摘要
经典金融学的核心是金融资产定价,而对金融衍生品进行合理的定价是研究的主要内容,也是金融数学最基本和最重要的研究领域之一。作为期权定价里程碑的Black-Scholes-Merton公式自1976年问世以来就得到了广泛的认可,Black和Merton也因为这个奠基性的工作于1997年获得了诺贝尔经济学奖。但是这个公式赖以成立的一个重要假设是标的资产服从几何布朗运动,然而大量的实证研究发现,标的资产在绝大多数情况下并不符合几何布朗运动的特性,而与分数几何布朗运动的特性相符合。为此很多学者提出用分数布朗运动来代替布朗运动。本文从三个方面讨论了Ito型分数金融市场下的期权定价问题。
●第一个方面是非完备市场中的期权定价问题。我们以带比例交易成本的期权定价问题为例,应用分数布朗运动随机积分理论和偏微分方程方法推导出了分数布朗运动驱动下带交易成本的欧式期权定价问题,得到了欧式期权价格的显式解。并证明了欧式期权看涨一看跌的平价公式,得到了与标准布朗运动条件下类似的一系列公式。作为本部分的结束,我们还考虑了带比例交易成本的永久美式看跌期权的定价问题,给出了它的显式定价公式,讨论了Hurst指数对期权价格的影响。
●第二个方面是随机利率下的期权定价问题。我们以利率服从分数Vasicek随机利率模型为例,讨论了期权定价问题。在假定标的资产价格和利率的运动过程服从几何分数布朗运动的条件下,利用风险对冲技术、分数布朗运动随机分析理论与偏微分方程方法,得到了分数Vasicek随机利率下欧式期权所满足的定价方程,获得了标的资产价格波动率是时间函数的情形下欧式看涨和看跌期权的一般定价公式以及它们的平价公式。
●第三个方面是跳-扩散模型下的期权定价问题。在这个问题中,我们利用复合泊松过程来刻划标的资产的随机跳跃,并且假设扩散过程是一个分数布朗运动。我们运用测度变换技巧和拟鞅定价方法,得到了欧式看涨期权定价的显式公式。
最后,我们将以上讨论得到的结果应用到实际的金融市场中。在当前CPI不断走高,“负利率”倒挂现象愈发突出、物价不断上涨的背景下,我们有针对性选择在当前国内金融市场上受到热捧的一些金融产品,如保本基金、可转换债券等作为我们的研究对象,并且对这两个结构性金融产品做了有关条款设计和定价机制方面的讨论。我们还考虑了结构化模型下的信用风险建模问题。
Financial asset pricing is the core issue of classical finance. The pricing theory of financial derivatives is the main content of financial asset pricing, also it is one of the most fundamental and substantial areas in mathematical finance. Option pricing theory has a long and illustrious history, but it also underwent a revolutionary change in 1973. The break-through in option valuation theory started with the publication of two seminal papers by Black&Scholes and Merton.
The Black-Scholes model has become the most popular method for option pricing and its generalized version has provided mathematically beautiful and powerful results on option pricing. Nevertheless, classical mathematical models of financial assets are far from perfect. One apparent problem exists in the Black-Scholes formulation, namely that financial processes are not Markovian in distribution. In fact, behavioral finance as well as empirical studies shows that there exists long-range dependence in stock returns and verifies that long-range dependence is one of the genuine features of financial markets. Behavioral finance also suggests the return distributions of stocks are leptokurtic and have longer and fatter tails than normal distribution and there exists long-range dependence in stock returns. These features have some differences with the standard brown motion, while are in accordance with the fractional brown motion. The fractional Black-Scholes model is a generalization of the Black-Scholes model, which is based on replacing the standard Brownian motion by a fractional Brownian motion in the Black-Scholes model.
This thesis is devoted to the financial derivatives pricing problem in a fractional ltd type financial market, and we want to establish the mathematical model for the financial market in fractional Brownian motion setting, by assuming the underlying asset price obey-ing the stochastic differential equation driven by fractional Brownian motion. Three topics are studied in this thesis.
·The first topic is the option pricing problem in incomplete markets. We focused on option pricing with proportional transaction costs. The problem is completely solved using the fractional Brownian motion theory and PDE approach, and general pricing formula for the European option with transaction costs is derived. Meanwhile, we get the explicit expression for the European option price with transaction costs and the call-put parity. The perpetual American put option pricing problem is also considered.
·The second topic is the option pricing problem when the risk-free interest rate is stochastic. In this part, we take fractional Vasicek model as an example of stochastic inter-est rate. We establish the mathematical model for the financial market in fractional Brown-ian motion setting. Using the risk hedge technique, fractional stochastic analysis and PDE method, we obtain the general pricing formula for the European option with stochastic in-terest rate. At the same time, we get a explicit expression for European option price with stochastic interest rate and the call-put parity. As we will show, the results in this part extend as well as improve previously known results.
·The third topic is the option pricing problem when underlying asset returns are dis-continuous. In this problem. We use compound Poisson process to characterize the jump, and we assume the underlying asset is driven by a mixture of both continues and jump pro-cesses, where we characterize the continues part by a fractional Browian motion. We call the process as the fractional jump-diffusion model. Using measure transformation technol-ogy and quasi-martingale approach, we derive an option pricing formula under fractional jump-diffusion model.
Finally, we apply these results to actual financial markets, including segregated funds pricing problem, convertible bonds pricing problem and credit risk modeling problem.
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