The Pricing Of Arithmetic Average Asian Options

Asian option is one of the Exotic options, it has three merits:the price is cheap; it can effectively protect the price of underlying assets from manipulation; it is much better for hedgers as relatively low risk. Therefore, Asian option has great advantage on the risk management, the risk control and so on. It has been one of the most active Exotic options in financial markets. The pricing of Asian options have also been gradually becoming a hot issue on the research of the derivative assets pricing.According to the different methods of payoff calculation, Asian option can be divided into two categories:geometric average Asian option and arithmetic average Asian option. Now, geometric average Asian options have explicit pricing formula, but there are some difficulties in the pricing of arithmetic average Asian options. The main problem is that arithmetic average assets price is no longer lognormal distribution. It can't directly use Black-Scholes formula. The arithmetic average Asian options have no explicit formula so far.The main goal of this paper is to study the pricing of discrete arithmetic average Asian options with transaction costs and continuous arithmetic average Asian options with transaction costs. We get an asymptotic relationship between arithmetic average Asian options with transaction costs and European options by the methods of no arbitrage and hedge risk. Therefore, the pricing of arithmetic average Asian options with transaction costs is converted into the pricing of European options, then we get the approximate pricing of arithmetic average Asian options with transaction costs.For the discrete arithmetic Asian options with transaction costs, firstly, we find the relationship between its expected return and the average expectations of each discrete point in time. Then, according to the pricing formula of the European options with transaction costs, we get the approximate pricing of the discrete arithmetic Asian options with transaction costs.For the continuous arithmetic average Asian options with transaction costs, firstly, we use a geometric Brown motion instead of the arithmetic average price approximately. Secondly, we calculate the expected rate of return and stock price volatility by "second moment" method, then we use these two parameters to construct a European option. Lastly, we get the approximate price of the continuous arithmetic average Asian options with transaction costs by the formula of the European options.