The hedging strategy for Asian option.pdf Introduction The world financial system is continuously developing, which causes its ever increasing fragmentation. The derivatives market as one of the elements of the system develops even more rapidly. The first derivatives were currency futures and forwards, emerged in the early 70's a little later there were options [1]. After the first option transaction, which took place in 1973 on the Chicago Board of Options Exchange [2], a revolution in the development of option trading began. By the end of the 1970's options were well studied on stock exchange and then new exotic options appeared. In the late 1980's and early 1990's exotic options became more in demand and their trade became more active in the over-the-counter market. Soon in the commodity and currency markets Asian options are becoming popular. Geman and Yor have considered Asian options in their work [3], such derivatives are based on the average prices of underlying assets. Using the Bessel processes authors found the value of the Asian option. Moreover, applying simple probabilistic methods they obtained the following results about these options: calculated moments of all orders of the arithmetic average of the geometric Brownian motion; obtained simple, closed form expression of the Asian option price when the option is "in the money". The exact pricing of fixed-strike Asian options is a difficult task, since the distribution of the average arithmetic of asset prices is unknown when its prices are distributed lognormally. The study of this problem is divided into three groups. A large number of works are connected with the numerical approach. Kemna and Vorst were among the first who solved the task [4]. In their work the pricing strategy includes Monte Carlo simulation with elements of dispersion reduction and improves the pricing method. Furthermore, the authors showed that the price of an option with an average value will always be lower than of a standard European option. Carverhill and Clewlow [5] used a fast Fourier transform to calculate the density of the sum of random variables, as convolution of individual densities. Then the payoff function is numerically integrated against the density. In this direction other authors continued to work, applying to the calculations improved methods of numerical simulation [6-8]. Unfortunately, these methods do not provide information on the hedging portfolio. The second approach, used by Ruttiens [9] and Vorst [10], is to change the geometric average price of the option. The third approach proposed by Levy [11] and accepted by some practitioners, suggests that the distribution of the arithmetic average is well approximated at least in some markets by a lognormal distribution, and therefore the problem is reduced to determining the necessary parameters. This problem is less complicated since the first two moments of the arithmetic mean are relatively simple. For a trader or an investor the main task is not only the saving but also the multiplication of its capital. Many risks can be avoided with the help of one popular and very effective technique - hedging. The option is hedged to protect its value from the risk of price movement of the underlying asset in an unfavorable direction. To solve the hedging problem stochastic calculus methods are used which became a powerful tool used in practice in the financial world. Stochastic calculus is a well-developed branch of modern mathematics with a "correct" approach to analyzing complex phenomena occurring on world stock markets. Book of A.N. Shiryaev [12] provides a complete and systemic view of ideas and techniques in stochastic finance. It should be noted that the task of options pricing and the construction of a hedging strategy is well studied for American and European options, for such derivatives there is a so-called delta strategy. But this technique is not developed for Asian options. In this paper we consider the financial portfolio with several risky assets. Based on the results presented in the article [14] we have solved the hedging problem for the Asian option using martingale methods. The main result of the paper is the formulas for the hedging strategy y(t) = (y1 (t),...,уd (t)) which are obtained with the help of the martingale representation and Ito's formula and is defined as Yi (t) = д G(t,5(t),S(t)), i = \d 5(t),S(t) e Rd , (1) f d d ~ / \ \ 1 i=1 x +1 i=1 y n i (v) - K d where G (t, x, y ) = E (2) In solving the problem we found the densities for the following random exponential variables n (v) = J0vexp{

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