The Pricing Of Barrier Options Under Vasicek Model

The option is a contract that the holder can buy or sell quantity of underlying assets at fixed time in the future. In resent years, options, as an important method to avoid risk or speculation have rapid developed. Because of the arbitrage-free principle in the market, the premium is large than zero obviously. So giving the fair and reasonable price becomes an important problem in the study foe financial theory. In 1973, F. Black and M.Scholes, the schlars in Chicage University raised Black-Scholes Option Pricing Model under ideal supposition, and got the analytic solution of this model, that is Black-Scholes fomula. Along with the development of financial market and the need of people for the disposition of risk and profit, various of derivative instrument appears, such as path dependent option,exotic option and so on. Going deep into the study of the pricing for these derivative instruments has profound the oretical and practical significance.This paper study weak path dependent options,that is, their final profits depend nom only on the price of the underlying assets in the expiration date, but alse on the barrier during the term of validity. We get the pricing formula of vanilla European options by using Ito lemma and martingale method first, then get the pricing formula of barrier options under random rate by using the property of conditional expectation and the price of stock. The main consequence is as follows:1. The pricing formula of the down-and-out call barrier option:2 The pricing formula of the up-and-out call barrier option.:3. The pricing formula of the down-and-in call barrier option:4. The pricing formula of the up-and-in call barrier option:and c(t, S_t,r_t, K) is the pricing formula of vanilla European options.