本文選題：混合分?jǐn)?shù)布朗運(yùn)動(dòng) + 交易費(fèi)��； 參考：《中國(guó)礦業(yè)大學(xué)》2017年碩士論文

【摘要】：回望期權(quán)是一種強(qiáng)路徑依賴(lài)性期權(quán),它在到期日的收益依賴(lài)于整個(gè)期權(quán)在有效期內(nèi)的風(fēng)險(xiǎn)資產(chǎn)的最大值或最小值,這樣使得回望期權(quán)在交割日的收益比較高,價(jià)格非常昂貴,更有研究的意義。當(dāng)前學(xué)術(shù)界對(duì)回望期權(quán)的研究幾乎都是建立在幾何布朗運(yùn)動(dòng)的假設(shè)下,然而在實(shí)際的金融交易中,資產(chǎn)收益率的分布呈現(xiàn)“尖峰厚尾”的特點(diǎn)而且資產(chǎn)價(jià)格出現(xiàn)間斷的不頻繁的“跳躍”情況,使得在幾何布朗運(yùn)動(dòng)下的研究與實(shí)際情況不符合。因此本文主要在混合分?jǐn)?shù)布朗運(yùn)動(dòng)、混合跳擴(kuò)散分?jǐn)?shù)布朗運(yùn)動(dòng)以及混合分?jǐn)?shù)布朗運(yùn)動(dòng)下帶交易費(fèi)三種模型下,分別對(duì)歐式回望看跌期權(quán)定價(jià)問(wèn)題進(jìn)行了研究。主要結(jié)果如下:(1)研究了標(biāo)的股票價(jià)格服從混合分?jǐn)?shù)布朗運(yùn)動(dòng)模型下的歐式回望期權(quán)定價(jià)問(wèn)題。利用伊藤公式,風(fēng)險(xiǎn)對(duì)沖方法得到了在該模型下期權(quán)價(jià)格所滿(mǎn)足的微分方程及其邊界條件,最后由有限差分方法,得到了該微分方程的數(shù)值解,并且通過(guò)實(shí)例驗(yàn)證了該數(shù)值解的有效性。(2)研究了更符合實(shí)際金融市場(chǎng)變化的問(wèn)題,在前面研究的基礎(chǔ)上加入泊松過(guò)程,建立混合跳-擴(kuò)散分?jǐn)?shù)布朗運(yùn)動(dòng)下歐式回望看跌期權(quán)定價(jià)模型。首先利用伊藤公式,風(fēng)險(xiǎn)對(duì)沖方法得到了在該模型下期權(quán)價(jià)格所滿(mǎn)足積分方程,通過(guò)泰勒展開(kāi),將積分方程轉(zhuǎn)化為偏微分方程,然后對(duì)偏微分方程分別進(jìn)行顯示差分與隱式差分,得到相應(yīng)的迭代方程。最后對(duì)偏微分方程進(jìn)行降維變化,對(duì)變換后的數(shù)學(xué)模型構(gòu)造顯式離散格式并理論分析其穩(wěn)定性和相容性,最后利用Matlab軟件對(duì)該格式進(jìn)行數(shù)值分析,討論市場(chǎng)參數(shù)對(duì)期權(quán)價(jià)值的影響。(3)研究了當(dāng)標(biāo)的股票價(jià)格由混合分?jǐn)?shù)布朗運(yùn)動(dòng)驅(qū)動(dòng),且支付固定交易費(fèi)用時(shí)歐式回望期權(quán)的定價(jià)問(wèn)題。首先運(yùn)用對(duì)沖原理得到該模型下歐式回望看跌期權(quán)價(jià)值所滿(mǎn)足的非線(xiàn)性偏微分方程及其邊界條件。然而求解非線(xiàn)性偏微分方程組的解析解相當(dāng)困難,又加上回望期權(quán)在到期日的執(zhí)行價(jià)格的不確定性,為此,通過(guò)變量替換將得到的偏微分方程進(jìn)行降維,然后通過(guò)對(duì)變換后的新方程構(gòu)造Crank-Nicols格式來(lái)其數(shù)值解。最后討論該數(shù)值格式的收斂性,交易費(fèi)比率、Hurst指數(shù)等對(duì)期權(quán)價(jià)值的影響。
[Abstract]:The return option is a strong path dependent option, whose return on the maturity date depends on the maximum or minimum value of the risk asset during the expiration period of the option, which makes the return on the delivery date relatively high, and the price is very expensive. More research significance. At present, almost all the researches in academic circles are based on the hypothesis of geometric Brownian motion, however, in the actual financial transactions, The distribution of return rate of assets shows the characteristics of "peak and thick tail" and the intermittent "jump" of asset price, which makes the research under geometric Brownian motion inconsistent with the actual situation. Therefore, in this paper, the pricing problem of European lookback put options is studied under the three models of mixed fractional Brownian motion, mixed jump diffusion fractional Brownian motion and mixed fractional Brownian motion with transaction costs. The main results are as follows: 1) in this paper, we study the pricing of European lookback options under the mixed fractional Brownian motion model. By using the Ito formula and the risk hedging method, the differential equation and the boundary conditions of the option price under the model are obtained. Finally, the numerical solution of the differential equation is obtained by the finite difference method. The validity of the numerical solution is verified by an example. The problem of the change of financial market is studied. The Poisson process is added on the basis of the previous research. The pricing model of European lookback put options under mixed hopping-diffusion fractional Brownian motion is established. Firstly, by using Ito formula and risk hedging method, the integral equation of option price under this model is obtained, and the integral equation is transformed into partial differential equation by Taylor expansion. Then the partial differential equations are shown and implicit respectively, and the corresponding iterative equations are obtained. Finally, the partial differential equation is reduced in dimension, the explicit discrete scheme is constructed for the transformed mathematical model, and its stability and compatibility are analyzed theoretically. Finally, the numerical analysis of the scheme is carried out by using Matlab software. This paper discusses the influence of market parameters on the value of options. We study the pricing problem of European lookback options when the underlying stock price is driven by a mixed fractional Brownian motion and pays a fixed transaction cost. Firstly, the nonlinear partial differential equation and its boundary conditions for the value of European lookback put options under this model are obtained by using the hedging principle. However, it is very difficult to solve the analytical solution of nonlinear partial differential equations and the uncertainty of the executive price of the lookback option on the maturity date. Therefore, the dimension of the partial differential equation will be reduced by replacing the variables. Then the Crank-Nicols scheme is constructed for the transformed new equation to obtain its numerical solution. Finally, the convergence of the numerical scheme and the influence of the transaction cost ratio and Hurst exponent on the option value are discussed.
【學(xué)位授予單位】：中國(guó)礦業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類(lèi)號(hào)】：F224;F830.9

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