英文标题：
《Pricing of high-dimensional options》
---
作者：
Alexander Kushpel
---
最新提交年份：
2015
---
英文摘要：
  Pricing of high-dimensional options is one of the most important problems in Mathematical Finance. The objective of this manuscript is to present an original self-contained treatment of the multidimensional pricing. During the past decades the Black-Scholes this model, which essentially is based on the log-normal assumption, has been increasingly criticised. In particular, it was noticed by Mandelbrot that empirical log-returns distributions are more concentrated around the origin and have considerably heavier tails. This suggests to adjust the Black-Scholes model by the introduction of the Levy processes instead of Brownian ones. This approach has been extensively studied in a univariate setup since the nineties. In the multivariate settings the theory is not so advanced. We present a general method of high-dimensional option pricing based on a wide range of jump-diffusion models. Namely, we construct approximation formulas for the price of spread options. It is important to get an efficient approximation for the respective density function, since the reward function has usually a simple structure. Instead of a commonly used tabulation approach, we use the respective m-widths to compare a wide range of numerical methods. We give an algorithm of almost optimal, in the sense of the respective m-widths, reconstruction of density functions. To demonstrate the power of our approach we consider in details a concrete class of Levy driven processes and present the respective rates of convergence of approximation formulas. The interrelationship between the theory and tools reflects the richness and deep connections in Financial Mathematics, Stochastic Processes, Theory of Martingales, Functional Analysis, Topology and Harmonic Analysis.
---
中文摘要：
高维期权的定价是数学金融学中最重要的问题之一。这篇手稿的目的是呈现一种原始的、独立的多维定价方法。在过去几十年中，Black-Scholes模型基本上基于对数正态假设，受到了越来越多的批评。特别是，Mandelbrot注意到，经验对数收益率分布更集中在原点周围，并且有相当大的尾部。这建议通过引入Levy过程而不是布朗过程来调整Black-Scholes模型。自20世纪90年代以来，这种方法在单变量系统中得到了广泛的研究。在多元环境下，这一理论并不那么先进。本文提出了一种基于跳扩散模型的高维期权定价方法。也就是说，我们构造了价差期权价格的近似公式。获得相应密度函数的有效近似值很重要，因为奖励函数通常具有简单的结构。与常用的制表方法不同，我们使用各自的m宽度来比较各种数值方法。我们给出了一个密度函数的近似最优重建算法。为了证明我们的方法的有效性，我们详细考虑了一类具体的Levy驱动过程，并给出了近似公式各自的收敛速度。该理论与工具之间的相互关系反映了金融数学、随机过程、鞅理论、泛函分析、拓扑与调和分析的丰富性和深刻联系。
---
分类信息：

一级分类：Quantitative Finance        数量金融学
二级分类：Mathematical Finance        数学金融学
分类描述：Mathematical and analytical methods of finance, including stochastic, probabilistic and functional analysis, algebraic, geometric and other methods
金融的数学和分析方法，包括随机、概率和泛函分析、代数、几何和其他方法
--
一级分类：Mathematics        数学
二级分类：Numerical Analysis        数值分析
分类描述：Numerical algorithms for problems in analysis and algebra, scientific computation
分析和代数问题的数值算法，科学计算
--
一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
--
一级分类：Quantitative Finance        数量金融学
二级分类：Pricing of Securities        证券定价
分类描述：Valuation and hedging of financial securities, their derivatives, and structured products
金融证券及其衍生产品和结构化产品的估值和套期保值
--

---
PDF下载：
-->