholes[1973]) is proposed. This model can calculate an option price as market transactions of derivatives. Black-Scholes model express time-series of stock price as geometric brown motion in stochastic differential equation. Option premium is culcurated from exercise price and time duration of option and geometric brown motion under risk-neutral probability. Appearance of Black-Scholes model expanded and grew option market at a rapid pace. For the achievement, Scholes and Marton won the novel prize.\newline But BS model does not represent all aspects of characterristics of a real market. And expansion of BS model is studied and proposed variety of models. Especially time-series of a stock price exhibits phenomenons like a price jump. And in order to modeling it, Jump Diffusion Model(Merton[1975]) using Poison Process to express jump phenomenons is proposed. \newline In this paper, I propose Corelational Jumps Model which have corelation of companies in stock price. A jump phenomeno of one company affect jumps of other corelational companies obeying correlation coefficients among companies. And it can culculate premiums of the companies together. In chapter 3, a dircted graph of corelational companies algorithm expain. Then in chapter 4, we simulate a proposed mode and expain its algorithm.

There are several types of options in the stock market. Eropean call option can not excuse in duration of T and its excusion price is K. Optin premium is calcurated under a risk-neutral probability. Eropean call option preminum is given by

E[x] express expected value of x. And Eropean put option premium is given by

Black-Schole model is given by

where μ present a draft parameter. it is a trend int the stock price. And σ is volatility. r is is the risk-free interest rate.N is gauss distribution.

The Poisson Process present random phenomenons happend as time sequence. It is widely used to model random points in time and space. Poison process is given by

where λ is the arrival intensity. k is a number something happen.

Under the mixed-exponential jump diffusion model (MEM), the dynamics of the asset price St under a risk-neutral measure1 P to be used for option pricing is given by

where r is the risk-free interest rate, σ the volatility, {N(t):t =0・・・} a Poisson process with rate λ, {W(t):t=0・・・} a standard Brownian motion.

Standart Jump Diffusion model occurs jump in one stock market and it does not affect other companies. In correlational Jumps model one jump among companies affects other stock price of a company obeying correlation coefficacies. Therefore equetions are given by

Where random_i is a n_th company. And U is discrete uniform distribution.Output_ij is a correration coefficacy from i company to j. it is from result of algorithm 1.

If download from Github you can use devtools by the commands:

> library(devtools)
> install_github("jirotubuyaki/Jdmbs")

Once the packages are installed, it needs to be made accessible to the current R session by the commands:

> library(Jdmbs)

For online help facilities or the details of a particular command (such as the function hmds) you can type:

> help(package="Jdmbs")

##Method This pakage has three method. And it is excused by:

> premium <- normal_bs(companies,simulation.length=180,monte_carlo=1000,start_price,mu,sigma,K,k,col);

> premium <- jdm_bs(companies,simulation.length=180,monte_carlo=1000,start_price,mu,sigma,K,k,col);

> premium <- jdm_new_bs(data,companies,simulation.length=180,monte_carlo=1000,start_price,mu,sigma,K,k,col);

New algorithm for option price is described and explain how to use. This package can produce a option price with related companies. And several improvements are planed. Please send suggenstion and report bugs to [email protected].

This activity would not have been possible without the support of my family and friends. To my family, thank you for lots of encouragement for me and inspiring me to follow my dreams. I am especially grateful to my parents, who supported me all aspects.