Pricing For Rainbow Barrier Option

In recent years, the financial derivatives market has developed with each passing day. In addition to the well-known European options and American options, a variety of exotic options have sprung up in order to meet the requirements of various investors. Rainbow barrier option is one of them. With respect to the same terms ordinary options, rainbow barrier option mainly has three advantages:firstly, the price is cheaper; secondly, it involves a variety of underlying assets, so the market price is not easy to be manipulated; thirdly, the terms of the option contract is flexible to meet the needs of different investors. So when rainbow barrier option appeared the financial market, it became the one of most actively traded exotic options. Nowadays they play an important role in the business fields of investment, risk aversion, and asset management, so pricing for them reasonably and effectively is particularly important for options market participants. On the one hand, the introduction of rainbow barrier feature makes the option price cheaper, but on the other hand, it also increased the difficulty of pricing for them. Pricing for rainbow barrier option has become an important topic in financial mathematics.In1973, Black and Scholes established the famous option pricing model, which called Black-Scholes model. The birth of the model led to the "second revolution on Wall Street" which is a milestone in the history of pricing reasonably for financial derivatives. However, the Black-Scholes model’s assumptions are too ideal to fully apply to the real financial markets. So many experts and scholars look for another way, for example, the binary tree method, finite difference methods, Monte Carlo simulation method, and equivalent martingale measure methods. They use the partial differential method, numerical method, or probability method to establish a variety of pricing models, and hope to get best model to match real financial markets.This paper mainly studies the pricing problem of rainbow single barrier European option. Rainbow barrier option is a multi-asset option. It generally can be divided into down-and-out call option, down-and-out put option, up-and-out call option, up-and-out put option, down-and-in call option, down-and-in put option, up-and-in call option, up-and-in put option. Based on a continuous time model, this article assumes that the underlying assets follow a geometric Brownian motion, and use the Girsanov theorem to change measures. Under the equivalent martingale measure, when barrier is fixed constants and exponential function, the paper use the risk neutral pricing theory and the density functions to get pricing formula of eight rainbow barrier options. In addition in real financial market, if knock-out options become invalid or knock-in options aren’t in effect in the validity period, the option holder will receive a certain amount of cash compensation. This article also discussed the pricing of eight rainbow barrier Options this case.