離散型雙邊界選擇權之定價

界限選擇權是現在市場上相當受歡迎的新奇選擇權之一, 與一般選擇權的不同特點在, 它是一種路徑相依的選擇權, 界限選擇權的收益是在某個具體或非具體的資產價格(標的物or 利率) 碰觸到了某個特定的水準時, 依此來當作是選擇權失效的依據。大多數用於評價界限選擇權的模型都是假設在連續的觀測時間; 在此假設下, 選擇權大多可表示成封閉解。許多現實中有關界限選擇權的合約觀測都是離散型的時間觀測點; 現實中現在還沒有公式可以定價這些離散型的選擇權, 儘管是利用Monte carlo method 直接去跑模型, 可能數值收斂的速度不快。參考的文獻中發現離散型界限選擇權利用連續型公式做修正所得到公式可得到相當卓越的精準度。本篇論文中首先說明利用標準的Black-Scholes Model 和diffusiontype 隨機微分方程來做為其數學模型來架構並描述股價的動態狀況,一樣利用測度轉換將其作拆解, 然後特別去針對某一個機率做推導, 最後求得離散型的雙界限選擇權價錢的公式, 再去利用Simpson 積分法則, 求得數值解。

關鍵字

界限選擇權；路徑相依選擇權；布朗運動有限區間機率；路徑跨越機率

並列摘要

Barrier option is one of the Exotic options which are very popular in the market. Differ from other simple options, barrier option is continuous time and path depend. The payoff of a barrier option depends on whether or not a specified asset price, index, or rate reaches a specified level during the life of the option. Most models for pricing barrier options assume continuous ,monitoring of the barrier; under this assumption, the option can often be priced in closed form. Many(if not most) real contracts with barrier provisions specify discrete monitoring constant; there are essentially no formulas for pricing these options, and even numerical pricing to run the model is probabily converge slowly. In this thesis, first introduce the standard Black-Scholes Model and diffusion-type Stochastic differential equation as mathmatical model , and use it to describe the stock dynamical system . Then pricing the discrete type double barrier option heuristically. Compared to the Monte Carlo Algorithm, our discrete formula is cost not so much time.

並列關鍵字

barrier option ； path-dependent options ； level crossing probabilities

參考文獻

邵翊哲(2010), 彩虹型雙界限選擇權之定價, 國立台北大學統研所碩士論文.

Helyette Geman and Marc Yor. (1996). Pricing and Hedging Doublebarrier

option:An Probability Approach. Mathematical Finance Vol

Karatzas, I., and S. Shreve (1991): Brownian Motion and Stochastic

Merton, R.C.(1973): ”Theory of Rational Option Pricing” Bell J. Econ.

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離散型雙邊界選擇權之定價

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