The Pricing Of Discrete Single Square Barrier Option With Time-Dependent Parameters

In recent years, with the development of financial market, financial derivatives have been favored by more and more investors. Among the derivatives which are trading in financial market, barrier option is one of the most active ones. Compared with continuous barrier option, discrete barrier option has the advantages of avoiding hitting barrier when the market closed. So, it is very important to study the pricing of discrete barrier option. In this article, we study a new kind of barrier option which changes the standard income structure, the discrete option-square barrier option. Our main work is studying the discrete option-square barrier option with time-dependent parameters and single barrier. In the past literatures, researchers have studies about the pricing of discrete option-square barrier option, but there are few studies about the discrete option-square barrier option with time-dependent parameters. In our article, we focus on the pricing of discrete down-and-out barrier European call option on a stock with time-dependent parameters and single barrier.We first calculate the stochastic differential equation and partial differential equation of discrete option-square barrier option according to A-hedging theorem and Ito formula. Then a set of transformations are employed totransform the partial differential equation with time-dependent parameters into the partial differential equation with constant parameters. At last, we convert the partial differential equation with constant parameters to heat equation. The solution of heat equation is the pricing model of discrete down-and-out barrier European call option on a stock with time-dependent parameters and single barrier. We consider the increasing sequence of monitoring dates. Our aim is to solving the partial differential equationOur main conclusions are as follows:Lemma 3.1.1 Consider Yt= St2, and St satisfies the stochastic differential equation, then Yt satisfiesProposition 3.1.2 Consider Yt= St2, and Yt satisfies the stochastic differential equation, also consider Cb is the price of discrete down-and-out barrier European call option on a stock with time-dependent parameters, then Cb satisfiesTheorem 3.3.1 The price of down-and-out barrier call option with strike price K and barrier level L at discrete monitoring date t=tn+1, is evaluated as follows...