最优增长投资组合与等价鞅测度之间的关系

本文研究了当基本价格过程为一类连续半鞅过程时log最优的自融资投资组合的财富过程与等价鞅测度之间的对应关系。结果显示在连续过程框架下，最小鞅测度就是相对熵最小的等价鞅测度。     

可积鞅测度的弱收敛

本文引入了可积鞅测度弱收敛的概念，并给出了可积鞅测弱收敛的一系列条件。     

马尔可夫交换Lévy过程的最小熵鞅测度

研究了由马尔可夫交换Lévy过程的随机指数所驱动的风险资产的期权定价问题,即市场的利率、风险资产的波动率以及N个状态的补偿子都依赖于不可观的经济状态,而这些经济状态服从于一个连续时间的隐马氏链模型.一般地,由马尔可夫交换Levy过程的随机指数所描述的市场是不完备的,因此,鞅测度不是唯一的.通过采用状态转换Esscher变换来确定等价鞅测度,并且证明了所得到的定价测度就是最小熵鞅测度.     

复测度B值鞅的原子分解

本文利用停时方法给出复测度B值鞅的原子分解，讨论了复测度B值鞅空间的包含关系．此结果是非负测度鞅的推广．     

复测度鞅变换的收敛性及其应用

在满足ｂ_∞~（Ｋ）∩ａ_１（Ｋ）条件的情况下,讨论了关于复测度ｄμ＝ωｄν的鞅变换,证明了复测度鞅变换的几乎处处收敛性定理。并且,作为该定理的一个应用,对复测度鞅的点态收敛性作了较精细的讨论。     

随机市场系数的M-V最优投资组合选择:一个鞅方法

通过引进凹函数U(x)以及等价鞅测度p^-，应用鞅的性质得到了随机市场系数情形下的M—V模型的最优投资策略以及有效前沿．     

一类特殊半鞅模型中无套利测度的刻划

本文研究金融市场中一类特殊半鞅模型,其价格过程具有X=LD的形式,这里L是局部有界鞅,D是可料有限变差过程.对这类模型我们导出其等价鞅测度存在的充分必要条件.另外,我们将[2]中的条件/△M/≤C推广到M为局部有界鞅,得到相应的结果.     

多叉树模型中鞅测度的刻画与构造

在无套利假设下,讨论了多叉树模型中鞅测度的构造问题.利用二叉树方法,构造了有限个符号测度.证明了-个概率测度为鞅测度的充要条件是它可以表示为这组符号测度的某个满足特定条件的凸线性组合.     

复测度拟鞅的若干性质

本文讨论了复测度拟鞅的若干性质.利用复测度鞅的相关结果,证明了关于复值函数Ψ的条件下,复测度拟鞅的弱型不等式及复测度拟鞅变换的收敛性.     

复测度鞅空间及其对偶

本文在Ψ满足（Ｋ）条件或　　　的条件下，讨论了关于复测度ｄμ＝Ψｄｖ的鞅空间　　　　和ａＫ＿ｐ．证明了它们之间的等价性以及与关于非负测度ｄｖ的相应鞅空间的同构性，给出了它们的对偶空间，特别地Ｈ１＝ＢＭＯ，最后证明了关于复测度软的均方算子Ｓ（ｆ）是弱（１，１）有界的。     

Optimal martingale measures for defaultable assets

We model a defaultable asset as solution to a stochastic differential equation driven by both a Brownian motion and the counting process martingale associated to the one-jump process. We discuss in this framework the minimal entropy martingale measure as well as the linear Esscher and the minimal martingale measure. In particular we deal with some rather delicate verification issues.     

The relative entropy in CGMY processes and its applications to finance

The CGMY market model generates infinite equivalent martingale measures (EMM). In order to price options, we need an adequate method to choose one EMM. This paper presents the relative entropy for CGMY processes, and apply it to choosing an EMM called the model preserving minimal entropy martingale measure.      

Application of Moore-Penrose inverse in deciding the minimal martingale measure

The Moore-Penrose inverse is an important tool in algebra.This paper shows that the MoorePenrose inverse is also an effcient technique in determining the minimal martingale measure if a security price follows a semi-martingale which satisfies some structure condition.We extend a result of Dzhaparidze and Spreij concerning the Moore-Penrose inverse to the case that the Moore-Penrose inverse of any matrix-valued predictable process is still predictable.Furthermore,we obtain an explicit formula of the minimal martingale measure by employing the Moore-Penrose inverse.Specifically,the minimal martingale measure in a generalized Black-Scholes model is found.     

有限状态多期模型下的最小熵等价鞅测度及其应用

采用有限状态多期模型描述股票价格变动过程,导出了有红利支付情形下的最小熵等价鞅测度,给出了股票价格变动趋势的风险中性预期与红利率和无风险利率之间相对大小的关系,从理论上证明了无风险利率大于股票红利率时,市场将呈现出一种向上的风险中性趋势;无风险利率小于股票红利率时,市场将呈现出一种向下的风险中性趋势;无风险利率等于红利率时,股票价格将围绕初始价格上下波动而没有明显的风险中性趋势.     

A note on convergence of option prices and their Greeks for Lévy models

We study the robustness of option prices to model variation after a change of measure where the measure depends on the model choice. We consider geometric Lévy models in which the infinite activity of the small jumps is approximated by a scaled Brownian motion. For the Esscher transform, the minimal entropy martingale measure, the minimal martingale measure and the mean variance martingale measure, we show that the option prices and their corresponding deltas converge as the scaling of the Brownian motion part tends to zero. We give some examples illustrating our results.     

On value preserving and growth optimal portfolios

In a discrete-time financial market setting, the paper relates various concepts introduced for dynamic portfolios (both in discrete and in continuous time). These concepts are: value preserving portfolios, numeraire portfolios, interest oriented portfolios, and growth optimal portfolios. It will turn out that these concepts are all associated with a unique martingale measure which agrees with the minimal martingale measure only for complete markets.     

On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps

We compute and then discuss the Esscher martingale transform for exponential processes, the Esscher martingale transform for linear processes, the minimal martingale measure, the class of structure preserving martingale measures, and the minimum entropy martingale measure for stochastic volatility models of the Ornstein–Uhlenbeck type as introduced by Barndorff-Nielsen and Shephard. We show that in the model with leverage, with jumps both in the volatility and in the returns, all those measures are different, whereas in the model without leverage, with jumps in the volatility only and a continuous return process, several measures coincide, some simplifications can be made and the results are more explicit. We illustrate our results with parametric examples used in the literature.     

The Minimal Entropy Martingale Measure and Numerical Option Pricing for the Barndorff–Nielsen–Shephard Stochastic Volatility Model

Abstract

We develop and apply a numerical scheme for pricing options in the stochastic volatility model proposed by Barndorff–Nielsen and Shephard. This non-Gaussian Ornstein–Uhlenbeck type of volatility model gives rise to an incomplete market, and we consider the option prices under the minimal entropy martingale measure. To numerically price options with respect to this risk neutral measure, one needs to consider a Black and Scholes type of partial differential equation, with an integro-term arising from the volatility process. We suggest finite difference schemes to solve this parabolic integro-partial differential equation, and derive appropriate boundary conditions for the finite difference method. As an application of our algorithm, we consider price deviations from the Black and Scholes formula for call options, and the implications of the stochastic volatility on the shape of the volatility smile.     

Minimal relative entropy for equivalent martingale measures by low-discrepancy sequence in Lévy process

In this paper, we focused on computing the minimal relative entropy between the original probability and all of the equivalent martin gale measure for the Lévy process. For this purpose, the quasiMonte Carlo method is used. The probability with minimal relative entropy has many suitable properties. This probability has the minimal Kullback-Leibler distance to the original probability. Also, by using the minimal relative entropy the exponential utility indifference price can be found. In this paper, the Monte Carlo and quasi-Monte Carlo methods have been applied. In the quasi-Monte Carlo method, two types of widely used lowdiscrepancy sequences, Halton sequence and Sobol sequence, are used. These methods have been used for exponential Lévy process such as variance gamma and CGMY process. In these two processes, the minimal relative entropy has been computed by Monte Carlo and quasi-Monte Carlo, and compared their results. The results show that quasi-Monte Carlo with Sobol sequence performs better in terms of fast convergence and less error. Finally, this method by fitting the variance gamma model and parameters estimation for the model has been implemented for financial data and the corresponding minimal relative entropy has been computed.