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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.

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 [

Kemma and Vorst [

The use of binomial tree model in option pricing has been very popular since the appearance of the pioneering work by Cox, et al. [

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. [

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 [

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 u i and d i 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.

In this paper, two assets has been considered in the market. One is the riskless asset (such as the bond) where the price B ( t ) t ≥ 0 satisfies

d B ( t ) = B ( t ) r ( t ) d t , B ( 0 ) = 1 (2.1)

where r ( t ) is the risk-free interest rate at time t. The other is the risky asset supposed to be the stock. ( Ω , F , { F t } t ≥ 0 , P ) is a complete probability space where the filtration { F t } t ≥ 0 follows the usual conditions. The [ 0, T ] is the time interval, where 0 and T represent the current date and the due date respectively. The stock pay dividend D i at discrete times t i , where t i = i h (h is a fixed positive constant), and the dividend takes the form D i = λ X i where λ is a constant.

The discrete dividend X ( t ) is assumed to follow the Geometric Brownian motion with Markov-modulated volatility:

d X t X t = μ d t + σ ( ξ t ) d W ( t ) (2.2)

where σ ( ⋅ ) is the volatility of dividend process and ξ t is a Markov chain with finite-state { 0,1,2 , ⋯ , k } , we assume the initial state to be 0, σ ( ξ t 0 ) = ξ ( 0 ) and p i j = p { ξ t + Δ t = j | ξ t = i } .

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 [

S t = E [ ∑ t m > t β ( t m ) D m ] / β ( t ) (2.3)

where β ( t ) = exp ( − ∫ 0 t r s d s ) is the discount factor with the interest rate r s .

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.

If the dividend process X ( t ) obeys the Geometric Brownian motion with Markov-modulated volatility, then we get

X t = X 0 exp { μ t − ∫ 0 t 1 2 σ ( ξ s ) 2 d s + ∫ 0 t σ ( ξ s ) d w s } . (3.1)

Taking the conditional expectation on the both sides of the above equation, we obtain:

E [ X t | F s ] = X s e ( t − s ) . (3.2)

From(2.3), we can get the stock price [

S t = E [ ∑ t m > t β ( t m ) D m ] / β ( t ) = ∑ m ≥ k e − r ( m h − t ) λ E t X m h = ∑ m ≥ k e − r ( m h − t ) λ X t e μ ( m h − t ) = λ X t e − ( r − μ ) ( k h − t ) 1 − e − ( r − μ ) h (3.3)

for t ∈ [ ( k − 1 ) h , k h ) , and

S t = E t [ ∑ i > m β ( t i ) D i ] / β ( t ) = ∑ i ≥ m e − r ( i h − t ) λ E t X i h = ∑ i ≥ m e − r ( i h − t ) λ X t e μ ( i h − t ) − λ X t = λ X t e − ( r − μ ) h 1 − e − ( r − μ ) h (3.4)

for t = m h , S t = S t − − D m , S t − = S t + D m = λ X t 1 − e − ( r − μ ) h .

In this section, we give new factors u i and d i instead of u and d proposed by Cox, Ross and Rubinstein [

Firstly, we assume that:

1) Discretizing the time period [ 0, T ] into n intervals of the same length Δ t = T N , t i = i Δ t , i = 0 , 1 , 2 , ⋯ , N .

2) The stock price S t i at t = i Δ t moves up to S t i + 1 = S t i u i or down to S t i + 1 = S t i d i with probabilities p or 1 − p , respectively.

3) The volatility σ ( ⋅ ) is not changed in each interval.

When the stock price S t i − 1 moves to S t i over the period [ t i − 1 , t i ] , the changing percentage of the stock price is denoted by Y i = S t i S t i − 1 . The first moment and the second moment of Y i in our original model can be given by:

E [ Y i ] = E [ X t i X t i − 1 e ( r − μ ) Δ t ] = E [ X 0 exp { μ i Δ t − ∫ 0 i Δ t 1 2 σ 2 ( ξ t ) d s + ∫ 0 i Δ t σ ( ξ t ) d w s } X 0 exp { μ ( i − 1 ) Δ t − ∫ 0 ( i − 1 ) Δ t 1 2 σ 2 ( ξ t ) d s + ∫ 0 ( i − 1 ) Δ t σ ( ξ t ) d w s } ] e ( r − μ ) Δ t = E [ exp { μ Δ t − ∫ ( i − 1 ) Δ t i Δ t 1 2 σ 2 ( ξ t ) d s + ∫ ( i − 1 ) Δ t i Δ t σ ( ξ t ) d w s } ] e ( r − μ ) Δ t = E [ e μ Δ t ] E [ exp { ∫ ( i − 1 ) Δ t i Δ t − 1 2 σ 2 ( ξ t ) d s + ∫ ( i − 1 ) Δ t i Δ t σ ( ξ t ) d w s } ] e ( r − μ ) Δ t = e μ Δ t e ( r − μ ) Δ t = e r Δ t , (4.1)

E [ Y i ] 2 = E [ X i Δ t X ( i − 1 ) Δ t e ( r − μ ) Δ t ] 2 = E [ X 0 2 exp { 2 μ i Δ t − ∫ 0 i Δ t σ 2 ( ξ t ) d s + ∫ 0 i Δ t 2 σ ( ξ t ) d w s } X 0 2 exp { 2 μ ( i − 1 ) Δ t − ∫ 0 ( i − 1 ) Δ t σ 2 ( ξ t ) d s + ∫ 0 ( i − 1 ) Δ t 2 σ ( ξ t ) d w s } ] e 2 ( r − μ ) Δ t = E [ exp { 2 μ Δ t − ∫ ( i − 1 ) Δ t i Δ t σ 2 ( ξ t ) d s + ∫ ( i − 1 ) Δ t i Δ t 2 σ ( ξ t ) d w s } ] e 2 ( r − μ ) Δ t

= E [ exp { 2 μ Δ t − ∫ ( i − 1 ) Δ t i Δ t σ 2 ( ξ t ) d s + ∫ ( i − 1 ) Δ t i Δ t 2 σ 2 ( ξ t ) d s } ] E [ exp { − ∫ ( i − 1 ) Δ t i Δ t 2 σ 2 ( ξ t ) d s + ∫ ( i − 1 ) Δ t i Δ t 2 σ ( ξ t ) d w s } ] e 2 ( r − μ ) Δ t = e 2 μ Δ t E [ exp { ∫ ( i − 1 ) Δ t i Δ t σ 2 ( ξ t ) d s } ] e 2 ( r − μ ) Δ t = e 2 μ Δ t ∑ j = 0 k p 0 j ( i − 1 ) e σ 2 ( j ) Δ t e 2 ( r − μ ) Δ t = e 2 r Δ t ∑ j = 0 k p 0 j ( i − 1 ) e σ 2 ( j ) Δ t . (4.2)

The first moment and the second moment of Y i in the binomial tree-based model could be written as follow:

E [ Y i ] = p u i + ( 1 − p ) d i , (4.3)

E [ Y i ] 2 = p u i 2 + ( 1 − p ) d i 2 . (4.4)

As the mean and the variance of Y i under the binomial tree-based model should be equal to that under the original model over the period [ t i − 1 , t i ] . We

match them and set up the system of equations. Let p = 1 2 , according to Rendleman and Bartter [

{ p u i + ( 1 − p ) d i = e r Δ t p u i 2 + ( 1 − p ) d i 2 = e 2 r Δ t ∑ j = 0 k p 0 j ( i − 1 ) e σ 2 ( j ) Δ t p = 1 2 (4.5)

From(4.5), we have:

u i = e r Δ t + e r Δ t ∑ j = 0 k p 0 j ( i − 1 ) e σ 2 ( j ) Δ t − 1 , d i = e r Δ t − e r Δ t ∑ j = 0 k p 0 j ( i − 1 ) e σ 2 ( j ) Δ t − 1 . (4.6)

If the dividend payment occurs in the m-th interval, then the changing percentage

of the stock price in this interval is Y m = X t m X t m − 1 e − ( r − μ ) ( h − Δ t ) . Furthermore, the first moment and second moment of Y i in our original model is given by:

E [ Y m ] = E [ X t m X t m − 1 e − ( r − μ ) ( h − Δ t ) ] = e r Δ t − ( r − μ ) h , (4.7)

E [ Y m ] 2 = E [ X t m X t m − 1 e − ( r − μ ) ( h − Δ t ) ] 2 = e 2 r Δ t − 2 ( r − μ ) h ∑ j = 0 k p 0 j ( m − 1 ) e σ 2 ( j ) Δ t . (4.8)

As the same analyzation in Case One, one has

{ p u m + ( 1 − p ) d m = e r Δ t − ( r − μ ) h , p u m 2 + ( 1 − p ) d m 2 = e 2 r Δ t − 2 ( r − μ ) h ∑ j = 0 k p 0 j ( m − 1 ) e σ 2 ( j ) Δ t , p = 1 2 . (4.9)

Then, we can obtain following solutions:

u m = e r Δ t − ( r − μ ) h + e r Δ t − ( r − μ ) h ∑ j = 0 k p 0 j ( m − 1 ) e σ 2 ( j ) Δ t − 1 , d m = e r Δ t − ( r − μ ) h − e r Δ t − ( r − μ ) h ∑ j = 0 k p 0 j ( m − 1 ) e σ 2 ( j ) Δ t − 1 . (4.10)

Following the same analysis method, the factors in other period can be get by using the similar techniques in Case One and Two.

The stock price path is shown in

1 2 . At time t 2 , the stock prices on the node of each path are S 2 1 = S 1 1 u 2 ,

S 2 2 = S 1 1 d 2 , S 2 3 = S 1 2 u 2 , S 2 4 = S 1 2 d 2 and the sum of the prices on the node of each path are M 2 1 = M 1 1 + S 2 1 , M 2 2 = M 1 1 + S 2 2 , M 2 3 = M 1 2 + S 2 3 , M 2 4 = M 1 2 + S 2 4 from top to bottom. The probability that the stock price passes through each

path is 1 4 . In general, at time t i , the stock prices on the node of each path are S i j = S i − 1 j + 1 2 u i when j is an odd number, and S i j = S i − 1 j 2 d i when j is an even number. The sum of the prices on the node of each path are M i j = M i − 1 j + 1 2 + M i j when j is an odd number, and M i j = M i − 1 j 2 + M i j when j is an even number from top to bottom. The probability that the stock price passes through each path is 1 2 i .

Thus, the price of Asian option can be computed as follows:

V ( X 0 , K , T , 0 ) = e − r T 1 2 N ∑ j = 1 2 N max ( M N j N + 1 − K , 0 ) . (4.11)

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 ξ t has only two states e 0 and e 1 to express prosperity and depression of economy respectively, the initial state ξ t 0 = e 0 . In these states, the values of the parameters are given as σ ( e 0 ) = 0.1 , σ ( e 1 ) = 0.3 . The transfer probability matrix is selected as

P = [ 0.7 0.3 0.2 0.8 ] , (5.1)

X 0 = 0.38 , r = 0.03 , μ = 0.02 , Δ t = 1 12 . We only consider the case of once dividend payment within the validity period of the option.

When the option expiry time takes from 1 12 to 15 12 , the price of the discrete

arithmetic average Asian option is shown in

In Figures 2-4, we can see that the dividend payment will reduce the price of the option, and the sooner the dividend payment, when the expiry time are same, the greater the impact on the option. For example, if the expiry time is fixed as one year, it could be seen form

Dividend pay in the third interval | Dividend pay in the sixth interval | ||||||
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N | K = 45 | K = 50 | K = 55 | N | K = 45 | K = 50 | K = 55 |

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | 5.3372 5.3866 5.2487 5.1858 5.1772 5.2081 5.2629 5.3388 5.4220 5.5142 5.6098 5.7079 5.8073 5.9071 6.0067 | 0.5378 0.7673 0.9371 1.1085 1.2884 1.4826 1.6580 1.8275 1.9951 2.1543 2.3091 2.4580 2.6022 2.7419 2.8773 | 0 0 0 0.0094 0.0751 0.1429 0.2374 0.3349 0.4436 0.5551 0.6659 0.7790 0.8916 1.0038 1.1154 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | 5.3372 5.3866 5.4358 5.4849 5.5412 5.5151 5.5249 5.5655 5.6212 5.6904 5.7688 5.8516 5.9383 6.0275 6.1178 | 0.5378 0.7673 1.0375 1.2873 1.5000 1.6578 1.8100 1.9640 2.1164 2.2644 2.4086 2.5498 2.6867 2.8203 2.9506 | 0 0 0 0.0316 0.1031 0.1798 0.2774 0.3788 0.4864 0.5964 0.7077 0.8194 0.931 1.0421 1.1525 |

Dividend pay in the ninth interval | Dividend pay in the twelfth interval | ||||||

N | K = 45 | K = 50 | K = 55 | N | K = 45 | K = 50 | K = 55 |

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | 5.3372 5.3866 5.4358 5.4849 5.5412 5.6182 5.7019 5.7964 5.8258 5.8728 5.9323 6.0000 6.0740 6.1520 6.2330 | 0.5378 0.7673 1.0375 1.2873 1.5000 1.7209 1.9220 2.1123 2.2480 2.3831 2.5166 2.6484 2.7777 2.9047 3.0292 | 0 0 0 0.0316 0.1031 0.1953 0.3119 0.4297 0.5371 0.6452 0.7551 0.8657 0.9755 1.0851 1.1938 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | 5.3372 5.3866 5.4358 5.4849 5.5412 5.6182 5.7019 5.7964 5.8947 5.9964 6.0995 6.1521 6.2129 6.2799 6.3512 | 0.5378 0.7673 1.0375 1.2873 1.5000 1.7209 1.9220 2.1123 2.2940 2.4652 2.6303 2.7521 2.8737 2.9938 3.1121 | 0 0 0 0.0316 0.1031 0.1953 0.3119 0.4297 0.5551 0.6810 0.8073 0.9161 1.0242 1.1318 1.2389 |

Dividend pay in the fifteenth interval | No dividend payment | ||||||

N | K = 45 | K = 50 | K = 55 | N | K = 45 | K = 50 | K = 55 |

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | 5.3372 5.3866 5.4358 5.4849 5.5412 5.6182 5.7019 5.7964 5.8947 5.9964 6.0995 6.2035 6.3069 6.4102 6.4718 | 0.5378 0.7673 1.0375 1.2873 1.5000 1.7209 1.9220 2.1123 2.2940 2.4652 2.6303 2.7876 2.9396 3.0859 3.1982 | 0 0 0 0.0316 0.1031 0.1953 0.3119 0.4297 0.5551 0.681 0.8073 0.9336 1.0584 1.1818 1.2869 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | 5.3372 5.3866 5.4358 5.4849 5.5412 5.6182 5.7019 5.7964 5.8947 5.9964 6.0995 6.2035 6.3069 6.4102 6.5124 | 0.5378 0.7673 1.0375 1.2873 1.5000 1.7209 1.9220 2.1123 2.2940 2.4652 2.6303 2.7876 2.9396 3.0859 3.2274 | 0 0 0 0.0316 0.1031 0.1953 0.3119 0.4297 0.5551 0.6810 0.8073 0.9336 1.0584 1.1818 1.3034 |

in the ninth interval. Form

The image of the above figures is consistent with the financial markets. The value of the option is reflected in two aspects, one is the intrinsic value and the other is the time value. The intrinsic value of the call option is equal to the stock

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

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.

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 determined by calculating the upper and lower factors. Finally, the validity of the method is verified by numerical calculation. However, the method is not feasible when the segmentation interval is relatively large.

This work was supported in part by the National Nature Science Foundations of China under Grant No. 61673103 and No. 61403248.

Fang, Y.Y., Shu, H.S., Kan, X., Zhang, X. and Zheng, Z.W. (2017) The Asian Option Pricing when Discrete Dividends Follow a Markov-Mo- dulated Model. Open Journal of Statistics, 7, 1067-1080. https://doi.org/10.4236/ojs.2017.76074