分形布朗运动下有交易成本的外汇期权定价

研究了有交易成本的分形Black-Scholes外汇期权定价问题.基于汇率的分形布朗运动分布假设,运用分形布朗运动的性质和随机微积分方法,得到了欧式外汇期权价格所满足的偏微分方程.最后,建立离散时间条件下的非线性期权定价模型,并且通过解期权价格的偏微分方程给出了有交易成本的欧式外汇期权定价公式.     

Black-Scholes微分方程的必要条件(Ⅰ)

我们知道,著名的Ｂｌａｃｋ－Ｓｃｈｏｌｅｓ微分方程是根据资产价格行为服从对数布朗运动导出来的,假设资产价格Ｓ服从对数布朗运动,即有这里Ｗ为标准布朗运动,σ为Ｓ的波动率,ｒ为无风险收益率,σ和ｒ均为常数．欧式看涨期权的价格函数　（ｔ,ｘ）则满足这里而Ｔ为有效期限,Ｋ为敲定价格,ｂ．,ｂ＊分别为期权敲出的资产价格对数下限、上限（只要资产价格对数高于ｂ＊或低于ｂ．,就把期权敲出）．由此可知,式（１）是Ｂｌａｃｋ－Ｓｃｈｏｌｅｓ微分方程（式（２））成立的充分条件．现在．我们通过分析式（２）的性质,来探索…     

债券定价及利率风险的市场价格的重构方法

假设利率变化的模型是由随机微分方程给出,则可以用推导Black-Scholes方程的方法来推出债券价格满足的偏微分方程,得到一个抛物型的偏微分方程.但是,在债券定价的方程中隐含有一个参数λ称为利率风险的市场价格.所谓债券定价的反问题,就是由不同到期时间的债券的现在价格来得到利率风险的市场价格.对随机利率模型下债券定价的正问题先给予介绍和差分数值求解方法,并介绍了反问题,且对反问题给出了数值方法.     

跳-扩散价格过程下有交易成本的期权定价研究

Black-Scholes模型成功解决了完全市场下的欧式期权定价问题.研究在不完全市场下的一类期权定价问题,即在假设交易过程有交易成本且标的资产价格服从跳-扩散过程下,推导出了在该模型下期权价格所满足的微分方程.     

一类跳跃扩散型股价过程组欧式未定权益定价

本文仅讨论一种类型的证券市场模型，其d种股票的价格过程满足一特殊的跳跃扩散型随机微分方程组，即市场风险源的个数与市场风险证券的个数相同，文章给出了这一模型下相应的跳跃扩散型倒向随机微分方程组适应解的存在唯一性定理及联系于我股票价格过程欧式未定权益（简记ECC）定价的基本公式，最后在常系数条件下导出了一种特殊形式欧式未定权益定价的Black-Scholes公式。     

美式看跌期权定价的一种混合数值方法

本文对美式看跌期权的定价提供了一种新的混合数值方法,即快速傅里叶变换法加龙格-库塔法.首先将美式看跌期权价格所满足的Black-Scholes微分方程定解问题转化为一个标准的抛物型初、边值问题,然后通过傅里叶变换,使之转换为一个不带股价变量的常微分方程初值问题,再利用龙格-库塔法对其进行数值求解.数值实验表明,本文算法是一种快速的高精度的算法.     

随机利率下分数跳-扩散Ornstein-Uhlenbeck期权定价模型

假设股票价格遵循分数布朗运动和复合泊松过程驱动的随机微分方程,短期利率服从HullWhite模型,建立了随机利率情形下的分数跳-扩散Ornstein-Uhlenbeck期权定价模型,利用价格过程的实际概率测度和公平保费原理,得到了欧式看涨期权定价的解析表达式,推广了Black-Scholes模型.     

具有违约风险的人寿保险的最优定价

本文探讨具有违约风险的人寿保险的最优定价.我们从Black-Scholes的期权定价模型出发,考虑风险管理和准备金的要求,根据一次支付和均衡支付这两种不同的假设分别建立两个优化模型,并且借助于优化技术获得最优解.数量化分析结果表明,两个模型的最优价格对于利息率参数以及非索赔成本的变化都不敏感.这说明这两个模型是稳定的,而且是实用的.     

股价最简微分方程,其解及与Black-Scholes模型的假设的相互关系

在系数的某种等价关系条件下,股价的两类数学表达式,一类是基于明确型描述的由类似固体力学方法导出的最简微分方程(S.D.E.)的解,另一类是基于不确定型描述(即统计理论)的Black-Scholes模型的假设(A.B-S.M.),即股价密度函数服从对数正态分布,可以是完全相同的.S.D.E.的解仅适用于股市的常规情形(无利好或利空消息,等),因此,A.B-S.M.的适用范围也相同.     

分数Black-Scholes模型下美式期权定价的积分方程式

为得到分数Black-Scholes模型下美式期权价格的公式,文章以看涨期权为例,应用偏微分方程法,推导期权价格的积分方程式.由于美式期权的价格可分解为欧式期权的价格和由于提前实施需要增付的期权金,而提前实施期权金与最佳实施边界的位置有关,所以为导出最佳实施边界所满足的方程,文章首先研究分数Black-Scholes方程的基本解,然后建立美式看涨期权的分解公式,推导最佳实施边界适合的非线性积分方程,从而得到美式看涨期权价格的积分方程式.美式看跌期权价格的积分方程式类似得到.     

Valuation of futures options with initial margin requirements and daily price limit

The paper presents a valuation model of futures options trading at exchanges with initial margin requirements and daily price limit, and this result gives an academic guidance to design trading rules at exchanges. Unlike the leading work of Black, certain trading rules are considered so as to be more fit for practical futures markets. The paper prices futures options with initial margin requirements and daily price limit by duplicating them with the help of the theory of backward stochastic differential equations （BSDEs, for short）. Furthermore, an explicit expression of the price Of the call （or the put） futures option is given and also is shown to be the unique solution of the associated nonlinear partial differential equation.     

A class of portfolio selection with a four-factor futures price model

Considering the stochastic exchange rate, a four-factor futures model with the underling asset, convenience yield, instantaneous risk free interest rate and exchange rate, is established. These processes follow jump-diffusion processes (Weiner process and Poisson process). The corresponding partial differential equation (PDE) of the futures price is derived. The general solution of the PDE with parameters is drawn. The weight least squares approach is applied to obtain the parameters of above PDE. Variance is substituted by semi-variance in Markowitzs portfolio selection model. Therefore, a class of multi-period semi-variance model is formulated originally. Then, a continuous-time mean-variance portfolio model is also considered. The corresponding stochastic Hamilton-Jacobi-Bellman (HJB) equation of the problem with nonlinear constraints is derived. A numerical algorithm is proposed for finding the optimal solution in this paper. Finally, in order to demonstrate the effectiveness of the theoretical models and numerical methods, the fuel futures in Shanghai exchange market and the Brent crude oil futures in London exchange market are selected to be examples.     

金融衍生产品的力学方法分析（Ⅱ）—期权市场价格基本方程

类似固体力学建立基本方程方法，根据期权特点，采用一些假设，建立期权市场价格基本方程：hv0(t)=m1vo^-1(t)-n1vo(t) F,式中h,m1,n1,F为常数，主要假设有：期权市场价格vo(t)的升降由市场供求决定；影响v0(t)的因素如行使价，期限，波幅等用正或反比关系；买和卖用相反规律。文中给出不同情况下基本方程的解，并和期货市场价基本方程的解vf(t)相比较，以及用隐函数存在定理证明vf与v0(t)存在一一对应关系，为研究期货vf对期权价vo(t)的影响提供理论依据。     

跳跃-扩散模型资产定价公式的数值计算方法

假定资产价格变化过程服从跳跃-扩散过程,那么基于它的欧式期权就满足一个偏积分-微分方程（PIDE）,本文利用差分法来离散这个PIDE方程,用两种迭代方法得到方程的数值解：基于雅可比正则分裂法和预条件共轭梯度法.     

Contingent claim pricing using probability distortion operators: methods from insurance risk pricing and their relationship to financial theory

This paper considers the pricing of contingent claims using an approach developed and used in insurance pricing. The approach is of interest and significance because of the increased integration of insurance and financial markets and also because insurance-related risks are trading in financial markets as a result of securitization and new contracts on futures exchanges. This approach uses probability distortion functions as the dual of the utility functions used in financial theory. The pricing formula is the same as the Black-Scholes formula for contingent claims when the underlying asset price is log-normal. The paper compares the probability distortion function approach with that based on financial theory. The theory underlying the approaches is set out and limitations on the use of the insurance-based approach are illustrated. The probability distortion approach is extended to the pricing of contingent claims for more general assumptions than those used for Black-Scholes option pricing.     

An extension of the Clark–Haussmann formula and applications

ABSTRACT

This work considers a financial market stochastic model where the uncertainty is driven by a multidimensional Brownian motion. The market price of the risk process makes the transition between real world probability measure and risk neutral probability measure. Traditionally, the martingale representation formulas under the risk neutral probability measure require the market price of risk process to be bounded. However, in several financial models the boundedness assumption of the market price of risk fails; for example a financial market model with the market price of risk following an Ornstein–Uhlenbeck process. This work extends the Clark–Haussmann representation formula to underlying stochastic processes which fail to satisfy the standard requirements. Our methodology is classical, and it uses a sequence of mollifiers. Our result can be applied to hedging and optimal investment in financial markets with unbounded market price of risk. In particular, the mean variance optimization problem can be addressed within our framework.     

计算股市的基本方程、理论和原理(Ⅰ)——基本方程

本文采用网络模型和类似于固体力学的方法论来研究计算股市。建立四个基本的联立方程,即:利率-流通量方程;股票买入、卖出方程;股价变化率方程;以及利率、股价及股价变化率方程。文中着重讨论利率-流通量方程的解及其简单应用,包括时间离散化时股市网络用Banach收缩映射定理证明最终趋向平衡状态,以及银行减息引起资金流动按指数型式衰减等。     

A non-Gaussian Ornstein–Uhlenbeck model for pricing wind power futures

The recent introduction of wind power futures written on the German wind power production index has brought with it new interesting challenges in terms of modelling and pricing. Some particularities of this product are the strong seasonal component embedded in the underlying, the fact that the wind index is bounded from both above and below and also that the futures are settled against a synthetically generated spot index. Here, we consider the non-Gaussian Ornstein–Uhlenbeck type processes proposed by Barndorff-Nielsen and Shephard in the context of modelling the wind power production index. We discuss the properties of the model and estimation of the model parameters. Further, the model allows for an analytical formula for pricing wind power futures. We provide an empirical study, where the model is calibrated to 37 years of German wind power production index that is synthetically generated assuming a constant level of installed capacity. Also, based on 1 year of observed prices for wind power futures with different delivery periods, we study the market price of risk. Generally, we find a negative risk premium whose magnitude decreases as the length of the delivery period increases. To further demonstrate the benefits of our proposed model, we address the pricing of European options written on wind power futures, which can be achieved through Fourier techniques.     

Pricing catastrophe options in discrete operational time

We employ a doubly-binomial process as in Gerber [Gerber, H.U., 1988. Mathematical fun with the compound binomial process. ASTIN Bull. 18, 161-168] to discretize and generalize the continuous “randomized operational time” model of Chang et al. ([Chang, C.W., Chang, J.S.K., Yu, M.T., 1996. Pricing catastrophe insurance futures call spreads: A randomized operational time approach. J. Risk Insurance 63, 599-616] and CCY hereafter) from a complete-market continuous-time setting to an incomplete-market discrete-time setting, so as to price a richer set of catastrophe (CAT) options. For futures options, we derive the equivalent martingale probability measures by benchmarking to the shadow price of a bond to span arrival uncertainty, and the underlying futures price to span price uncertainty. With a time change from calendar time to the operational transaction-time dimension, we derive CCY as a limiting case under risk-neutrality when both calendar-time and transaction-time intervals shrink to zero. For a cash option with non-traded underlying loss index, we benchmark to the market reinsurance premiums to span claim uncertainty, and with a time change to claim time, we derive the cash option price as a binomial sum of claim-time binomial Asian option prices under the martingale measures.     

可违约债券在随机波动率假定下近似定价公式的求解

在假设标的资产价格的波动率是一个快速均值回复OU过程的函数的条件下,导出相应的可违约债券价格公式所应满足的偏微分方程,并利用Taylor级数展开得到一组Poisson方程.求解这些方程,得到非完全市场下固定补偿率的债券价格的近似表达式,然后在不同的补偿率规定上作了一些修正和推广.