状态转换环境下期权定价及其应用研究

发布时间：2018-03-14 19:21

本文选题：状态转换　切入点：期权定价　出处：《华南理工大学》2013年硕士论文　论文类型：学位论文

【摘要】：期权是金融市场最重要的金融衍生品之一，它赋予了持有者以约定的价格和时间交易商品或者证券的权利。期权被广泛应用到了套期保值、投资组合构造、公司员工激励、兼并重组等实践应用中，是推动金融创新的重要力量。我国对金融创新日益重视，期权产品在我国有广泛的应用前景。金融衍生品定价，特别是期权定价，是近百年金融学术研究的热点问题。自1973年Black-Scholes期权定价理论面世以来，现代期权定价理论已经发展成为了金融工程的一个重要分支。然而，由于真实金融市场的复杂性，期权定价模型依然存在一些缺陷和不足，仍需进一步发展完善。金融市场不仅存在长记忆性和模糊性，还存在不同市场状态的相互交替，如股票市场中“牛市”和“熊市”的更替。状态转换模型是刻画金融市场状态转换的有效方式，本文在前人研究的基础上运用随机方法和数值方法进一步研究状态转换环境下期权定价问题，并将状态转换下期权定价理论应用到可转债定价研究中，旨在完善和扩展状态转换期权定价理论。为此，本文研究内容和结论主要包括：首先，本文将几何布朗运动下最小二乘蒙特卡罗模拟法引入到状态转换几何布朗运动驱动的美式期权定价中，构造了状态转换下美式期权的最小二乘蒙特卡罗模拟方法并给出了具体的算法步骤。将状态转换模型驱动的普通美式看跌期权三叉树方法、有限差分方法（Crank-Nicolson法）、普通最小二乘模拟、改进最小二乘模拟四种方法的定价结果和计算耗时等进行比较分析。比较结果表明，状态转换最小二乘蒙特卡罗模拟方法有较高的准确度，而引入拟蒙特卡罗技术、随机数重排、对偶技术等可以降低模拟结果的方差。虽然最小二乘模拟在计算效率上不具优势，但却能够方便地处理具有美式特征的复杂期权。其次，考虑到真实金融市场存在的长记忆性，本文同时考虑了股票市场的状态转换特性和长记忆性，建立了状态转换分数布朗运动驱动欧式期权定价模型。本部分首先推导了状态转换分数Ito公式和基于Esscher变换的等价鞅测度，并基于此推导得到了状态转换混合分数布朗运动驱动的欧式期权价值的Black-Scholes公式和Black-Scholes偏微分方程。本文还介绍了基于Black-Scholes偏微分方程的有限差分方法用于求解期权价值。通过数值算例和分析表明，状态转换混合分数布朗运动下欧式期权价值受马尔科夫生成矩阵即状态转换程度的影响非常显著，，而Hurst指数对期权价格的影响还依赖于到期时间。最后，我们还将状态转换混合分数几何布朗运动驱动的欧式期权定价模型应用到了欧式股本权证定价问题的建模中。最后，将状态转换下美式期权定价理论和数值算法引入到具有美式期权特征的可转债定价中，研究含违约风险、股权稀释作用和债务杠杆作用情况下状态转换可转债定价问题。首先推导了状态转换驱动下可转债的Black-Scholes偏微分方程，然后探讨了可转债的股权稀释效应和债务杠杆作用，并建立了可转债定价的有限差分方法、三叉树方法等数值算法。数值算例表明，三叉树方法和有限差分方法能较好计算可转债价值且各有优缺点，状态转换强度、违约强度等对可转债价值有显著影响。
[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.

【学位授予单位】：华南理工大学
【学位级别】：硕士
【学位授予年份】：2013
【分类号】：F830.9;F224

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