In the over-the-counter (OTC) markets, the holders of may contracts are vulnerable to counterparty credit risk. Because of this issue, an vulnerable options should be considered. In addition, in a financial environment, the pricing of path-dependent options yields many interesting mathematical challenges. In this paper, we study the pricing of vulnerable path dependent options using double Mellin transforms and the method of images to investigate an explicit (closed) form pricing formula. In financial market, the derivation of the closed solutions on the financial instruments is very important. Apart from the path-dependent options, we will deal with the pricing of European option under another financial model. Under stochastic elasticity of variance (SEV) or generalized constant elasticity of variance(GCEV), we will study the approximated closed solution on the option price by using multiscale analysis. Obtaining the closed solutions or the analytic solutions allows us to implement option`s data fitting more easily and effectively. Zoom과 동시에 진행:https://snu-ac-kr.zoom.us/j/8156716391