本文選題：狀態(tài)轉(zhuǎn)換　切入點(diǎn)：期權(quán)定價　出處：《華南理工大學(xué)》2013年碩士論文　論文類型：學(xué)位論文

【摘要】：期權(quán)是金融市場最重要的金融衍生品之一，它賦予了持有者以約定的價格和時間交易商品或者證券的權(quán)利。期權(quán)被廣泛應(yīng)用到了套期保值、投資組合構(gòu)造、公司員工激勵、兼并重組等實(shí)踐應(yīng)用中，是推動金融創(chuàng)新的重要力量。我國對金融創(chuàng)新日益重視，期權(quán)產(chǎn)品在我國有廣泛的應(yīng)用前景。 金融衍生品定價，特別是期權(quán)定價，是近百年金融學(xué)術(shù)研究的熱點(diǎn)問題。自1973年Black-Scholes期權(quán)定價理論面世以來，現(xiàn)代期權(quán)定價理論已經(jīng)發(fā)展成為了金融工程的一個重要分支。然而，由于真實(shí)金融市場的復(fù)雜性，期權(quán)定價模型依然存在一些缺陷和不足，仍需進(jìn)一步發(fā)展完善。金融市場不僅存在長記憶性和模糊性，還存在不同市場狀態(tài)的相互交替，如股票市場中“牛市”和“熊市”的更替。狀態(tài)轉(zhuǎn)換模型是刻畫金融市場狀態(tài)轉(zhuǎn)換的有效方式，本文在前人研究的基礎(chǔ)上運(yùn)用隨機(jī)方法和數(shù)值方法進(jìn)一步研究狀態(tài)轉(zhuǎn)換環(huán)境下期權(quán)定價問題，并將狀態(tài)轉(zhuǎn)換下期權(quán)定價理論應(yīng)用到可轉(zhuǎn)債定價研究中，旨在完善和擴(kuò)展?fàn)顟B(tài)轉(zhuǎn)換期權(quán)定價理論。為此，本文研究內(nèi)容和結(jié)論主要包括： 首先，本文將幾何布朗運(yùn)動下最小二乘蒙特卡羅模擬法引入到狀態(tài)轉(zhuǎn)換幾何布朗運(yùn)動驅(qū)動的美式期權(quán)定價中，構(gòu)造了狀態(tài)轉(zhuǎn)換下美式期權(quán)的最小二乘蒙特卡羅模擬方法并給出了具體的算法步驟。將狀態(tài)轉(zhuǎn)換模型驅(qū)動的普通美式看跌期權(quán)三叉樹方法、有限差分方法（Crank-Nicolson法）、普通最小二乘模擬、改進(jìn)最小二乘模擬四種方法的定價結(jié)果和計(jì)算耗時等進(jìn)行比較分析。比較結(jié)果表明，狀態(tài)轉(zhuǎn)換最小二乘蒙特卡羅模擬方法有較高的準(zhǔn)確度，而引入擬蒙特卡羅技術(shù)、隨機(jī)數(shù)重排、對偶技術(shù)等可以降低模擬結(jié)果的方差。雖然最小二乘模擬在計(jì)算效率上不具優(yōu)勢，但卻能夠方便地處理具有美式特征的復(fù)雜期權(quán)。 其次，考慮到真實(shí)金融市場存在的長記憶性，本文同時考慮了股票市場的狀態(tài)轉(zhuǎn)換特性和長記憶性，建立了狀態(tài)轉(zhuǎn)換分?jǐn)?shù)布朗運(yùn)動驅(qū)動歐式期權(quán)定價模型。本部分首先推導(dǎo)了狀態(tài)轉(zhuǎn)換分?jǐn)?shù)Ito公式和基于Esscher變換的等價鞅測度，并基于此推導(dǎo)得到了狀態(tài)轉(zhuǎn)換混合分?jǐn)?shù)布朗運(yùn)動驅(qū)動的歐式期權(quán)價值的Black-Scholes公式和Black-Scholes偏微分方程。本文還介紹了基于Black-Scholes偏微分方程的有限差分方法用于求解期權(quán)價值。通過數(shù)值算例和分析表明，狀態(tài)轉(zhuǎn)換混合分?jǐn)?shù)布朗運(yùn)動下歐式期權(quán)價值受馬爾科夫生成矩陣即狀態(tài)轉(zhuǎn)換程度的影響非常顯著，，而Hurst指數(shù)對期權(quán)價格的影響還依賴于到期時間。最后，我們還將狀態(tài)轉(zhuǎn)換混合分?jǐn)?shù)幾何布朗運(yùn)動驅(qū)動的歐式期權(quán)定價模型應(yīng)用到了歐式股本權(quán)證定價問題的建模中。 最后，將狀態(tài)轉(zhuǎn)換下美式期權(quán)定價理論和數(shù)值算法引入到具有美式期權(quán)特征的可轉(zhuǎn)債定價中，研究含違約風(fēng)險、股權(quán)稀釋作用和債務(wù)杠桿作用情況下狀態(tài)轉(zhuǎn)換可轉(zhuǎn)債定價問題。首先推導(dǎo)了狀態(tài)轉(zhuǎn)換驅(qū)動下可轉(zhuǎn)債的Black-Scholes偏微分方程，然后探討了可轉(zhuǎn)債的股權(quán)稀釋效應(yīng)和債務(wù)杠桿作用，并建立了可轉(zhuǎn)債定價的有限差分方法、三叉樹方法等數(shù)值算法。數(shù)值算例表明，三叉樹方法和有限差分方法能較好計(jì)算可轉(zhuǎn)債價值且各有優(yōu)缺點(diǎn)，狀態(tài)轉(zhuǎn)換強(qiáng)度、違約強(qiáng)度等對可轉(zhuǎn)債價值有顯著影響。
[Abstract]:Option is one of the most important financial derivatives in financial markets, it gives the holder of the agreed price and time of goods or securities trading rights. The option is widely applied to the hedging portfolio, the company employees incentive, mergers and acquisitions and other practical applications, is an important force to promote the financial innovation of China's growing importance. The financial innovation option and hasbroad application prospect in our country.
Pricing of financial derivatives, especially the option pricing, is a hot issue in academic research in finance of nearly a hundred years since the launch in 1973 Black-Scholes option pricing theory, modern option pricing theory has become an important branch of financial engineering. However, because of the complexity of the real financial market, option pricing model still has some defects and shortcomings still, to be further developed. The financial market not only has long memory and fuzziness, alternating with each other there are different market conditions, such as the stock market "bull" and "bear market" for more. State transition model is the effective way to describe the financial market transition, based on the previous research by random method and numerical method for further study the option pricing problem under the environment of state transition, and the state transition under the option pricing theory is applied to the pricing of convertible bonds. The purpose of this study is to improve and expand the pricing theory of state transition options.
First of all, the American option pricing of geometric Brown motion under the Least Squares Monte Carlo simulation method is introduced to the state transition of geometric Brown motion driven, construct the Least Squares Monte Carlo simulation under the state of American option conversion method and gives the specific steps of the algorithm. The state transition of ordinary American option trinomial tree model driven method, finite difference method (Crank-Nicolson method), the ordinary least squares simulation, four ways of improving the pricing results and computation time were compared. The comparison results show that the least squares simulation, state transition Least Squares Monte Carlo simulation method has higher accuracy, and the introduction of Quasi Monte Carlo method, random number rearrangement, dual technology can reduce the variance of the results. Although the least squares simulation in computational efficiency does not have the advantage, but it can easily deal with American Complex options with features.
Secondly, considering the long memory of the real financial market, this paper also considers the stock market state conversion characteristics and long memory, established the model of European option pricing driven state transition fractional Brown motion. This part firstly deduces the state conversion fraction of Ito formula and the equivalent martingale measure based on Esscher transform, and based on this derivation the Black-Scholes formula and the Black-Scholes state transition of European option value of mixed fractional Brown motion driven by partial differential equations. This paper also introduces the finite difference Black-Scholes based on partial differential equation method is used to solve the option value. Through numerical examples and analysis show that the state transition mixed fractional Brown motion under the European option value by Markov matrix is generated the influence of state transition is very significant, while the influence of Hurst index on the option price also depends on the maturity Finally, we apply the European option pricing model with state transition and mixed fractional geometric Brown motion to the modeling of European equity warrants pricing.
Finally, the state transition under the American option pricing theory and numerical algorithm is introduced to the American option with the characteristics of the pricing of convertible bonds, convertible bonds with default risk, the pricing problem of transition state equity dilution and debt leverage situation. Firstly, the state transfer under the drive of convertible bonds Black-Scholes partial differential equation, and then explore the convertible bond dilution effect and debt leverage, and established the finite difference method of pricing of convertible bonds, numerical algorithm of the trinomial tree method. Numerical examples show that the trinomial tree method and finite difference method can calculate the value of convertible bonds and each has advantages and disadvantages, state transition strength, strength of breach of contract can significantly affect the value of the bonds.

【學(xué)位授予單位】：華南理工大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2013
【分類號】：F830.9;F224

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