Pricing Stochastic Barrier Options under Hull-White Interest Rate Model

A barrier option valuation model with stochastic barrier which was regarded as the main feature of the model was developed under the Hull-White interest rate model.The purpose of this study was to deal with the stochastic barrier by means of partial differential equation methods and then derive the exact analytical solutions of the barrier options.Furthermore,a numerical example was given to show how to apply this model to pricing one structured product in realistic market.Therefore,this model can provide new insight for future research on structured products involving barrier options.

A Boundary Element Formulation for the Pricing of Barrier Options

In this article, we derive a boundary element formulation for the pricing of barrier option. The price of a barrier option is modeled as the solution of Black-Scholes’ equation. Then the problem is transformed to a boundary value problem of heat equation with a moving boundary. The boundary integral representation and integral equation are derived. A boundary element method is designed to solve the integral equation. Special quadrature rules for the singular integral are used. A numerical example is also demonstrated. This boundary element formulation is correct.

Study on Quantum Finance Algorithm:Quantum Monte Carlo Algorithm based on European Option Pricing

As one of the major methods for the simulation of option pricing,Monte Carlo method assumes random fluctuations in the distribution of asset prices.Under certain uncertainties process,different evolution paths could be simulated so as to finally yield the expectation value of the asset price,which requires a lot of simulations to ensure the accuracy based on huge and expensive calculations.In order to solve the above computational problem,quantum Monte Carlo(QMC)has been established and applied in the relevant systems such as European call options.In this work,both MC and QM methods are adopted to simulate European call options.Based on the preparation of quantum states in QMC algorithm and the construction of quantum circuits by simulating a quantum hardware environment on a traditional computer,the amplitude estimation(AE)algorithm is found to play a secondary role in accelerating the pricing of European options.More importantly,the payoff function and the time required for the simulation in QMC method show some improvements than those in MC method.

考虑国际金融风险影响的上证50ETF期权定价研究

近年来如何刻画国际金融风险对中国市场的影响,成为学术界的热门热点之一。已有文献大多集中于研究国际股票市场之间的风险溢出效应,较少关注国际股票市场对中国期权市场的风险外溢效应。本文将标普500ETF走势嵌入上证50ETF的收益率过程,构建IFR_BS模型(BS Model with the Impact of International Financial Risk);然后应用特征函数微扰法和Fourier-Cosine定价方法,推导出该模型下欧式期权的近似解析定价公式。数值实验和实证结果表明:(1)IFR_BS模型可以较好地刻画上证50ETF收益率分布的“尖峰”、“肥尾”和“有偏”等统计特征。(2)考虑国际金融风险溢价的IFR_BS模型下的期权定价公式,可以解决BS模型对短到期期权尤其是短到期深度OTM期权估值不足的问题。

Modified Differential Transform Method for Solving Black-Scholes Pricing Model of European Option Valuation Paying Continuous Dividends

.Option pricing is a major problem in quantitative finance.The Black-Scholes model proves to be an effective model for the pricing of options.In this paper a com-putational method known as the modified differential transform method has been em-ployed to obtain the series solution of Black-Scholes equation with boundary condi-tions for European call and put options paying continuous dividends.The proposed method does not need discretization to find out the solution and thus the computa-tional work is reduced considerably.The results are plotted graphically to establish the accuracy and efficacy of the proposed method.

On the Convergence of a Crank-Nicolson Fitted Finite Volume Method for Pricing European Options under Regime-Switching Kou’s Jump-Diffusion Models

In this paper,we construct and analyze a Crank-Nicolson fitted finite volume scheme for pricing European options under regime-switching Kou’s jumpdiffusion model which is governed by a system of partial integro-differential equations(PIDEs).We show that this scheme is consistent,stable and monotone as the mesh sizes in space and time approach zero,hence it ensures the convergence to the solution of continuous problem.Finally,numerical experiments are performed to demonstrate the efficiency,accuracy and robustness of the proposed method.

CONVERGENCE RATES OF MOVING MESH RANNACHER METHODS FOR PDES OF ASIAN OPTIONS PRICING

This paper studies the convergence rates of a moving mesh implicit finite difference method with interpolation for partial differential equations （PDEs） with moving boundary arising in Asian option pricing. The moving mesh scheme is based on Rnnacher timestepping approach whose idea is running the implicit Euler schemes in the initial few steps and continuing with Crank-Nicolson schemes. With graded meshes for time direction and moving meshes for space direction, the fully discretized scheme is constructed using quadratic interpolation between two consecutive time level for the PDEs with moving boundary. The second-order convergence rates in both time and space are proved and numerical examples are carried out to confirm the theoretical results.

Lattice Boltzmann methods for solving partial differential equations of exotic option pricing

This paper establishes a lattice Boltzmann method （LBM） with two amending functions for solving partial differential equations （PDEs） arising in Asian and lookback options pricing. The time evolution of stock prices can be regarded as the movement of randomizing particles in different directions, and the discrete scheme of LBM can be interpreted as the binomial models. With the Chapman-Enskog multi-scale expansion, the PDEs are recovered correctly from the continuous Boltzmann equation and the computational complexity is O（N）, where N is the number of space nodes. Compared to the traditional LBM, the coefficients of equilibrium distribution and amending functions are taken as polynomials instead of constants. The stability of LBM is studied via numerical examples and numerical comparisons show that the LBM is as accurate as the existing numerical methods for pricing the exotic options and takes much less CPU time.

A Power Penalty Approach to Numerical Solutions of Two-Asset American Options

This paper aims to develop a power penalty method for a linear parabolic variational inequality(Ⅵ) in two spatial dimensions governing the two-asset American option valuation.This method yields a two-dimensional nonlinear parabolic PDE containing a power penalty term with penalty constantλ>1 and a power parameter k>0.We show that the nonlinear PDE is uniquely solvable and the solution of the PDE converges to that of theⅥat the rate of order(?)(λ^(-k/2)).A fitted finite volume method is designed to solve the nonlinear PDE,and some numerical experiments are performed to illustrate the usefulness of this method.

The Operator Splitting Method for Black-Scholes Equation

The Operator Splitting method is applied to differential equations occurring as mathematical models in financial models. This paper provides various operator splitting methods to obtain an effective and accurate solution to the Black-Scholes equation with appropriate boundary conditions for a European option pricing problem. Finally brief comparisons of option prices are given by different models.

A Fixed Point Method for the Linear Complementarity Problem Arising from American Option Pricing

For American option pricing, the Black-Scholes-Merton model can be discretized as a linear comple- mentarity problem （LCP） by using some finite difference schemes. It is well known that the Projected Successive Over Relaxation （PSOR） has been widely applied to solve the resulted LCP. In this paper, we propose a fixed point iterative method to solve this type of LCPs, where the splitting technique of the matrix is used. We show that the proposed method is globally convergent under mild assumptions. The preliminary numerical results are reported, which demonstrate that the proposed method is more accurate than the PSOR for the problems we tested.

Rough Heston Models with Variable Vol-of-Vol and Option Pricing

In this paper,a rough Heston model with variable volatility of volatility(vol-of-vol)is derived by modifying the generalized nonlinear Hawkes process and extending the scaling techniques.Then the nonlinear fractional Ric-cati equation for the characteristic function of the asset log-price is derived.The existence,uniqueness and regularity of the solution to the nonlinear fractional Riccati equation are proved and the equation is solved by the Adams methods.Finally the Fourier-cosine methods are combined with the Adams methods to price the options.

求解双资产欧式看跌期权定价问题的差分方法

研究了求解双资产极大看跌期权的差分方法.首先通过变换,将无界区域上的期权定价问题转换为偏微分方程的初边值问题,然后构造了一种隐式的差分格式.论证了差分解的惟一存在性,在L_(∝)模意义下差分解的绝对稳定性,并且给出了差分解的误差估计.数值实验验证了所构造差分格式的有效性.

一类偏积分微分方程的时间隐显方法研究

期权定价方程是现代金融理论的重要研究工具.随着期权市场的快速发展,对期权定价理论的研究由简单的Black-Scholes方程转变为带跳扩散方程.以Merton提出的带跳扩散方程为研究对象,其对应的是一个偏积分微分方程,利用隐显中点方法对时间进行离散.通过Matlab编写相应程序,数值模拟实验结果表明,该方法是稳定的和收敛的.

Option Pricing by Mean Correcting Method for Non-Gaussian Lvy Processes

For a non-Gaussian Levy model, it is shown that if the model exists a trivial arbitrage-free interval, option pricing by mean correcting method is always arbitrage-free, and if the arbitrage-free interval is non-trivial, this pricing method may lead to arbitrage in some cases. In the latter case, some necessary and sufficient conditions under which option price is arbitrage-free are obtained.

期权定价方法综述

介绍了期权定价理论的产生和发展 ;然后对期权定价方法及其实证研究进行了较详细的分类综述 ,突出综述了既适用于完全金融市场 ,又适用于非完全的金融市场的确定性套利定价方法、区间定价方法和ε-套利定价方法 ;最后 ,对各种方法的条件和特点进行了讨论和评价 .

基于发电主市场价格信息的辅助服务定价方法

在分析随机性的基于容量市场的辅助服务期权定价方法的基础上,提出了与发电主市场价格信息直接联系的随机电价模型以及期权定价模型,形成了基于发电主市场价格的通过期权交易获取辅助服务的方法。最后通过算例进行了说明。

O-U过程下不确定执行价格的亚式期权定价

文章利用能反映股票预期收益率波动变化的指数Omstein-Uhlenback过程,来描述期权标的股票价格的变化规律;在无风险利率依赖于时间参数的情况下,利用鞅方法和随机微分方程,研究了具有不确定执行价格的几何平均亚式期权的定价问题,得到了股票遵循指数O-U过程且具有不确定执行价格的几何平均亚式看涨及看跌期权的定价公式。

规定水平下重置期权的有限差分解

利用Black-Scholes偏微分方程,结合重置期权与关卡期权的关系,建立了规定水平下的重置期权定价模型,最后运用C-N格式和θ法构造该模型的有限差分格式.

基于期权理论的多元相机决策权利定价模型

期权理论可以很好地对投资和筹资中的相机决策权利进行定价,格型法则是研究复杂期权定价问题最常用的数值方法之一.文章在期权理论的基础上,根据风险中性原则,通过确定合理的涨跌幅度和涨跌概率,建立了评价多元相机决策权利价格的多维格型模型,并保证了模型的一致性和稳定性.最后,通过数值评价验证了模型的计算有效性.