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The pricing problem of option pricing formula with dividends average Asian option under fractional Brownian motion is studied in this paper. Then by solving the partial differential equation, the pricing formula and call-put parity of the geometric average Asian option with dividend payment and transaction costs are obtained. At last, the influences of Hurst index and maturity on option value are discussed by numerical examples.

Option pricing theory has been an unprecedented development since the classic Black-Scholes option pricing model [ 1 ] was proposed. Asian options are a kind of common strong path-dependent options, whose value depends on the average price of the underlying asset during the life of the option. Fusai and Meucci [ option pricing formula with dividends ] have discretely studied Asian option pricing problem under the Levy process. Vecer [ 3 ] has got the unified algorithm of the Asian option value based on the basic theory of stochastic analysis.

However, the empirical analysis shows that there is a long-term correlation between the underlying asset prices, so that the geometric Brownian motion is not considered as an ideal tool to describe the process of asset price.

Since fractional Brownian motion has the properties of self-similarity, thick tail, and long-term correlation, that fractional Brownian motion has become a good tool to depict the process of underlying asset price. According to the standard Brownian motion, Mandelbrot and Van Ness [ 5 ] obtained a stochastic integral form of fractional Brownian motion.

Based on the wick product, Duncan et al. In the reality of the securities market, investors were faced with considerable and nonignorable transaction costs and Leland [ 8 ] firstly examined the problems of option pricing and hedging with transaction costs.

Due to infinite variation of geometric Brownian motion, transaction costs would become infinite in the continuous time completely hedging strategy. So Leland suggested that no-arbitrage assumption is replaced by Delta hedging strategy under the condition of discrete time occasions and transaction costs.

The model was then extended by Hoggard et al. Guasoni [ 10 ] studied the standard option with transaction costs under the fractional Brownian motion, but he did not obtain option pricing formula. Then Liu option pricing formula with dividends Chang and others [ 11 ] extended the option pricing with transaction costs under fractional Brownian motion and provide an approximate solution of the nonlinear Hoggard-Whalley-Wilmott equation.

But these studies are usually aimed at European standard options. In this paper, Asian option pricing problems with transaction costs and dividends under fractional Brownian motion are studied.

Firstly, the partial differential equation satisfied by geometric average Asian option value is obtained on the basis of no-arbitrage principle. Then the analytic expressions of option value and parity formula are presented by solving the partial differential equation. At last, the influences of Hurst exponent and maturity on option value are discussed by numerical examples. Definition 1 see [ 17 ]. Let be a complete probability space on which a standard fractional Brownian motion with Hurst exponent is continuous, centered Gaussian processes with covariance functions.

Suppose that stochastic process satisfied the following equation: Suppose that stochastic process ; then, for anyone has where is geometric average of between the time period of. In this paper, the following basic assumptions were needed. Let denote the transaction cost per unit dollar of transaction, where is a constant. To buy or sell option pricing formula with dividends of the underlying asset need pay proportional transaction costs ; note that denotes buying the underlying asset and denotes selling.

Let denote the value of the geometric average Asian call at timewhere is option pricing formula with dividends average of underlying asset in. Then the value of the portfolio at time is.

After the time intervalthe change in the value of the option pricing formula with dividends is as follows: **Option pricing formula with dividends** ; then, 5 becomes where The mathematical expectation of transaction costs is obtained in the following form: By assumption ivthe following relation holds: By 6 and **option pricing formula with dividends**one has Substituting 10 into 9the following partial differential equation is obtained: Substituting and into 11the following equation is obtained: Suppose that the underlying asset option pricing formula with dividends satisfied 3 ; then, the value of the geometric average Asian call at time, satisfies the following mathematical model: Theorem 3 is obtained for the long position of the option.

If the short position of option is considered, similarly, we can also **option pricing formula with dividends** the mathematical model 14 and only the corresponding modified volatility is given by the following form: Let Lewhich is called fractional Leland number [ 8 ]. For the long position of a single European Asian option, its final payoff is or and they are both convex function, soand noticingthus. However, for the short position of a single European Asian option, its final payoff at maturity is or and they are both concave function, so that.

So for a single European Asian option, 13 and 15 can be represented as. Suppose that the underlying asset price satisfied 3 ; then, the value,of the geometric average Asian call with strike pricematurityand transaction fee rate at time is where. By Theorem 3the value,of the geometric average Asian call satisfies the following model: Let ; then Combined with the boundary conditions of the call option,the model 19 can be converted to Let which satisfied the conditions and then we have Substituting 23 into 21we can get Set Combining with the terminal conditionswe have where Thus the model 21 is converted into the classic heat conduction equation Its solution is After variable reduction, we have where So the value of geometric average Asian call option at time is obtained.

Suppose the underlying asset price satisfies 3 ; then the option pricing formula with dividends betweenthe value of geometric average Asian call option, andthe value of put option with strike pricematurityand transaction fee rate at timeis where are the same as above. Let Then is suitable for the following terminal question in: We let ; then is suitable for the following in: Set the form solution of problem of 36 is Substituting 37 into 36 and comparing coefficients, one has Takingthe solutions of 38 **option pricing formula with dividends** Thus The parity formula between call option and put option is.

Pricing formula of the geometric average Asian put options can be obtained by Theorems 6 and 7. Suppose satisfies 3 ; then the value,of the geometric average Asian put option with strike pricematurityand transaction fee rate at time is In particular, if, are all constant and price formula 17 is reduced to the following formula [ 19 ]: If risk-free interest ratedividend yieldand volatility are option pricing formula with dividends constant, then the price formulas of geometric average Asian call and put option with strike pricematurityand transaction fee rate under fractional Brownian motion at time are, respectively, where The rest of symbols are the same as Theorem 6.

Noticing that ifis standard Brownian motionthe corresponding underlying asset price follows geometric Brownian motion, one has the following results. If risk-free interest rate and dividend yield are the functions of time and volatility is constant, then the price formula with respect to geometric average Asian call option pricing formula with dividends with strike pricematurityand transaction fee rate under standard Brownian motion at time is where.

We discuss the impact option pricing formula with dividends Hurst index and transaction rates on the Asian option value by numerical examples.

Assume that the parameter selection is as follows: We calculate the value of the option by using the price formula of The relationships between the value of call option or put option and the underlying asset price with different Hurst index are given in Figures 1 and 2respectively. From Figures 1 and 2the relationship between Hurst index and Asian option value is negative. Furthermore, the impact on the call option value decreases with the option pricing formula with dividends of the underlying asset price, but the impact on the put option value decreases with the decrease of the underlying asset price.

By Figures 3 and 4we can get the change trend of the value of Asian call and put options with the change of maturity and Hurst index at the same time. The option value increases with the maturity increases, but the value of the call option increases faster than the value of a put option increase.

Asian options are popular financial derivatives that play an essential role in financial market. Pricing them efficiently and accurately is very important both in theory and practice. We have investigated geometric average Asian option pricing formula with dividends valuation problems with transaction costs under the fractional Brownian motion.

Meanwhile, the pricing formula and call-put parity of the geometric average Asian option with transaction costs are derived by solving PDE. At last, the influences of Hurst exponent and maturity on option value are discussed through numerical examples.

The authors declare that option pricing formula with dividends is no conflict of interests regarding the publication of this paper. The authors would like to thank the anonymous referees for valuable suggestions for the improvement of this paper. All remaining errors are the responsibility of the authors.

Home Journals About Us. Journal of Applied Mathematics. Subscribe to Table of Contents Alerts. Table of Contents Alerts. Abstract The pricing problem of geometric average Asian option under fractional Brownian motion is studied in this paper. Introduction Option pricing theory has been an unprecedented development since the classic Black-Scholes option pricing model [ 1 ] was proposed. Then the value of the portfolio at time is After the time intervalthe change in the value of the portfolio is as follows: So for a single European Asian option, 13 and 15 can be represented as 3.

Option Pricing Formula Theorem 6. Suppose that the underlying asset price satisfied 3 ; then, the value,of the geometric average Asian call with strike pricematurityand transaction fee rate at time is where Proof. Let ; then Combined with the boundary conditions of the call option,the model 19 can option pricing formula with dividends converted to Let which satisfied the conditions and then we have Substituting 23 into 21we can get Set Combining with the terminal conditionswe have where Thus the model 21 is converted into the classic heat conduction equation Its solution is After variable reduction, we have where So the value of geometric option pricing formula with dividends Asian call option option pricing formula with dividends time is obtained Theorem 7.

Set the form solution of problem of 36 is Substituting 37 into 36 and comparing coefficients, one has Takingthe solutions of 38 are Thus The parity formula between call option and put option is Pricing formula of the geometric average Asian put options can be obtained by Theorems 6 and 7. If risk-free interest rate and dividend yield are the functions of time and volatility is constant, then the price formula with respect to geometric average Asian call option with strike pricematurityand transaction fee rate under standard Brownian motion at time is where 4.

Numerical Example We discuss the impact of Hurst index and transaction rates on the Asian option value by numerical examples. The values option pricing formula with dividends the call option with different.

The values of the put option with different. Relationship among Hurst index, maturity, and call option value. Relationship among Hurst index, maturity, and put option value. View at Google Scholar G. View option pricing formula with dividends Google Scholar J. View at Google Scholar T.

View at Google Scholar P.