Pricing of discrete barrier options based on an analytical method

The problem o f analytically pricing the discrete monitored European barrier options is studied under the assumption of the Black-Scholes market.First,using variable transformation,the mean vector and covariance matrix of multi-dimensional marginal distribution are given.Secondly,the analytica pricing formulas of the discrete monitored upknock-out European call option and the discrete monitored down-knock-out European put option a e obtained by using the conditional probability and the characteristics o f the multidimensional normal distribution.Finally,the effects of the discrete monitoring barriers on the prices of the barrier optionsare discussed and analyzed.The research results state that the price o f the discrete monitored up-knock-out European call option mcreases with the increase in the up barrier,a d the price o f the discrete monitored down-knock-out European put option decreases with the increase in the down barrier.

Random Timestepping Algorithm with Exponential Distribution for Pricing Various Structures of One-Sided Barrier Options

The exponentially-distributed random timestepping algorithm with boundary test is implemented to evaluate the prices of some variety of single one-sided barrier option contracts within the framework of Black-Scholes model, giving efficient estimation of their hitting times. It is numerically shown that this algorithm, as for the Brownian bridge technique, can improve the rate of weak convergence from order one-half for the standard Monte Carlo to order 1. The exponential timestepping algorithm, however, displays better results, for a given amount of CPU time, than the Brownian bridge technique as the step size becomes larger or the volatility grows up. This is due to the features of the exponential distribution which is more strongly peaked near the origin and has a higher kurtosis compared to the normal distribution, giving more stability of the exponential timestepping algorithm at large time steps and high levels of volatility.

Numerical Methods for Discrete Double Barrier Option Pricing Based on Merton Jump Diffusion Model

As a kind of weak-path dependent options, barrier options are an important kind of exotic options. Because the pricing formula for pricing barrier options with discrete observations cannot avoid computing a high dimensional integral, numerical calculation is time-consuming. In the current studies, some scholars just obtained theoretical derivation, or gave some simulation calculations. Others impose underlying assets on some strong assumptions, for example, a lot of calculations are based on the Black-Scholes model. This thesis considers Merton jump diffusion model as the basic model to derive the pricing formula of discrete double barrier option;numerical calculation method is used to approximate the continuous convolution by calculating discrete convolution. Then we compare the results of theoretical calculation with simulation results by Monte Carlo method, to verify their efficiency and accuracy. By comparing the results of degeneration constant parameter model with the results of previous models we verified the calculation method is correct indirectly. Compared with the Monte Carlo simulation method, the numerical results are stable. Even if we assume the simulation results are accurate, the time consumed by the numerical method to achieve the same accuracy is much less than the Monte Carlo simulation method.

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.

An Accurate FFT-Based Algorithm for Bermudan Barrier Option Pricing

An efficient and accurate numerical method, which is called the CONV method, was proposed by Lord et al in [1] to price Bermudan options. In this paper, this method is applied to price Bermudan barrier options in which the monitored dates may be many times more than the exercise dates. The corresponding algorithm is presented to practical option pricing. Numerical experiments show that this algorithm works very well for different exponential Lévy asset models.

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.

The valuation of barrier options under a threshold rough Heston model

In this paper,we propose a novel model for pricing double barrier options,where the asset price is modeled as a threshold geometric Brownian motion time changed by an integrated activity rate process,which is driven by the convolution of a fractional kernel with the CIR process.The new model both captures the leverage effect and produces rough paths for the volatility process.The model also nests the threshold diffusion,Heston and rough Heston models.We can derive analytical formulas for the double barrier option prices based on the eigenfunction expansion method.We also implement the model and numerically investigate the sensitivities of option prices with respect to the parameters of the model.

PRICING BARRIER OPTIONS UNDER STOCHASTIC VOLATILITY FRAMEWORK

Abstract Option pricing problem plays an extremely important role in quantitative finance. In com- plete market, Black-Scholes-Merton theory has been central to the development of financial engineering as both discipline and profession. However, in incomplete market, there are not any replicating port- folios for those options, and thus, the market traders cannot apply the law of one price for obtaining a unique solution. Fortunately, the authors can get a fair price via local-equilibrium principle. In this paper, the authors apply the stochastic control theory to price the exotic option-barrier options, and analyze the relationship between the price and the current positions. The authors get the explicit expression for the market price of the risk. The position effect plays a significant role in option pricing, because it can tell the trader how many and which direction to trade with the market in order to reach the local equilibrium with the market.

American Barrier Option Pricing Formulas for Currency Model in Uncertain Environment

Option pricing problem is one of the central issue in the theory of modern finance.Uncertain currency model has been put forward under the foundation of uncertainty theory as a tool to portray the foreign exchange rate in uncertain finance market.This paper uses uncertain differential equation involved by Liu process to dispose of the foreign exchange rate.Then an American barrier option of currency model in uncertain environment is investigated.Most important of all,the authors deduce the formulas to price four types of American barrier options for this currency model in uncertain environment by rigorous derivation.

Pricing Continuously Monitored Barrier Options under the SABR Model:A Closed‐Form Approximation

The stochastic alpha beta rho(SABR)model introduced by Hagan et al.(2002)is widely used in both fixed income and the foreign exchange(FX)markets.Continuously monitored barrier option contracts are among the most popular derivative contracts in the FX markets.In this paper,we develop closed-form formulas to approximate various types of barrier option prices(down-and-out/in,up-and-out/in)under the SABR model.We first derive an approximate formula for the survival density.The barrier option price is the one-dimensional integral of its payoff function and the survival density,which can be easily implemented and quickly evaluated.The approximation error of the survival density is also analyzed.To the best of our knowledge,it is the first time that analytical(approximate)formulas for the survival density and the barrier option prices for the SABR model are derived.Numerical experiments demonstrate the validity and efficiency of these formulas.

Adaptation to Climate Change in the Pastoral and Agropastoral Systems of Borana, South Ethiopia: Options and Barriers

The pastoral and agropastoral systems of the Borana in southern Ethiopia are highly vulnerable to climate change and its impacts. Assistance to enable these smallholders to successfully adapt to future climate change in locally relevant ways can be usefully informed by the analysis and better understanding of past and ongoing adaptation. We conducted farm household surveys, focus group discussions, expert consultations and secondary data collation in 2012 in the Borana. The study employed a combination of Pressure-State-Response (PSR) framework to analyse how climate change put pressure on pastoral and agropastoral farming systems and livelihoods, and Pelling’s (2011) typological framework to analyse local adaptation responses. Results showed that pastoral and agropastoral households, their communities and institutions adopted a wide range of adaptation options primarily through adjusting their farming practices and diversifying into non-pastoral livelihoods. The smallholders primarily pursued a resilience approach to adaptation with short term goals intended to avoid system disruptions instead of long-term transformational approaches that significantly address the root causes of vulnerability. A range of barriers constrained local adaptive capacity and shaped routes for adaptation. Adaptation pathways that address critical barriers to adapt, integrate indigenous institutions into adaptation and link adaptation with local development process are necessary to bring long-term and non-marginal, major changes that reduce vulnerability and ensure co-benefit of improving livelihoods.

G-理论下的欧式向下敲出看涨期权定价

用G-几何布朗运动描述标的股票价格变化,假设波动率属于某一确定区间,通过G-Girsanov变换得到风险中性测度下欧式向下敲出看涨期权的定价区间。实验对比了布朗运动与G-布朗运动的路径,讨论了G-二次变差的参数对期权价格区间的影响,并对期权价格进行敏感性分析,验证了模型的合理性。

Toxic Currency Options in Poland as a Consequence of the 2008 Financial Crisis

The paper is on toxic foreign exchange options problem which occurred in Poland just prior to and after the outbreak of the recent crisis. Especially Polish enterprises were severely stricken by transactions on fx and interest rate derivatives contracted with their banks. Poland was the only EU country which did not precipitate into recession during the financial crisis beginning in 2008. However, the toxic fx and interest rate derivatives transmitted the shockwaves from global financial markets into Poland. Huge dimensions of losses resulted in conflicts between banks and their customers, who claimed just being cheated by the financial institutions. The article deeply researches into reasons for such developments on Polish fx over-the-counter derivatives market. As a case study, an authentic strategy has been presented. The contract was concluded between the construction company and one of the biggest commercial banks in Poland. Because the case study may be representative for many other cases, the analysis includes exact pricing of option strategy and therefore reveals inequality of the contract. The consequences of non-implementing the MiFID directive in the context of derivatives offering to non-financial customers were also touched in the paper.

The Utilization of Exotic Options in the Formation of Structured Products

This paper deals with the issue of investment certificate formation in the financial market. Investment certificate is a type of structured products, the value of which is derived from the value of an underlying asset. The underlying asset is usually a share in a company, a basket of shares, or an entire index, etc.. It can be stated that for every estimated development of an asset （growth, fall, and stagnation） or for every attitude to risks （conservative or aggressive investors）, there is a suitable kind of certificate. The main objective is to perform an analysis of the structured product--Austria/Germany Bond 3 and its guarantee certificate construction using digital-barrier options. The authors have found an alternative opportunity to the purchase of this certificate, i.e., investment in a bank deposit, together with a purchase of cash or nothing down and four-knock-out call options and a sale of cash or nothing down and four-knock-out put options. The authors prove that the alternative investment has the same profit profile as the certificate. The authors made this analysis with the objective to contribute to the intellectualization of investors.

Optimum development options and strategies for water injection development of carbonate reservoirs in the Middle East

Through the research on several carbonate reservoirs developed in the Middle East, the basic characteristics of different types of carbonate reservoirs are determined, and a set of high-efficiency water injection development options and strategies are presented. Hidden baffles and barriers exist in carbonate reservoirs in the Middle East, so the reservoirs could be divided into different separated development units based on the baffles and barriers characteristics. Flexible and diverse profile control techniques such as high angle wells and simple and applicative zonal water injection have been introduced to improve the control and development degree of reservoirs. Three principal water injection development methods suitable for different carbonate reservoirs in the Middle East are proposed, including the combination of crestal gas injection and peripheral water injection, bottom interval injection and top interval production(buoyancy underpinning), and "weak point and strong plane" area well pattern. Based on the characteristics of very low shale content, fast and far pressure transmission in the Middle East carbonate reservoirs, a large well-spacing flood pattern is recommended, and reasonable development strategies have been made such as moderate water injection rate and maintaining reasonable production pressure drawdown and voidage replacement ratio, so as to maximize the recovery of reservoirs in the none or low water cut period.

基于O-U特征的Bachelier模型的期权定价

随着“负油价”现象的出现,对于期权等衍生产品来说,标的资产负价格意味着经典Black-Scholes模型失效。本文对原始的Bachelier模型进行修改,保留标的资产价格可以为负的特点,并且使其具有Ornstein-Uhlenbeck随机过程特征。基于修正的Bachelier模型结合蒙特卡洛数值算法对欧式期权、美式期权以及障碍期权进行定价,进一步扩展其期权定价应用范围。通过数值模拟,基于修正的Bachelier模型在期权定价上表现出很高的计算精度,基于O-U特征的Bachelier模型的期权定价可以作为Black-Scholes模型期权定价的替代方案,指导期权等衍生品定价决策。

蒙特卡洛模拟法在障碍期权定价中的应用

实际金融市场中,障碍期权标的资产价格运动并不服从理想的几何布朗运动,而是存在跳跃过程。为研究障碍期权的定价问题,考虑蒙特卡洛模拟法来进行障碍期权标的资产的路径模拟,同时使用资产价格的跳扩散模型为期权进行定价。由于蒙特卡洛模拟法会产生较大的误差,所以加入方差减少技术中的重要性抽样来对障碍期权重新定价,以进一步提高定价算法的精确度和运行速度。

基于Bayesian MCMC方法的美式障碍期权模拟定价

在最小二乘蒙特卡洛(LSM)方法基础上,提出用加权最小二乘与蒙特卡洛(MC)方法相结合,得到加权最小二乘蒙特卡洛(WLSM)方法,研究了障碍期权模拟定价的问题.假设标的资产价格过程遵循几何布朗运动,分析了是否支付红利和生成期权价格路径的问题.使用随机化Faure序列替换LSM方法中伪随机数,给出了WLSM方法在美式障碍期权定价的算法步骤.使用R语言对美式障碍上升敲出看跌期权(up-and-out put)在支付红利的情形下进行数值模拟,结果表明此方法与其他定价模型方法相比,定价更准确,说明该方法具有可行性和有效性.

次扩散过程驱动下的欧式障碍期权定价

障碍期权是一种重要的新型期权,在现有的定价模型中,标的资产价格的驱动源为布朗运动和分数布朗运动,无法描述标的资产常值周期性的特点.本文把标的资产价格变化的常值周期性的特征纳入障碍期权定价模型中,建立了次扩散机制下的欧式障碍期权定价模型.运用Delta对冲技巧和伊藤公式得到了欧式下降敲出看跌障碍期权满足的偏微分方程.应用变量替换的方法,借助Possion公式给出了欧式下降敲出看跌障碍期权的显示定价公式,最后给出了相关数值计算结果.

电力市场环境下燃气轮机调峰交易模式研究

基于燃气轮机启停迅速、出力便于调节且启停成本较低的特性,针对电网急需调峰容量的状况,提出了电力市场环境下燃气轮机与系统调度中心签订障碍期权合约参与启停调峰、签订普通中长期双边合约参与深度调峰的思路。首先确定了参与调峰的交易主体、交易方式等内容,然后据此制定了可行的交易机制,接着分析了调峰合约的实施过程,最后确定了调峰交易的相关价格。分析表明,该交易模式既可使系统以经济的方式调度启停调峰容量,关键时刻增加系统的深度调峰能力;又避免了交易者的反复竞价,提升了交易的执行效率。算例分析表明了该市场交易机制的有效性和可行性。