本文關(guān)鍵詞： 特征函數(shù) 等鞅變換 數(shù)值計(jì)算　出處：《廈門大學(xué)》2014年碩士論文　論文類型：學(xué)位論文

【摘要】：亞式期權(quán)非常流行,它是最受歡迎的新型期權(quán)之一。雖然亞式期權(quán)經(jīng)過(guò)了眾多的學(xué)者與研究者反復(fù)的研究,但是至今也沒有形成一個(gè)統(tǒng)一的定價(jià)表達(dá)式。在研究的過(guò)程中,研究者們也提出了許多的定價(jià)方法,現(xiàn)在為大家所接受的定價(jià)方法有幾何平均近似算術(shù)平均法、蒙特卡洛模擬法、偏微分方程數(shù)值解法、特征函數(shù)法。作為學(xué)術(shù)界研究的重要課題的亞式期權(quán)是路徑依賴的期權(quán),因此其定價(jià)問(wèn)題變得異常的復(fù)雜,這也使得它成為了研究中的難題。本文在特征函數(shù)法和數(shù)值計(jì)算的基礎(chǔ)上,推導(dǎo)了亞式期權(quán)價(jià)格的表達(dá)式,得到了數(shù)值結(jié)果,為亞式期權(quán)的定價(jià)提供了新的思路。 本文的創(chuàng)新之處在于在標(biāo)的資產(chǎn)價(jià)格變化中引入了跳躍與波動(dòng)率的變化,并且在用特征函數(shù)進(jìn)行定價(jià)的基礎(chǔ)上,利用數(shù)值計(jì)算的方法討論了期權(quán)價(jià)與跳躍、波動(dòng)率的關(guān)系。跳躍是無(wú)處不在的,然而跳躍又具有隨機(jī)性與偶然性,這使得它成為生活中極其重要的現(xiàn)象,也使得它成為了學(xué)術(shù)研究中極為重要的研究課題。波動(dòng)率隨著時(shí)間的變化而變化的觀點(diǎn)得到了學(xué)術(shù)界的認(rèn)可,同時(shí)也為實(shí)際生活中的數(shù)據(jù)所證實(shí)。本文采用等鞅變換得到等價(jià)的隨機(jī)過(guò)程,然后利用特征函數(shù)法得到兩組常微分方程組和亞式期權(quán)價(jià)格的形式上的表達(dá)式,最后通過(guò)利用數(shù)值計(jì)算的方法得到了期權(quán)價(jià)格。 在特征函數(shù)法與等鞅變換的基礎(chǔ)上,我們得到了亞式期權(quán)價(jià)格的表達(dá)式。我們也通過(guò)數(shù)值計(jì)算的方法來(lái)研究了期權(quán)價(jià)格與各種變量的變化關(guān)系,在設(shè)定一系列的初值后,本文通過(guò)微分方程的數(shù)值解法得到了常微分方程組的數(shù)值解,同時(shí)通過(guò)數(shù)值積分得到了亞式期權(quán)的價(jià)格。在數(shù)值計(jì)算的基礎(chǔ)上,我們得到以下結(jié)論：第一,跳躍的存在會(huì)影響亞式期權(quán)的價(jià)格;第二,跳躍強(qiáng)度的大小會(huì)影響亞式期權(quán)的價(jià)格；第三,波動(dòng)率的變化會(huì)影響期權(quán)價(jià)格。
[Abstract]:Asian option is very popular, it is one of the most popular new options, although Asian option has been studied repeatedly by many scholars and researchers. However, there is not a unified pricing expression. In the process of research, researchers also put forward many pricing methods, which are now accepted as geometric mean approximate arithmetic average. Monte Carlo simulation method, partial differential equation numerical solution, characteristic function method. As an important subject of academic research, Asian option is path dependent option, so the pricing problem becomes very complicated. Based on the eigenfunction method and numerical calculation, the expression of the price of Asian option is derived, and the numerical results are obtained. It provides a new idea for the pricing of Asian options. The innovation of this paper lies in the introduction of jump and volatility in the change of underlying asset price, and on the basis of pricing with characteristic function, the paper discusses the option price and jump by numerical calculation method. The relationship of volatility. Jump is everywhere, but jump has randomness and contingency, which makes it an extremely important phenomenon in life. It has also become an extremely important research topic in academic research. The point of view that volatility changes with time has been recognized by the academic community. In this paper, the equivalent stochastic process is obtained by using the equal-martingale transformation, and then the formal expressions of two sets of ordinary differential equations and the price of the Asian option are obtained by using the eigenfunction method. Finally, the option price is obtained by numerical calculation. On the basis of the eigenfunction method and the equal-martingale transformation, we obtain the expression of the price of the Asian option. We also study the relationship between the price of the option and the variables by numerical calculation. After a series of initial values are set, the numerical solution of ordinary differential equation system is obtained by numerical solution of differential equation, and the price of Asian option is obtained by numerical integration. We get the following conclusions: first, the existence of jump will affect the price of Asian options; Secondly, the price of Asian option will be affected by the strength of jump. Third, volatility changes will affect option prices.
【學(xué)位授予單位】：廈門大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2014
【分類號(hào)】：F224;F830.91

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