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期权定价模型研究永安期货研究院金融期货部：周博王晓宝衍生品市场中，期权的成功很大程度得益于其定价模型的标准化，这使得大众对期权的公允价达到了一致的认可，从而交易顺利进行。
按照执行方式划分，期权分为欧式期权和美式期权；按照标的物性质划分，又分为现货期权和期货期权。
本文主要研究商品类美式期货期权，因此，本部分重点放在做市商常用的美式期货期权的定价公式研究。
一、最小二乘蒙特卡洛模拟期权定价模型Tilley（1993）最早提出了将蒙特卡洛方法应用于美式期权定价一种解决办法，但由于这些解决办法存在某些缺陷，没有得到广泛的应用。
在这方面的突破性研究当属Longstaff和Schwartz（2001），他们引入最小二乘法来确定每一时刻衍生证券的连续价值和相关变量价值之间的最佳拟合关系，并以此判断在该时刻是否提前履行期权。
目前，最小二乘蒙特卡洛模拟（Least Square Monte Carlo）已经成为使用蒙特卡洛模拟方法来进行美式期权定价的标准方法。
（一）LSM模拟算法基本思想最小二乘蒙特卡洛模拟的基本思想是：与传统的蒙特卡洛模拟类似，将期权的到期剩余时间划分为有限个时间间隔，并生成随机的标的资产价格路径样本，利用最小二乘法对样本路径在各时刻的截面数据进行回归求得期权的持有期望报酬，并将其与在该时刻提前行权的收益相比较，相对较大值即为该时刻的期权价值，如果行权价值大于持有的期望价值，则立即行权为最优策略，否则，继续持有期权。
（二）LSM模拟算法的算法实现步骤LSM模拟方法的基本步骤如下：首先，生成标的资产价格的样本路径；其次，从期权到期日开始逆向求解，得到每条样本路径上的最优期权执行时间和相应的期权收益；最后，将每条样本路径的期权收益用无风险利率贴现，然后取它们的均值即得到模拟的期权价值。
下面我们以单一标的资产美式看跌期权定价为例，说明LSM模拟方法的算法实现步骤。
第一步：生成标的资产价格样本路径根据期权理论，我们假设期权的到期日为T，执行时间为T∗，则对美式期权而言，T∗∈[0,T]，即期权可以在到期日前的任意时刻执行。
基于深度学习算法的欧式股指期权定价研究——来自50ETF期权市场的证据谢合亮;游涛【期刊名称】《统计与信息论坛》【年(卷),期】2018(033)006【摘要】深度学习(Deep Learning)在人工智能领域取得了巨大的成就,在学界和业界都激起了深度学习的热潮.根据金融数据的时序特征,将深度学习中循环神经网络(RNN)引入期权定价模型,构建了一种基于长短记忆神经网络(LSTM)的新的期权定价模型,并利用50EFT看涨期权和看跌期权进行实证分析.研究结果表明:LSTM期权定价模型比经典的Black-Scholes蒙特卡洛方法具有更高的定价精确性.%Deep learning (DL )has made great achievements in the field of artificial intelligence ,which has aroused the learning boom in the academic community as well as the industry .According to the characteristics of time series of financial data ,the recurrent neural network(RNN) of deep learning will be introduced in option pricing model ,and constructed a new option pricing model by using the long short-term memory (LSTM ) with 50EFT call options and put options data .The empirical results demonstrate that LSTM option pricing model is more accurate than the classical Black-Scholes Monte Carlo method .【总页数】8页(P99-106)【作者】谢合亮;游涛【作者单位】中央财经大学统计与数学学院,北京100081;深圳证券交易所综合研究所,深圳 518028【正文语种】中文【中图分类】F224.0【相关文献】1.基于沪深300指数的欧式股指期权定价研究 [J], 张天凤;金梦迪;忠桥2.股指期权合约规格设计研究——基于全球股指期权市场对比的视角 [J], 程志富; 陈晶3.交易成本调整能够改善期权市场质量吗?——来自上证50 ETF期权市场的经验证据 [J], 程志富;陈晶4.短期MAX动量效应与衰减——基于上证50ETF期权市场的证据 [J], 熊海芳;黄超5.短期MAX动量效应与衰减--基于上证50ETF期权市场的证据 [J], 熊海芳;黄超因版权原因，仅展示原文概要，查看原文内容请购买。
The study on option pricing problemScience and Technology College of Ningbo University, Ningbo, Zhejiang, ChinaKeywords: option;option pricing;the basic way of option pricing; stock option pricing Abstract.Uncertain pricing is one core of financial mathematics study, it involves the theories of modern finance such as asset pricing theory, investment combination theory and risk management theories, as well as stochastic analyzing and optimizing theory of modern mathematics. Effective investment of risky assets is the key to financial derivative securities for the correct valuation.In order to adapt to the continuous development of financial markets, we need to have singular conduct an in-depth study of options, in order to meet investor preferences better.浅谈期权定价问题关键词:期权；期权定价；期权定价基本方法；股票的期权定价方法中文摘要.期权定价问题已经成为金融工程研究的核心问题之一，它涉及现代金融学的资产定价理论、投资组合研究、风险管理理论以及现代数学中的随机分析、优化理论等学科。
Option pricing analysis, implied volatility application and verification of Hang Seng Index OptionsI. IntroductionOptions is some kind of agreement that the parties to reach. The buyer of the option pay some of the costs the option seller, and to achieve the right to buy or sell a certain number of base stocks (assets) in a future date or before the expiry of agreement. European option refers to the party of the call option only can exercise the option on the option expiration day.The main difference of modern finance and traditional finance theory in its study is the transition from qualitative analysis to quantitative analysis. Mathematical finance can be considered as the most representative aspects in the quantitative analysis of the modern financial industry. Quantitative analysis can not be separated from applications of calculation software. Matlab is one of the most popular numerical computing software, which integrates the high-performance numerical computing and data visualization and provides a large number of internal functions. In recentyears, it is widely used and provides strong support in financial quantitative analysis. For a long time, the option pricing model has a very important role in the financial engineering. In this paper, we use the Matlab, the realization of the European option implied volatility and its application in practice, and set the Hong Kong Hang Seng Index Options for example, to give a validation for the Black-Scholes-Merton option pricing model and analyze the theory of option pricing and the actual price’s difference and reason.Ⅱ.The Black-Scholes-Merton option pricing model and MATLAB realization1. B-S-M modelAssuming the stock price process S (t) follow the geometric brown sports at time t as follows:dS(t)=mS(t)dt+sS(t)dW(t)The asset price of no risk R (t) obeys the following equation:dR(t)=rR(t)dtWhere r, m, s> 0 are constants, m is the expected return rate of the stock, s is a stock price volatility, r is the risk-free asset yields rate and 0 <r <m; dW (t) is a standardBrown motion and can be obtained by (1):]),)(2/()([ln :)(ln 2t T s t T s m t S F T S ---+European call option is a contract, and its role is that it makes the contract holder has the right to purchase an asset with a predetermined price at a certain time T in the future.In the risky world, the target assets is the stock described by formula (1), the expected value of the maturity of European call option with no paying out a is ：]0,)([max(^X T S E -. Where, E represents a risk expectations under neutral conditions, the European call option price with no paying out a dividend c equal to the present value of discounting this expected value at the risk-free rate ：}]0,)([max{^)1(X T s E e c T r -=-- For a risk-neutral world, it only can get the return rate of risk-free. So, it needs to replace m in lnS (T) distribution by r:]),)(2/()([ln :)(ln 2t T s t T s r t S F T S ---+Therefore, the expression of the theoretical price of the European option can be drawn according to the BSM option-pricing model:European call option ：)()(2)(1d N Xe d N S C t T r t t ---= European put option ：)](1[*)](1[*12)(d N S d N Xe P t T r t ---=--In which ，2/1221)())(2()ln(t T t T r r X S d t --++=σ,2/1212)(t T d d --=σSt: The market prices of target assetX: Execution pricer: Risk-free interest rateσ: The target asset price volatilityT-t: Time to expiry2. MATLAB implementationFrom the European option price formula based on previous subsection, we can see that, if all parameters are known, calculate the price of the European call option can generally be divided into three steps (without bonus):Firstly, calculate and get d1, d2, which will use a logarithmic function; Secondly ,computing N (d1), N (d2), and it need to view the normal distribution table to obtain the actual value; finally, the European option price can be obtained by substituted into the formula. The model has been established in the financial database in MATLAB.If the user needs to use, then can calculate European option prices with convenient internal function, the function named blsprice.>> [call, put] = blsprice (price, strike, rate, time,volatility)As shown in Equation, do the option pricing just need directly input various parameters. Price is the stock price, Strike is the execution price, Rate behalf of the risk-free rate, Time is the time to maturity, which is the option duration (unit: years), Volatility indicates the standard deviation of the calibration assets. Output parameters: Call is used to represent a European call option price; Put is used to represent a European put option price.We can use an example to look at the BSM model calculation results:Consider a non-dividend stock, if the stock price is 50, the standard deviation of the volatility is 0.2, the risk-free rate is 5%, the option's strike price is 60Yuan, the implementation period is 18 months, using MATLAB to calculate European option prices.>> [call, put] =blsprice (50, 60, 0.05, 1.5, 0.2)Output：Call=2.8012Put =8.4658If the price process of purchasing a copy of the target stock satisfies the formula, the dividends with no paying of European call and put option price are respectively 2.8012 Yuan and 8.4658 Yuan.Ⅲ.The application of implied volatility1. Overview of the implied volatilityFrom the Black-Scholes model, we can see that the volatility is a constant. But that is not the case, the actual market volatility can not be directly observed, and changes over time, so people introduced implied volatility, which is obtained through back stepping option prices observed in the market with the Black-Scholes formula. The implied volatility is essentially contains market investors' average estimate of future asset volatility, and has an extremely important role in the options market.The implied volatility is determined by the options market price, and it is the real India shot of the market price, and effective market price is the product of the balance relationship between supply and demand, so the implied volatility is an important indicator of risk. Historical volatility reflects the options target securities’fluctuations in the past period of time, the option publishers and investors only can use historical volatility as a reference in the early time of options issuing.In general, the higher the implied volatility of options means greater risk. Options investors can do the equity trading in the direction by change the option target asset prices, and can also get profit from changes in the target asset price volatility. Generally, fluctuation can not rise or fall with unlimited, it will bound in a range, and the investor can buy options at low implied volatility and sell them to get higher profits when there is a higher volatility. The price of the option is overvalued or not, mainly to see the relationship of comparison between the implied volatility securities and the historical volatility. The implied volatility is the prediction of the underlying assets (stocks or Indices) for a period in the future and it changes in the same direction with warrants.. Generally, the implied volatility is different from the historical volatility, but that should have little difference. If the implied volatility is much higher than the historical volatility, indicating that the option is overvalued.Options implied volatility changes in two aspects:Ifhigher historical volatility of the underlying stock, the higher the implied volatility of the related warrants; if lower historical volatility of the underlying shares, warrants implied volatility is relatively low.Especially in warrants issued, the historical volatility of the underlying shares will be as one of the basis considering by the Issuer to determine the implied volatility of the warrants, in order to determine the price of the Warrants.In addition, the relationship between supply and demand will also affect the implied volatility, implied volatility is a reflection of the relationship between supply and demand in a way for warrants .When investors demand a certain warrants, the warrants prices will be artificially high, and implied volatility will reach a high level, and even much higher than the actual volatility of the underlying stocks.2.Calculation of stock price volatility.Then for the stock price, the calculation of stock price volatility is based on the stock price within a certain period of time , set (n +1) as the number of observations, Si is the stock price at the end of the i-Th time interval.Let Ui=ln ( Si/Si-1 ), for St=Si-1eUi, so Ui is the continuously compounded rate of return after i-th time interval , the Ut standard deviation is t σ ,which is the daily fluctuation rate of the stock price in this period of time . The estimated value is:∑=--=n i i i U U n 12)(11σ U Is the average value of Ui.After calculate the daily volatility of the stock price, you can use the following formula to calculate the annual volatility of the stock price.Stock annual volatility (σ) = stock price daily volatility * (number of trading days per year) 1/2.Ⅳ.Option Model Application ——Hong Kong's Hang Seng index option price calculationWe adopt the Hang Seng closing index in 2010.02.26 to 2010.05.25 to calculate its volatility. Through the calculating and appropriate adjustments, the follow data can be obtained by:We use the overnight lending rate of banks in Hong Kong as the risk-free rate.For dividend rate, because dividends are roughly fixed at the same time on the market, so we will use a different dividend rate in the calculation of different maturity option price. Due within a month, we will use 0 as the dividend rate; due within three months, we will use a 3% dividend rate.In the next, we use matlab tool to calculate option prices expired in June 2010:The contrast between the option price calculation results and Hang Seng Index Options offer in JuneⅤ.ConclusionThis article discusses the problem of option pricing, and build its option pricing model using matlab. After that, discussed the implied volatility of options and researched using Matlab Financial Toolbox to analyze the implied volatility problem. Finally, we use the Hong Kong Hang SengIndex’s practical situation to conduct a check on the BSM model, and use the model in matlab to calculate the value of the Hang Seng Index Options, conduct a comparison with actual market quotes, and reached the following conclusions: Maturity shorter options contract prices closer to the actual market price, while the long-term price contracts have a relatively great deviation with the actual price of the options. Overall, the BSM model is not suitable for the direct application to the actual market price. The causes of the deviation are as follows:1)BSM option-pricing model requires artificial competent reckon that some of the input variables as the basis to calculate the price of an option, because it is man-made estimates, its relevance and reliability are lacking.2) The option’s expiration time is later, the greater market volatility maybe happens in the middle period. Therefore, there are very obvious differences among investors, and the natural pricing will be more difficult. Conversely, the earlier the expiration of the option, the market changes and the volatility will be smaller. So if there is less difference of investors, setting price is relatively easy.3) Market expected dividend dividends may vary with thecurrent level.4) The transaction price is also affected by factors such as the impact of market supply and demand, investor’s expectation and so on.5)The level of the dividend rate expected by stock market may be different from the current level.。
浅论数学金融学中关于期权定价的问题作者：崔连香来源：《金融经济·学术版》2012年第06期摘要：期权是指对未来选购某种商品的选择权，简单得说就是购买方向出售方支付一定的定金后，获得在一个约定到期日内按提前协定价格购买或出售一定数量商品标的资产权力。
在我国金融业发展过程中，金融期权不但能有效地转移金融风险，还能保护广大投资者的资金安全，使广大投资者能立于不败之地，所以金融期权是一种非常具有发展前途的金融创新工具。
我们通过对金融行业发展的研究发现，期权的定价模型，一直都被认为期权理论中的一个难点。
对于金融期权一些书籍只是简单的介绍，没有使用数学方法深层次推导，本文简单地分析了数学金融学中的期权定价问题，阐明了研究这一问题的有力工具是倒向随机微分方程和正倒向随机微分方程。
关键词：股票市场;期权定价;数学金融1997年10月14日,瑞典皇家科学院将第二十九届诺贝尔经济学奖授予美国哈佛大学教授罗伯特·默顿(Robert C.Merton)和迈伦·肖尔斯(Myron S.Scholes)，以鼓励他们在数学金融学方面的杰出贡献。
因此，引起最近这十几年来人们对数学金融学关注。
金融数学(mathematics of finance)是运用数学理论和方法研究金融经济运行规律的一门新学科，在国际上称为数理金融学。
1、数学在金融学的定量研究中起着重要作用Robert C.Merton所写名著Continuous-TimeFinance中,Merton自己写道:“现代金融学中的数学模型包含了概率论和最优化理论的一些最漂亮的应用。
科学中漂亮的东西未必一定实用,而科学中实用的东西又并非都是漂亮的，指数学金融学却两者俱全，可见对其的评价。
1997年诺贝尔经济学奖的得主们经过反复研究发现，股票市场价格遵循带漂移的几何布朗运动的规律,用较深的数学知识就是随机过程和随机微分方程，终于设计出比较科学的、各类期权定价公式。
期权定价理论期权定价是所有金融应用领域数学上最复杂的问题之一。
第一个完整的期权定价模型由Fisher Black和Myron Scholes创立并于1973年公之于世〔有关期权定价的进展历史大伙儿能够参考书上第358页，有爱好的同学也能够自己查找一下书上所列出的经典文章，只是这要求你有专门深厚的数学功底才能够看明白〕。
B—S期权定价模型发表的时刻和芝加哥期权交易所正式挂牌交易标准化期权合约几乎是同时。
不久，德克萨斯仪器公司就推出了装有依照这一模型运算期权价值程序的运算器。
现在，几乎所有从事期权交易的经纪人都持有各家公司出品的此类运算机，利用按照这一模型开发的程序对交易估价。
这项工作对金融创新和各种新兴金融产品的面世起到了重大的推动作用。
为此，对期权定价理论的完善和推广作出了庞大奉献的默顿和Scholes在1997年一起荣获了诺贝尔经济学奖〔Black在1995年去世，否那么他也会一起获得这份殊荣〕。
原始的B—S模型仅限于这类期权：资产可用于卖出期权；能够评估价值，资产价格行为随时刻连续运动。
随后建立在原始的B—S模型上的研究以及许多其他期权定价模型的变体相继显现，用于处理其他类型的标的资产以及其他类型的价格行为。
在大多数情形下，期权定价模型的推倒基于随机微积分〔Stochastic Calculus〕的数学知识。
没有严密的数学推演，演示这种模型只是摸棱两可的。
但是，这并非要紧的问题，因为确定期权公平价格的必要运算已自动化，且达到上述目的的软件在大型运算机及微机中均可获得。
因此，在那个地点，我只简单介绍一下B—S模型的关键几个要素，至于具体的数学推导〔专门复杂〕，感爱好的同学能够在课后阅读一下相关资料〔一样差不多上在期权定价理论章节的附录中〕。
第一，我们来回忆一下套利的含义套利套利〔arbitrage〕通常是指在金融市场上利用金融产品在不同的时刻和空间上所存在的定价差异、或不同金融产品之间在风险程度和定价上的差异，同时进行一系列组合交易，猎取无风险利润的行为。
投资者认知偏差与实物期权的行为定价叶德珠 陈 莹摘 要:以投资者理性及诸多严格假设条件为前提,B l a c k—S c h o l e s公式解决了金融期权的定价问题。
但在实物期权实践中,投资者存在诸多系统性认知偏差,从而可能导致定价偏误。
本文从比较静态分析的角度,讨论了过分乐观、过度自信、短视、损失规避等认知偏差对投资者实物期权定价的影响,并以这种行为定价思想逻辑地解释了投资、公司治理等实物期权实践中一些典型的异常现象。
关键词:认知偏差 实物期权 行为定价一 引 言B l a c k-S c h o l e s公式对金融期权的定价实现,是建立在市场无套利前提条件之上,依靠运用风险中性复制等技术而取得的。
因此,在这个定价过程中,投资者的偏好以及效用函数的差异都可以被忽略不计,投资者存在的非理性认知偏差,对期权定价结果几乎没有影响。
实物期权理论脱胎于金融期权定价理论,也依据B l a c k-S c h o l e s公式进行估值。
但与标准化设定的金融期权定价过程相比,实物期权在标的资产价格、执行价格、期限、波动率等关键要素方面都存在较大的模糊性(M y e r s,1977,T r i g e o r g i s,1995),其定价存在一个较明显的浮动区间。
因此,投资者存在的过分乐观、过度自信、短视、锚定等认知偏差,就会通过这些关键要素的非标准化取值而显著地影响到实物期权的价值。
目前实物期权分析广泛应用于实践,但对它的定价则主要还是套用金融期权定价方法,以此得到的理论价值容易与其实际价值表现脱节,从而误导市场。
为纠正这种定价偏误,本文借助行为金融学研究成果,尝试对投资者的认知偏差进行系统确认和技术表达,从比较静态分析的角度讨论这些认知偏差对实物期权定价的影响,以反映投资者的行为特征,使实物期权得到更准确定价。
并将这种更为接近现实的行为定价技术分析应用于实践,解释投资与公司治理等领域的一些实物期权“异常”现象。
股票的期权定价理论介绍和相关的数值分析康书隆2002级数量经济硕士研究生内容摘要：期权是人们为了规避市场风险而创造出来的一种金融衍生工具.期权之所以能够规避市场风险是因为金融证券的收益同相应的金融衍生物的收益总是负相关的。
理论和实践均表明，只要投资者合理的选择其手中证券和相应衍生物的比例，就可以获得无风险利率，从而获得无风险收益。
这种组合的确定有赖于对衍生证券的定价。
上个世纪七十年代初期，Black 和Scholes 通过研究股票价格的变化规律，运用套期保值的思想，成功的推倒出了无分红情况下股票期权价格所满足的随机偏微分方程。
从而为期权的精确合理的定价提供了有利的保障。
这一杰出的成果极大的推进了金融衍生市场的稳定，完善与繁荣。
本文首先将尝试着阐述期权定价理论产生的背景，过程及其带来的重大意义；在其后部分，我们将分析这一理论的数学基础以及Black---Scholes 随机微分方程的推导过程；最后我们将运用有限插分的方法来求解Black---Scholes 随机微分方程。
之所以这样做，是为了弥补Black---Scholes 随机微分方程解析解只能够对欧式期权进行定价的不足。
最后，我们将定量分析执行价格的变化和股票平均波动率变化对期权价格的影响。
并且绘制出一系列的图形帮助人们理解这种影响。
从而对于人们理解一些参数的变化对于期权价格的影响有一定的帮助。
关键词：维纳过程，伊藤过程，Black_Scholes 方程, 期权。
一、期权定价理论产生的背景，思想和重大意义１.1: 期权定价理论产生的背景Black-Scholes期权定价模型将股票期权价格的主要因素分为四个:预期股票价格、交割成本、股票价格波动幅度和时间。
其成功之处在于:第一,提出了风险中性(即无风险偏好)概念,且在该模型中剔除了风险偏好的相关参数,大大简化了对金融衍生工具价格的分析;第二,该型创新地提出了可以在限定风险情况下追求更高收益的可能,创立了新的金融衍生工具——标准期权。
