Finite element method for pricing rainbow option

彩虹期权的有限元方法

金宵

11749927

硕士

计算数学

李景治

2019-05-09

2019-07-10

哈尔滨工业大学

深圳

As a derivative tool, options are a great advantage of versatility and can be applied to a variety of investment strategies. Options can diversify risks, help strengthen the overall anti-risk capabilities of the financial market, and enhance the soundness of the financial system. It is precisely because of the continuous innovation and development of the domestic option futures market that strengthening the study of the options market is particularly important for improving the capital market business. With the rapid development of financial markets and the increasing number of options, the option pricing theory is also constantly improving. In the 1970s, since the birth of the Black-Scholes option pricing formula, scholars have substantially increased their research results on option pricing formulas. These include European options, Asian options, American options and so on. In recent years, the international financial market has emerged a large number of new varieties, variant options derived from the changes, combinations, and derivatives of standard options. Multi-factory options are one of them. For example, rainbow options, basket options, etc. are all such options. We derive general analytic approximations for pricing European rainbow options on N assets.Option trading is a brand new derivative product and effective risk management tool based on futures trading. It has unique economic functions and high investment value. One is that options are more conducive to spot business operations and hedging. By purchasing options, they can avoid the risk of additional margin in futures trading. Second, options are conducive to the development of order agriculture and solve the ”three ruralissues”. The government guides and encourages farmers to enter the options market by providing farmland with financial subsidies for the option premiums and payment processing fees. Third, futures investors can use options to circumvent market risks. Options can be futures ”reinsurance”. Different combinations of the two can constitute a variety of trading strategies with different risk preferences, providing investors with more trading options.Rainbow option, first proposed by Margrabe(1978), is a kind of option which involves many kinds of risk assets, and the aim is to get the best return on a variety of assets. Margrabe derives an exact solution for pricing an exchange option, under the assumption that the price of two assets follow correlated log-normal processes. Since a linear combination of log-normal processes is no longer log-normal,we can reduce the dimension of the correlated log-normal processes to one only when the strike of a two-asset option is zero. Johnson(1987) extends the two-asset rainbow option pricing formula provided by Stulz(1982) to the general case of N cases. Rubinstein derived the rainbow option pricing formula under the risk-neutral assumption, which depends on the lowest or highest price of the underlying asset. Hucki and kolokoltsov used game theory to study the pricing of rainbow options with fixed transaction costs. Meng et al. propose a rainbow option pricing method based on sine series expansion. Dockendorf et al. study the pricing of European rainbow options by stochastic co-integration model.Rainbow options differ from traditional American and European options in that they are excellent tools for hedging risks arising from holding multiple assets. Rainbow options are most commonly used to assess natural resources because they depend on both the price and quantity of natural resources in stock. Rainbow options are divided into three categories: better-of options, outer performance options, maximum call options and minimum call options. In this paper, we try to use numerical method to value European rainbow options.Option pricing is an ancient issue. As early as 1900 years, Louis Bachelier published the dissertation ”Speculative Trading Theory”. It was recognized as a milestone in modern finance. For the first time in his dissertation, a random walk model was used to give a random model of stock price operation. In this paper, he mentioned the pricing of options.In 1964, Paul Samuelson (Nobel Prize winner in economics) amended L.Bachelier’s model. Replace the stock price in the original model with the stock return. If St represents the stock price, then dSt/St represents the return of the stock. The stochastic differential equation proposed by P.Samuelson corrects the unreasonable situation in the original model that makes the stock price negative.Based on this model, P.Samuelson also studies the pricing of call options. Suppose V is the value of option, S is the stock price, K is the strike price, T is the maturity time, then the value of V is related to αc, αs. Here αc, αs are the mathematical expected value of the return of the original asset St and the option price Vt at time t = T, respectively. These two quantities depend on the individual preferences of investors. Therefore, although this formula is beautiful, it can not be used in actual transactions.In 1973, Fishcher Black and Myron Sholes established the pricing formula of call options. Here αc, αs do not appear, but instead riskless interest rate r. The innovation of this formula is that it does not depend on the preferences of investors. It leads all investors to a risk-neutral world in which risk-free interest rates are returned.Black-Scholes model is based on the ideal market which is quite different from the real market. In the past twenty years, economists have tried to find a more realistic option pricing model under these conditions. Many excellent results have been achieved, which greatly enrich the option pricing theory.Since the 1990s, especially in recent years, many economists have made extensive research on Option Pricing in incomplete markets, abnormal price jumps of underlying assets, or the variance of underlying assets return is not constant, and many important research results have been achieved. Among them, it is worth mentioning that it greatly enriches the relevant theory of option pricing model. On the basis of classical models, many new models are put forward. Starting from the Black-Scholes model, the option pricing theory has a history of nearly 40 years. With the development of modern financial markets, the market has created a lot of complex options products, along with stocks, bonds, foreign exchange and futures. Rainbow option is an important new financial derivative product. In the development process of market, the rate of return and price of the option products based on the underlying assets have also changed, and the pricing problem have always been one of the core problems of financial mathematics.The numerical methods of option pricing are divided into five categories: analytical solutions, tree methods, numerical methods for partial differential equations, Monte Carlo methods, and Fourier transform methods.(1) Analytical solution methodActually, according to the known stochastic differential equation (SDE) model, we then solve the process of the expression of this random process function. If we can find the analytical solution of this SDE, then the price of an European pathless-dependent option is The discounted expected value at the time of final value. This is an analytical solution to the option pricing.The advantage of this kind of method is obvious. Once the analytical solution exists, then the option price formula of the calculation speed is very fast, no matter to do fitting and optimization will be efficient on quality, and shortcomings of this kind of method is obvious. That is, for most of the model and the most exotic options, analytical solution may not exist.(2) Tree methodInform you of the volatility of a target asset, then you can construct a Binary tree up and down fluctuations of N segments. Then use the inverse to get the option price for the initial time.The tree model has the advantage that any continuous-time model cannot replace it. That is every pricing, in the tree model, regardless of American, European, pathdependent, singular, through the Backward Induction Principle The price is always accompanied by an explicit hedging strategy. In the continuous-time model, the problem of the continuous-time hedging strategy is a Backward Stochastic Differential Equation (BSDE) problem.On the other hand, the disadvantages of the tree model are also obvious. The high-dimensional problem tree model cannot be solved. Therefore, for the problems of multiple target assets, especially assets with a correlation coefficient, we can only appeal to PDE model.(3) PDE methodIn fact, different random models correspond to different PDE. BS PDE is just a PDE expression for a single asset that conforms to the geometric Brownian motion stochastic model. As for options, we often know the payoff of their final maturity date, so we use payoff function as the final value of PDE. If there are analytic solutions to PDE, the optimal solution is also the analytic solution. However, if the analytic solution does not exist, we must resort to numerical methods.The disadvantages of PDE methods are two main problems: path dependence problem and high dimension problem. Many of the PDE forms of path-dependent problems are troublesome or even unspeakable. If the numerical method of PDE rises from the plane grid to the spatial grid, it is not only complicated in complexity, but also more difficult to control in edge value conditions.(4) Monte Carlo methodThe Monte Carlo method is the most widely used method at present. Because there is no option price with advanced exercise attribute is actually an expectation, we can simulate the many routes and use the average number to estimate the real expectation. For American-style or Bermuda-type options with advanced exercise attributes, its option price is actually a random optimization problem. However, the shortcomings of Monte Carlo are also obvious: because we need to simulate millions of paths, and we need to do path calculations for exotic options, Americans need to do more regression. The Monte Carlo method has become synonymous with long computing time.(5) Fourier methodThe Fourier method is also called the eigenfunction method. For many models, their eigenfunctions are often expressed explicitly. We can use the inverse Fourier transform to obtain the density of the original random variables, and thus achieve the purpose of solving the option price.In this paper, we derive efficient numerical methods for pricing European rainbow options on two assets. We can also express the option as a sum of risk assets and risk-assets exchange options. The underlying asset prices are assumed to follow log-normal process_x0002_es. Based on the principle of no arbitrage, stochastic differential equation and Black Scholes model, we obtain the partial differential equation for pricing two-asset rainbow options. The continuous and discontinuous Galerkin discrete Finite Element method was applied to outer-performance options, better-of options and maximum call options respectively. The error analysis of the finite element method of pricing model was carried out, and the rationality and effectiveness of the method were verified by numerical examples.

期权作为一种衍生工具,具有通用性强的优点,可以应用于多种投资策略。 期权可以分散风险,有助于增强金融市场整体抗风险能力,增强金融体系的稳定性。 正是由于国内期权期货市场的不断创新和发展，加强对期权市场的研究对于完善资本市场业务尤为重要。随着金融市场的快速发展和期权数量的不断增加， 期权定价理论也在不断完善。20世纪70年代，布莱克-斯科尔斯期权定价公式诞生以来，学者们对期权定价公式的研究成果大大增加。这些期权包括欧式期权、亚式期权、美式期权等等。近年来，国际金融市场出现了大量的新品种、奇异期权和标准期权衍生品。多资产期权是其中之一。例如，彩虹期权、一篮子期权等等都是这样的多资产期权。我们推导出N种资产的欧式彩虹期权定价模型并研究其解析解、数值解。期权交易是基于期货交易的一种全新的衍生产品和有效的风险管理工具。 期权具有独特的经济功能和较高的投资价值。一是因为期权更有利于现场交易业务和风险对冲。通过购买期权，交易者可以避免在期货交易中追加保证金的风险。 二是有利于发展有序农业,解决“三农”问题。政府引导和鼓励农民进入期权市场，向他们提供关于期权保证金和交易产生的手续费的财政补贴。第三,期货投资者可以利用期权规避市场风险。期权可以是期货“再保险”的手段。两者的不同组合可以构成多种具有不同风险偏好的交易策略，为投资者提供更多的交易选择。彩虹期权最早由Margrabe于1978年提出，是一种涉及多种风险资产的期权， 其目的是从多种资产投资中获得最佳的收益。在假设两种资产的价格遵循相关的对数正态分布过程的前提下，Margrabe推导出了一个交换期权定价的精确解。 由于对数正态分布过程的线性组合不再是对数正态分布过程，因此只有当双资产选项的敲定价为零时，我们才能将相关的对数正态分布过程的维数减少到一维。 1987 年，Johnson将Stulz在1982 年提出的双资产彩虹期权定价公式推广到一般的多种资产的情况。Rubinstein在风险中性假设下推导出彩虹期权定价公式，该公式依赖于标的资产的最低或最高价格。Hucki和kolokoltsov利用博弈论研究了具有固定交易成本的彩虹期权定价问题。Meng等人提出了一种基于正弦级数展开的彩虹期权定价方法。Dockendorf等采用随机协调模型研究欧洲彩虹期权的定价问题。彩虹期权不同于传统的美式和欧式期权，因为它们是对冲多重资产风险的优秀工具。彩虹期权最常用来评估自然资源，因为它们同时依赖于自然资源的价格和库存数量。彩虹期权分为三类: 择优期权、利差期权、极大看涨期权和极小看涨期权。我们尝试用数值方法对欧式彩虹期权进行定价研究。 期权定价是一个古老的问题。早在 1900 年的时候,Louis Bachelier就发表了论文《投机交易理论》。它被公认为是现代金融的一个里程碑。在他的论文中首次采用随机游走模型给出了股票价格运行的随机模型。在那篇论文中，他提到了期权的定价问题。1964年，诺贝尔经济学奖得主 Paul Samuelson 对 Louis Bachelier的模型进行了修正。用股票的回报代替原模型中的股票价格。如果表示股票价格， 那么表示股票的回报。由P.Samuelson提出的随机微分方程修正了原模型中使股价为负的不合理情况。基于该模型， P.Samuelson 还研究了看涨期权的定价。假设是期权的价值，是股票价格,是执行价，是到期日，那么的价值和，有关，它们分别是原始资产以及期权价格在时的回报数学期望值。 这两个数量取决于投资者的个人偏好。因此， 尽管这个公式很漂亮， 但不能在实际交易中使用。1973年，Fishcher Black和Myron Sholes建立了看涨期权的定价公式。公式中， 没有出现，取而代之的是无风险利率。这个公式的创新之处在于它不依赖于投资者的偏好。它将所有投资者引向一个以无风险利率作为回报的风险中性的世界。Black-Scholes模型建立在与真实市场相差较大的理想市场基础上， 近二十多年来， 经济学家们试图在放松这些条件的情况下， 寻求更贴切实际市场的期权定价模型， 取得了许多优秀成果， 极大地丰富了期权定价理论。 90年代以来特别是近几年，很多经济学家对不完整市场、标的资产的价格存在异常变动跳跃或者标的资产收益率的方差不为常数等情况下的期权定价问题进行了广泛研究，取得了许多重要研究成果。其中最值得一提的就是极大地丰富了期权定价模型方面的相关理论，在经典的Black-Scholes模型基础上,提出了许多新的模型。期权定价理论从Black-Scholes模型出发， 已有近40年的历史。随着现代金融市场的发展，股票、债券、外汇和期货市场衍生了许多复杂的期权产品。 在市场发展过程中， 标的资产的期权产品在收益率和价格上也发生了变化， 其定价问题一直 是金融数学的核心问题之一。期权定价的数值方法分为五类: 解析解法、树方法、偏微分方程数值方法、蒙特卡罗方法和傅里叶变换方法。 （1）解析解法根据已知的随机微分方程(SDE)模型， 求解该随机过程的函数表达式。如果我们能找到这个SDE的解析解， 那么欧式无路径依赖期权的价格就是最终时刻价值的贴现期望值。 这是对期权定价的一种解析解方法。这种方法的优点是显而易见的，一旦解析解存在，那么期权价格公式的计算速度非常快，拟合和优化也是高效高质的。这种方法的缺陷是显而易见的， 那就是，对于大多数的模型和奇异期权来说，不存在解析解。(2)树方法如果已知目标资产的波动性， 然后就可以构造一个 N段上下波动的二叉树。然后用逆向归纳法得到初始时刻的期权价格。模型的优点是任何连续时间模型都不能替代它。 即每一种期权的定价问题，无论美式期权、 欧式期权、奇异期权还是路径依赖期权， 通过树模型的逆向归纳原则， 价格总是伴随着明确的套期保值策略。 在连续时间模型中，连续时间套期策略问题是一个倒向随机微分方程问题(BSDE)。另一方面， 树模型的缺点也很明显。高维问题无法利用树模型求解。 因此， 对于多风险资产的期权定价问题， 尤其是具有相关系数的多种风险资产， 我们只能求助于PDE模型。 (3) PDE方法事实上，不同的随机模型对应着不同的偏微分方程(PDE)。Black-Scholes 偏微分方程只是一个符合几何布朗运动随机模型的单个资产的PDE 表达式。对于期权，我们通常知道其在最终到期日的收益函数，因此我们使用收益函数作为PDE的终值。如果PDE有解析解，那么最优解也是解析解。然而，如果解析解不存在，我们必须求助于数值方法。PDE方法的缺点主要有两个方面：路径依赖和高维问题难度大的缺点。路径相关问题的PDE 形式很多都是很麻烦甚至难以描述的， 如果PDE 的数值方法从平面网格上升到空间网格，不仅形式上复杂，而且在边值条件上也更难以控制。 (4)蒙特卡罗方法蒙特卡罗方法是目前应用最广泛的方法。因为没有提前实施属性的的期权价格实际上是一个数学期望，所以我们可以模拟期权的多条路径，用平均值来估计实际期权价格的期望值。对于具有提前实施属性的美式期权或百慕大式期权，其期权价格实际上是一个随机优化问题。 然而，蒙特卡罗的缺点也是显而易见的： 因为我们需要模拟数以百万计的路径，对奇异期权进行路径计算，美式期权则需要做更多的回归。蒙特卡罗方法已经成为长计算时间的代表。 (5) 傅里叶方法傅里叶方法也称为特征函数法。 对于许多模型，它们的特征函数通常是显式表示的。利用傅里叶反变换得到原始随机变量的密度分布函数，从而达到求解期权价格的目的。本文推导了一种欧式彩虹期权定价的数值方法，这种期权基于两种风险资产。 部分彩虹期权也可分解为风险资产和交换期权的组合。假设风险资产满足对数正态分布的随机过程模型。基于无套利原理、随机微分方程和Black-Scholes模型， 得到了两种风险资产的彩虹期权定价的偏微分方程。分别将连续和不连续的Galerkin离散化有限元方法应用于利差期权、择好期权和极大看涨期权。对定价模型的有限元法进行了误差分析， 并通过数值算例验证了该方法的合理性和有效性。

rainbow option Black-Scholes model finite element method

彩虹期权 Black-Scholes模型 有限元方法

英语

联合培养

南方科技大学

Jin X. Finite element method for pricing rainbow option[D]. 深圳. 哈尔滨工业大学,2019.