The Asian Option Pricing when Discrete Dividends Follow a Markov-Modulated Model

This paper is concerned with the pricing problem of the discrete arithmetic average Asian call option while the discrete dividends follow geometric Brownian motion. The volatility of the dividends model depends on the Markov-Modulated process. The binomial tree method, in which a more accurate factor has been used, is applied to solve the corresponding pricing problem. Finally, a numerical example with simulations is presented to demonstrate the effectiveness of the proposed method.


Introduction
The option is a contract that gives the owner a right to purchase or sell a certain amount of asset (the underlying asset) at the agreed price (the strike price) within the prescribed time limit, see [1] [2] [3]. It is a financial instrument based on the futures, which gives the owner the right without the obligation. Asian options are path-dependent securities whose payoffs depend on the average of the underlying asset price during the life of the option, see [4] [5] [6] [7]. The the path-dependent option under the framework of the binomial tree by introducing the state variables. Rogers and Shi [14] calculated the Asian options with finite difference method. Hishida and Yasutomi [15] revealed the relationship between European Options and Asian Options, and obtained the approximate price of the Asian option based on the relationship. Boyle and Potapchik [16] provided a summary of the different methods of pricing Asian options and gave the approaches for computing price sensitivities, the methods discussed include Monte Carlo simulation, finite difference approach and various quasi analytical approaches and approximations.
The use of binomial tree model in option pricing has been very popular since the appearance of the pioneering work by Cox, et al. [17] and Rendleman and Bartter [18]. The main advantage of the method is the ease of implementation.
The first binomial tree model for pricing Asian option was proposed by Hull and White [13] in 1993. Based on the binomial trees models, considerable research results have been reported on the Asian option pricing problem. For example, Klassen [19] proposed a modified binomial approach the smallest possible number of average values at each node based on the Hull and White model.
Massimo, et al. [20] associated a set of representative averages chosen among all the effective averages realized at each node of the tree, and then use backward recursion and linear interpolation to compute the option price. Dai, et al. [21] proposed a more representative average prices in terms of the function of the arithmetic average price. Hsu and Lyuu [22] proposed a quadratic-time convergent binomial method based on the Lagrange multiplier to choose the number of states for each node of a tree. Kolkiewicz [23] proposed a method of hedging path-dependent options in a discrete-time setup under the Black-Scholes model.
In finance, most scholars usually model the stock prices directly, and most of them do not consider the dividend income from holding the stock, this practice contradicts with the fact that the stock pays dividends. Even if the payment of dividends is considered, the dividends are basically considered to be distributed continuously, which contradicts with the fact that dividends are distributed in discrete form. Arbitrage pricing theory (APT) tells us that the price of a stock in a company should be equal to the present value of the future dividend payments.
From this point of view, it is more appropriate to model the dividend process according to the arbitrage pricing theory such that the stock price process becomes an evolutionary result of the dividend process.
Driffill, et al. [24] considered the American option pricing while the dividends  [29] presented a study of American floating strike lookback options written on dividend-paying assets.
In the above references, both the dividend model and the binomial tree model have attracted considerable research interest, but few people combine these two models together to consider the pricing problem of discrete arithmetic average Asian option, moreover, the release of discrete dividends with binomial tree method has also not been investigated. Based on the above analysis, this paper considers the important impact of the market status, volatility with the dividend process and market risk. With the help of the arbitrage pricing theory and the dividend discount model, in this paper, a model with the Markov-modulated dividend process is proposed, and the binomial tree method is used to discuss the pricing of the discrete arithmetic average Asian option under the dividend model. Markov chain is employed to describe the changing rule of the stochastic volatility of the dividend model. Different from most of the existing results, this paper selects the binomial tree model which is from Rendleman and Bartter [18] not from Cox, et al. [17]. The reason for choosing this model is that p is assumed to be equal to 1 2 when we calculate the upper factor and lower factor. The upper factor and lower factor are also different from the general ones, our factors are changed in different intervals since the volatility of dividend process is a random variable. The variable factors make the price of the option more accurate. The paper is divided into the following several parts: in Section 2, some preliminary theoretical knowledge including market models are given; in Section 3, the stock price under the dividend model is given based on the arbitrage pricing theory; in Section 4, we compute the change percentage of the stock price under the original model and the binomial tree model, and the factors i u and i d are computed based on the change percentage, furthermore, the expression of the discrete arithmetic average Asian option is given under the binomial tree model; in Section 5, a simulation example is given to demonstrate the effectiveness of the proposed results; finally, conclusions are drawn in Section 6.

Preliminary Knowledge
In this paper, two assets has been considered in the market. One is the riskless asset (such as the bond) where the price ( ) 0 The discrete dividend ( ) X t is assumed to follow the Geometric Brownian motion with Markov-modulated volatility: where ( ) σ ⋅ is the volatility of dividend process and t ξ is a Markov chain with finite-state { } 0,1, 2, , k , we assume the initial state to be 0, ( ) ( ) Here, we assume that the approached model working in the appropriate pricing measure. The APT implies that the stock price is given by the expected sum of all future dividends appropriately discounted, so that we have the following basic formula [25] ( ) is the discount factor with the interest rate s r .
We assume that: 1) The announcement and payment times always coincide for the dividends.
2) The dividend process satisfies suitable growth conditions ( r µ > ) so that the above sum is always finite.

The Stock Price under the Dividend Process
If the dividend process ( ) X t obeys the Geometric Brownian motion with Markov-modulated volatility, then we get Taking the conditional expectation on the both sides of the above equation,

Binomial Tree Based Models for Pricing Asian Option
In this section, we give new factors i u and i d instead of u and d proposed by Cox, Ross and Rubinstein [17].
2) The stock price i t S at t i t = ∆ moves up to with probabilities p or 1 p − , respectively.
3) The volatility ( ) σ ⋅ is not changed in each interval.

Case One: No Dividend Payment within the Validity Period of the Option
When the stock price The first moment and the second moment of i Y in the binomial tree-based model could be written as follow:

Case Two: Once Dividend Payment within the Validity Period of the Option
If the dividend payment occurs in the m-th interval, then the changing percentage As the same analyzation in Case One, one has Then, we can obtain following solutions:

Compute the Price of Asian Option
The stock price path is shown in Figure 1.

Numeral Calculations
In this section, we shall present the numerical results to demonstrate the effectiveness of the proposed model. In order to simplify the simulation, we assume that the chain t ξ has only two states 0 e and 1 e to express prosperity and depression of economy respectively, the initial state      price minus the outstanding value of the option. Obviously, the dividend led to the reduction of the intrinsic value. The value of the stock is the discount to all future cash flows, dividends are nothing more than a part of the current value of cash. For example, the ten dollars stock pay one dollar dividend, resulting in a lower stock price, stock price drops to nine dollars naturally. So when the value of time does not change, the payment of the dividend leads to the decrease in the intrinsic value of the option, which further results in the decrease in the option price.
In Figure 5, we can see that the option price is higher if the strike price is lower for other fixed influencing factors. This is because when the strike price is lower the possibility of the execution is bigger, and thus the option price is higher. Conversely, the option price will be lower, but will not be negative. The change of strike price is also changing the intrinsic value of the option.
In addition, both in above figures, we can see that the price of options is getting higher and higher as the expiry time increasing. This is because the expiry time changes the time premium of the option.

Conclusion
In this paper, the binomial tree method is used to calculate the price of the discrete arithmetic average Asian option, and the binomial tree model is