Application of Generalized Binomial Distribution Model for Option pricing of Distribution Model for pricing.”

In this work, the Generalized Binomial Distribution (GBD) combined with some basic financial concepts is applied to generate a model for determining the prices of a European call and put options. To demonstrate the behavior of the option prices (call and put) with respect to variables, some numerical examples and graphical illustration have been given in a concrete setting to illustrate the application of the obtained result of the study. It was observed that when there is an increase in strike prices, it leads to decrease in calls option price 𝐶𝐶 (0) and increase in puts option price 𝑃𝑃 (0), . Decrease in interest rate leads to decrease in calls option price 𝑃𝑃 (0) , and increase in puts option price 𝑃𝑃 (0) , and decrease in expiration date leads to decrease in calls option price 𝐶𝐶 (0) and decrease in puts option price 𝑃𝑃 (0) . It was also found that the problem of option price can be approached using Generalized Binomial Distribution (GBD) associated with finance terms.


Introduction
This paper focuses on a particular type of derivative security known as an option .A contract which gives a buyer the right but obligation to buy or sell an underlying asset or instrument at a specified strike price on a specified date is called an option. To determine its value at any given point in time; one would like to know the value at the time the option is created before the future behavior of the underlying assets is known. Determining an option value is commonly called option pricing. It is also known that both Black-Scholes, CRR model and Binomial model can be used to determine an option's value under a certain conditions on their parameters. If the conditions on the parameters of the generalized Binomial distribution are satisfy, then the generalized Binomial distribution can be combined with financial terms to determine the option value .For example: Chandral et al [1]  Cheng -few lee et al [2] showed how the Binomial distribution is combined with some basic finance concepts to generate a model for determining the price of stock option to be of the form;  where is the interest rate, n is the number of years for the option to expire, [0, − − ] is the payoff value and = 0,1, 2, … .
With a little additional effort Cox et al [3] gave a complete formula in more convenient way to be of the form below for all ≥ max  (4) where is the risk free rate and 1 − are the neutral probabilities, is the rate at which the stock prices go up and is the rate at which the stock prices go down, is the strike price and is a positive integer.
It can be seen clearly in equation (3) that the parameters satisfy the following conditions I.
If the above conditions on the parameters of Binomial model are also satisfied by the generalized Binomial distribution then the generalized Binomial can be combined with financial terms to determine the call price of an option.
In this paper we use a generalized Binomial distribution together with financial terms to evaluate and monitor the behavior of a call and put option with respect to variables in comparison with Cox et al [3].
The proposed model is of the form

Method
The tools for giving the result are the generalized Binomial distribution, Dwass identity with financial terms and Wealth Equation.
The Generalized Binomial distribution in this study was first presented by Dwass (1979). It is a discrete distribution that depends on four parameters ̂, � , N and , where A and B are positive, N is a positive integer and is an arbitrary real number, satisfying ( − 1) ≤̂+ � . And Tereapabolan [1] gave Dwass identity of the form Let X be the generalized Binomial random variable. Then following Terepabolan [1], its probability function is of the form Tereapabolan and Wongkasem [10] pointed out the three special cases of the distribution in (6) 1ˆˆ.
. 1ˆ, For Wealth Equation Stockbridge [7] introduced a powerful and general equation for replicating portfolios with the following assumptions: i. The initial values of the stock is (0) ( (0) is the stock price at t=0). ii. At the end of the period, the prices is either going up or down with factors and that is, (0) with where and are the factors of going up and down respectively. iii. The movement can also be traced from a view point of tossing a die, which results to a head and tail. If it results to a head at a time = 1, we have 1 ( ) = (0) and if it results to a tail at a time = 1, we have 1 ( ) = (0) . iv. One dollar invested in the money market at time zero will yield 1 + dollar at time one, where is the interest rate. Conversely one dollar borrowed from the money market at time zero will result in a debt of 1 + at time one. v. The price either increases, by > 1 or will decrease by < 1. vi. The price of an option is dependent on the following variables: a.
where 0 , 0 are the number of shares of the stock and the unit of the bond respectively.
Then the following holds.
Let be a derivative security paying off at = 2. We deduce the following equation called Wealth Equation

Main Result
The following theorem present a generalized Binomial distribution model for option pricing in term of = 0 and = Theorem 3.1: For = ( 0 , 0 ) ∈ ℝ 2 , where 0 is the number of shares of stock and 0 is the unit of the bond at time = 0, then there exist value of 0 and 0 such that the wealth of the portfolio at time = 0 and = , is And for ∈ {0} equation ( ) hold and also for ∈ ℕ equation ( ) hold Here ( ) denote the replicating portfolio at t = 0, and ( ) denote the replicating portfolio at maturity, for = 1,2,3 ….
Solving the equations (13) smultaneously, we obtain And substituting 0 into equation (13) which is the number of shares of stock at = 1, we further obtain which gives the unit of the bond at = 1.

American Journal of Applied Mathematics and Statistics
Since no arbitrage principle holds, it implies (0) = (0) by lemma 2.1. Thus Now we are interested in the case where there is more than one period for the option to expire and = 2 for the call to be exercised. After one period, the stock price can either be (0) or (0) between the first and second periods.
The stock price can once again go up by or down by , so the possible prices of the stock for the two periods (0) . We can also trace the movement of the stock price from = 1 to = 2 from the perspective of tossing a coin, and the outcome of the coin toss determines the price of the stock at = 1.
We assume the coin toss need not to be fair which implies the probability of head need not to be half .We assume only that the probability of head, which we take to be ˆ0Â

A B
> + and the probability of getting a tail take to Now if we repeatedly toss the coin and whenever we get the head the stock price moves up by the factor whereas whenever we get a tail, the stock price moves down by a factor . When Where (0) is the unknown call price. Hence = 2 ( ), = 2 ( ) and = 2 ( ) are payoff call at expiration = 2. Hence can be determined by defining = 1 ( ) and = 1 ( ), with one period to expiration when the stock price is either (0) = 1 ( ) or (0) = 1 ( ). Cox et al [3] and Chandra et al [6] gave expression for respectively as Now solving also equation ( 25) (26) we obtain the value of 1 and 1 ( ) = as where are the payoff of the stock at = 2. Using the Dwass identity to generalized the payoff values of the stock as follows By Binomial theory expansion, we have that

A B + and 1ˆÂ
A B − + used in this paper is in extension with ̂ � used in Cox et al [4] and Chandral et al [2], and satisfies the property of a probability which implies that 1 1.
gives the chance of the up and down movement of the stock and call prices. The European put option follows exactly the same derivation as the European call option, by induction method we obtain ( ) with the following pay-off values ( ) ( )

Numerical Illustration
The following illustrative examples are used to validate the theoretical results Given also the payoff to be of the form . , then 2 (2) = = 44, 2 (1) = = 0, and 2 (0) = = 0. Using the above information, we obtain and values are the possible prices of the option before the expiration .
Using the same information to find the price of European call and put option as follows Using the model for call option which is as given as .ˆn  .ˆn        Table 2 above show that decrease in interest rate leads to decrease in calls price, and increase in puts price. Case 2: Varying the Strick price: Keeping the following variables constant, = 1.2, = 0.8, = 2, (0) = 100, = + 1 = 10% + 1.   Table 3 Show that increase in strike prices will leads to decrease in calls price and increase in puts price. Case 3: Varying the Expiration date: Keeping also the following variables constant = 1.2, = 0.8, = 100, (0) = 100, = + 1 = 10% + 1.    Table 4 show that decrease in expiration date leads to decrease in calls price and a slight decrease in puts price.     Table 5 show that increase in stock price leads to increase in calls price and decrease in puts price.

Discussion of Results
It is found that the problem of option price can be approached using generalized Binomial distribution associating it with finance terms which gives the same numerical results with Chandral et al [2] using the same information. Table 1, It is clear that when the call option is in-themoney implies � (0)1 > 1 � and � (0)3 > 3 � the call option gets higher value, when the put option is in-themoney, implies � (0)1 < 1 � and � (0)3 < 3 � get higher value.When the call option is out-of -the money, implies � (0)1 < 1 � and ( 3 < 3 ) it loses value, put option is out -of -the money, implies � (0)1 > 1 � and � (0)3 > 3 � also loses value. This is in agreement with Adam [1] options that are in-the-money have a higher value compared to options that are out-of-the money. Figure 1 and Table 2, show that increase in interest rate leads to increase in calls price and decrease in puts price .It is also observed that as the interest rate tends to zero, there will be a point of intersection of the prices, which will make both call and put of equal price. Which implies (0) ≥ (0) .This agrees with Adam [1] when interest rate rise, a call option value will also rise and put option value will fall. For Figure 2 and Table 3; when there is increase in stock price, call price decreases and put price will increase. It is also observed that as the strike price keeps increasing, there will be an equal price thereby having a point of intersection of prices. Chandral et al [2] have it that (0) is a non increasing and (0) is a non-decreasing function of . Clearly from Table 3 and Figure 2 (0) ≥ (0) . Figure 3 and Table 4, shows that decrease in expiration will lead to decrease in both calls and puts price. This is in agreement with Nyustern [7] both calls and puts become more valuable as the time to expiration increase and loses more value as time decreases. It is observed in Figure 3 that the prices will always be in a parallel price form, meaning there will be no point of intersection of price. Implies (0) > (0) . Figure 4 and Table 5 show that increase in stock prices leads to increase in calls price and decrease in puts price. From Figure 4, it is clear that when there is increase in stock prices there will be an equal price of call and put. (point of intersection), such that (0) = (0) . In general (0) ≥ (0) . Nystern [7] an increase in the asset will increase the alue of the calls, puts on the other hand, becomes less valuable as the value of the asset increases. which agrees with the study.

Conclusion
Whenever stock price movement is confirmed to be discrete, in movement, the price of the option can be evaluated using generalized Binomial distribution (GBD). And the behaviour of the price of an option (call and put) is influence and dependent on the following.
i. The strike price K ii. The expire time T iii. The risk free rate r iv. The underlying price (0)