Lower-bound Formulas For The Price Of Asian Options In The Market With Jumps

In the recent decades, financial derivatives market is developing rapidly. Especially in the early nineties many financial institutions introduced some options which have new trading approaches and whose prices are very flexible and efficient with the need of the market. These options called new types of options. Asian options is one of them. It was first launched by American Bankers Trust Company in Tokyo of Japan. There are two types of Asian options, fixed-strike and floating-strike Asian options. The payoffs of them depend on the average price of the underlying assets within validity period. Usually we consider of arithmetic average price. It's hard to get their closed pricing formulas.This paper borrowed the idea of Kuan-Wen Chen and Yuh-Dauh Lyuu who gave the pricing formulas of fixed-strike and floating-strike Asian options through working out the lower-bound of price in complete market. We deduced the pricing formulas of two types of Asian options in incomplete market.In complete market we gave the method of the pricing formulas of Asian options by Kuan-Wen Chen and Yuh-Dauh Lyuu. They supposed that the underlying assets follow geometric Brownian motion. The price of Asian options was just the maximal lower bound of this inequality. Then the problem was transformed into how to solve the maximal lower bound. During this process, according to the characteristics of the expression of Asian options, they collected a set. This set can subtly help us obtain the lower bound. Furthermore this lower bound was the pricing formulas of Asian call options. Finally, numerical experiments showed that this formula, whether the test was on the efficiency or accuracy have good results.Moreover this paper extended the method of Kuan-Wen Chen and Yuh-Dauh Lyuu gave into the market with jumps. Supposed that the price of the underlying assets follows jump-diffusion process. That is to say the price of the underlying assets is composed of Brownian motion with drift and composite Poisson process. By solving the lower bound of the prices, transform the problem of pricing of options into how to solve the lower bound of the price. During this process, we should look for the changes of the theorem and lemma when the market is changed. Afterwards use the related theorem and lemma in the market with jumps to deduce the lower bound of the price of Asian options. Finally, numerical experiments show that the lower bound of the price of Asian options is meaningful.