双边敲出障碍期权定价模型

本文针对经理通过操纵股价牟利和非公司经营业绩下滑 ,股价下跌给经理期权收益造成损失两方面的问题 ,提出采用限制股票价格变化的方式 ,计算经理股票期权收益 ,构建了双障碍敲出期权用于经理激励 ,并给出了与 Black- Scholes公式的对比分析 .     

认股权证的等价鞅测度定价模型与数值方法

本文对认股权证应用等价鞅测度方法进行定价 .推导出计算更自然、更简单的类似 Black- Scholes模型的认股权证定价公式 .给出了一种比较好的数值计算方法 .并讨论了认股权证在中国证券市场的发展状况 .     

市场利率波动对期权价值的影响

本文在股价服从指数 O- U过程模型假设下 ,考虑到市场利率波动与股价波动的相关性 ,着重分析了市场利率的波动对期权价值的影响 ,并将所得结果与 Black- Scholes定价模型进行了比较     

跳跃扩散型汇率过程的外汇期权定价

在完全外汇市场环境下 ,讨论了外汇汇率过程受 Brown运动和 Poisson过程共同驱动时外汇欧式未定权益的定价问题 ,并在常系数情形下获得了欧式外汇期权 Black- Scholes定价公式及其套期保值策略 ,最后给出了一种多汇率过程的线性组合式未定权益的定价     

诺贝尔经济学奖与数学（续三）

1997年的诺贝尔经济学奖授予两位美国经济学家RobertC.Merton(1944-)和 Myron S.Scholes(1941-)以奖励他们的确定衍生证券价值的新方法. 实际上,这里应该共享这份荣誉的经济学家还有Fisher Black(1938-1995),可惜他在两年前不幸去世,而这里所说的新方法就是关于衍生证券定价的Black-Scholes理论,它以Black与Scholes在1973年发表他们著名的期权定价公式作为起点,又由Merton进一步完善和系统化.这一理论被誉为“华尔街的第二次革命”.目前在全世界的证券市场,每天都有成千上万的投资者     

诺贝尔经济学奖与数学（续三）

1997年的诺贝尔经济学奖授予两位美国经济学家Robert C.Merton(1944-)和 Myron S.Scholes(1941-)以奖励他们的确定衍生证券价值的新方法。     

Taylor公式的若干推广

在分析Taylor公式结构特征的基础上讨论Taylor公式的推广.指出通过每一个线性微分算子和每一组线性泛函都可得到函数的类似于Taylor公式的表达式,通常的Taylor公式只是微分算子和线性泛函的特殊选取.并证明类似的结果可以推广到抽象的Hilbert空间.     

风险中性过程的非参数估计

无套利定价理论说明，任何衍生证券定价都可以在基础资产价格(或收益)的风险中性过程基础上进行，而且方差函数的估计是估计风险中性过程的关键问题，由于方差不可观测，采用特定的参数模型将是危险的。本文主要讨论时间序列模型下条件方差函数的非参数估计，对核估计和局部多项式估计给出确定窗宽的M-图方法，并给出时间序列模型下衍生证券定价的风险中性调整方法，最后作了模拟计算。     

美式看跌期权定价中的小波方法

本文采用有限差分格式和 Daubechies正交小波 ,提出了一种求解 Black- Scholes方程数值解新算法 .为美式看跌期定价提供了一条新的途径 .利用小波基的自适应性和消失矩特性 ,使偏微分算子矩阵和小波级数稀疏化 ,大大减少了计算量 .     

修正的期权定价模型及定价公式

通过一系列函数变换，求出了修正的Black—Scholes欧式定价模型方程的解，并对股票与国债的投资组合进行了分析。     

Professor V. Lakshmikantham,Editor Stochastic Analysis and Applications (1983–2012)

We use the stochastic calculus of variations for the fractional Brownian motion to derive formulas for the replicating portfolios for a class of contingent claims in a Bachelier and a Black–Scholes markets modulated by fractional Brownian motion. An example of such a model is the Black–Scholes process whose volatility solves a stochastic differential equation driven by a fractional Brownian motion that may depend on the underlying Brownian motion.     

The Black and Scholes equation with stochastic volatility. Variational methods

We give a variational analysis and numerical simulations for a Black and Scholes equation with stochastic volatility.     

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.     

Fractional stochastic integration and Black–Scholes equation for fractional Brownian model with stochastic volatility

We modify the Hu-Øksendal and Elliot-van der Hoek approach to arbitrage-free financial markets driven by a fractional Brownian motion that is defined on a white noise space. We deduce and solve a Black–Scholes fractional equation for constant volatility and outline the corresponding equation with stochastic volatility. As an auxiliary result, we produce some simple conditions implying the existence of the Wick integral w.r.t. fractional noise.     

American Options Exercise Boundary When the Volatility Changes Randomly

The American put option exercise boundary has been studied extensively as a function of time and the underlying asset price. In this paper we analyze its dependence on the volatility, since the Black and Scholes model is used in practice via the (varying) implied volatility parameter. We consider a stochastic volatility model for the underlying asset price. We provide an extension of the regularity results of the American put option price function and we prove that the optimal exercise boundary is a decreasing function of the current volatility process realization. Accepted 13 January 1998     

Homotopy analysis method for option pricing under stochastic volatility

In this paper, the homotopy analysis method, whose original concept comes from algebraic topology, is applied to connect the Black–Scholes option price (the good initial guess) to the option price under general stochastic volatility environment in a recursive manner. We obtain the homotopy solutions for the European vanilla and barrier options as well as the relevant convergence conditions.     

American option pricing under stochastic volatility: an empirical evaluation

Over the past few years, model complexity in quantitative finance has increased substantially in response to earlier approaches that did not capture critical features for risk management. However, given the preponderance of the classical Black–Scholes model, it is still not clear that this increased complexity is matched by additional accuracy in the ultimate result. In particular, the last decade has witnessed a flurry of activity in modeling asset volatility, and studies evaluating different alternatives for option pricing have focused on European-style exercise. In this paper, we extend these empirical evaluations to American options, as their additional opportunity for early exercise may incorporate stochastic volatility in the pricing differently. Specifically, the present work compares the empirical pricing and hedging performance of the commonly adopted stochastic volatility model of Heston (Rev Financial Stud 6:327–343, 1993) against the traditional constant volatility benchmark of Black and Scholes (J Polit Econ 81:637–659, 1973). Using S&P 100 index options data, our study indicates that this particular stochastic volatility model offers enhancements in line with their European-style counterparts for in-the-money options. However, the most striking improvements are for out-of-the-money options, which because of early exercise are more valuable than their European-style counterparts, especially when volatility is stochastic.     

A closed form solution for vulnerable options with Heston’s stochastic volatility

Over-the-counter stock markets in the world have been growing rapidly and vulnerability to default risks of option holders traded in the over-the-counter markets became an important issue, in particular, since the global finance crisis and Eurozone crisis. This paper studies the pricing of European-type vulnerable options when the underlying asset follows the Heston dynamics. In this paper, we obtain a closed form analytic formula of the option price as a stochastic volatility extension of the classical Heston formula and find how the stochastic volatility effect on the Black–Scholes price as well as on the decreasing speed of the option price with credit risk depends on moneyness.     

Pricing and hedging GLWB in the Heston and in the Black–Scholes with stochastic interest rate models

Valuing Guaranteed Lifelong Withdrawal Benefit (GLWB) has attracted significant attention from both the academic field and real world financial markets. As remarked by Forsyth and Vetzal (2014) the Black and Scholes framework seems to be inappropriate for such a long maturity products. They propose to use a regime switching model. Alternatively, we propose here to use a stochastic volatility model (Heston model) and a Black–Scholes model with stochastic interest rate (Hull–White model). For this purpose we present four numerical methods for pricing GLWB variables annuities: a hybrid tree-finite difference method and a Hybrid Monte Carlo method, an ADI finite difference scheme, and a Standard Monte Carlo method. These methods are used to determine the no-arbitrage fee for the most popular versions of the GLWB contract, and to calculate the Greeks used in hedging. Both constant withdrawal and optimal withdrawal (including lapsation) strategies are considered. Numerical results are presented which demonstrate the sensitivity of the no-arbitrage fee to economic, contractual and longevity assumptions.     

A regression-based numerical scheme for backward stochastic differential equations

Based on Fourier cosine expansion, two approximations of conditional expectations are studied, and the local errors for these approximations are analyzed. Using these approximations and the theta-time discretization, a new and efficient numerical scheme, which is based on least-squares regression, for forward–backward stochastic differential equations is proposed. Numerical experiments are done to test the availability and stability of this new scheme for Black–Scholes call and calls combination under an empirical expression about volatility. Some conclusions are given.