金融衍生产品中美式与亚式期权定价的数值方法研究
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摘要
早期的金融市场上只有四种金融工具：银行存款、汇票(银行承兑汇票)、债券和股票[朱利安·沃姆斯利(2003)]。最早的银行存款产生于13世纪，所知最早的是卢卡的瑞塞迪银行(Ricciardi of Luca)，它在1272-1310年给英国王室提供了40万英镑的借款，虽然这笔借款的拖欠导致了该公司的破产，但同时也留下了一个早期的金融风险的例子。汇票(Bill of Exchange，Draft)作为以支付金钱为目的，并且可以流通转让的债权凭证几乎和银行存款同时代产生，而各种类型的债券(Bond)作为定期获得利息，到期偿还本金及利息的凭证是到16世纪才出现的。第一只真正意义上的政府债券是1555年的Grand Parti of FrancisⅠ，它不是面对少数银行发行的，而是面向所有的投资者。股票则起源于1600年的英国东印度公司，最初的股票是出海前向人集资，航次终了将个人的出资及该航次的利润交还出资者的凭证。第一家永久性的股份公司是成立于1602年的荷兰东印度公司。1613年起，该公司改为四航次才派发一次利润，这也正是“股东”和“派息”的前身。
由此可以看出，早期的金融投资相对简单，投资者遵循的是“低买高卖”，“不要把所有的鸡蛋放在一个篮子里”之类的朴素的投资哲学。直到20世纪后半叶，金融市场的发展才出现急剧上升的趋势，数量工具也在金融市场中崭露头脚。
1900年，Louis Bachelier发表了他的学位论文“Théorie de la Spéculation”(投机交易理论)[Bachelier(1900)]。它被公认为现代金融学的里程碑。他在论文中首次利用随机游走的思想给出了股票价格运行的随机模型，在这篇论文中，他就提到了期权的定价问题。
七十年代初，Black和Scholes取得了一个重大的突破[Black(1989)]，掀起了第二次金融变革。他们推导出了基于无红利支付股票的任何衍生证券的价格必须满足的微分方程
Fisher Black和Myron Scholes在他们那篇突破性的论文[Black & Scholes(1973)]中成功求解了他们的微分方程，得到了欧式看涨期权和看跌期权的精确公式。RobertMerton随即将模型推广到了更一般的范畴[Merton(1971)][Merton(1973)]。因为他们的的杰出工作，Robert Merton和Myron Scholes一起分享了1997年的诺贝尔经济学奖。
金融市场的迅速发展也推动了相关的数学工具的迅猛发展[巴克斯特、伦尼(2006)]。近年来在金融数学方面应用越来越广泛的倒向随机微分方程(BSDE)的线性形式首先由(Bismut(1978)]在1978年引入，1990年[Pardoux & Peng(1990)]研究了Lipschitz条件下非线性倒向随机微分方程解的存在唯一性定理。[Duffie & Epstein(1992b)]在研究随机微分效用过程中也独立地引进了倒向随机微分方程的一个特别典型的情况。他们发现可以用它来描述不确定经济环境下的消费偏好(即计量经济学的基础—效用函数理论)随后El Karoui和Quenez发现金融市场的许多重要的衍生证券(如期货期权等)的理论价格都可以用倒向随机微分方程解出，特别是上面提到的Black-Scholes公式正可以被归纳为BSDE线性情况下的一种特殊形式。可参见[Duffie & Lions(1992)]，[Duffie，Geoffard，& Skiadas(1994)]，[El Karoui & Quenez(1995)]，[El Karoui，Peng & Quenez(1997)]。
求解BSDEs一般有两种途经：一种途径是用随机方法直接对BSDEs求解，如Monte Carlo方法；另一种途径则是通过求解PDE来求解BSDEs。第二种途径的理论基础是[Peng(1991)]获得的一类与正向随机微分方程(FBSDEs)的解耦合的倒向随机微分方程(BSDE)的解与一类拟线性二阶偏微分方程(PDE)的解的对应关系，即非线性Feynman-Kac公式。Feynman-Kac公式将与SDE耦合的BSDE的解和PDE的解联系起来。这样一方面可以使用相对比较成熟的PDE方法来解BSDE，另一方面可以反过来使用BSDE的随机算法来解决PDE中的问题。见[Ma，Protter & Yong(1994)]，[Ma & Yong(1995)]，[Duffie，Ma，& Yong(1995)]，[Ma，Protter，San Martín & Torres(2002)]，[Peng & Xu(2006a)]等。随后[Hu & Peng(1995)]又通过概率方法获得了一类完全耦合的SDE和BSDE解的存在唯一性。
在实际市场中，更多的衍生证券定价是得不到确切的解析公式的，这时候就可以使用数值计算方法。已有的金融衍生产品的定价方法种类非常丰富，如二叉树方法，Monte Carlo随机模拟方法以及有限差分方法、特征方法等各种PDE中的数值方法(可参见[Karatzas & Shreve(1998)]、[Seydel(2004)]、[姜礼尚(2003)])。其中经典的二叉树方法作为一种简便易行的期权估价方法由[Cox，Ross & Rubinstein(1979)]在著名的CRR模型中引入，随后扩展出三叉树方法以及切片法(可参见[Parkinson(1977)]和[Boyle(1988)])，并不断改进(见[Hull & White(1988)][Levy，Avellaneda & Paras(1994)][Tian(1993)]等)。二叉树方法和切片法的收敛性证明分别由[Lamberton(1998)]和[Amin & Khanna(1994)]给出。至于Monte Carlo方法，[Douglas，Ma and Protter 1996]，[Ma，Protter and Yong 1994]针对正倒向随机微分方程(FBSDE)情形。[Bouchard，Ekeland and Touzi 2002]考虑了非线性情形下的蒙特卡洛(Monte-Carlo)方法。最近，[zhao，Chen & Peng(1998)]通过蒙特卡洛(Monte-Carlo)方法研究提出了一类解倒向随机微分方程的高精度数值格式。
通过BSDE与PDE的联系，[Ma，Protter & Yong(1994)]提出了一种用PDE来解一类正倒向随机微分方程的“四步法”，随后[Douglas，Jr．，Ma & Protter et al．(1996)]提供了一种特征线差分方法来解上述问题，本论文就是基于“四步法”这种途径分别考虑了美式期权和亚式期权的定价问题：
欧式期权价格可以通过解一个混合导数的高维抛物形偏微分方程(组)的初边值问题得到，通常用数值方法求解，常用的方法有差分方法、有限元方法、有限体积方法等，可参见[李荣华&冯果忱(1996)][李德元&陈光南(1995)]。与欧式期权不同，由于美式期权可在期权有效期的任何一个时刻执行，美式期权实际上是一个非线性问题。该问题可转化为一个自由边界问题[姜礼尚(2003)]或者依赖于时间的线性互补问题(linear complementarity problem，LCP)[Oxsterlee(2003)]。解决线性互补问题现有很多数值方法，如[Clarke & Parrott(1996)]提出了近似线性互补方程的PFAS方法，其优点是该多层格式的迭代次数与网格数是不相关的，并且得到了有效的数值解，但是其执行难度是非常大的。[Oxsterlee(2003)]应用并改进了PFAS格式，有效的避免了数值解的震荡。[Zvan，Forsyth & Vetzal(1998)]对于美式期权的可提前执行约束采用惩罚方法来解决，在原偏微分不等式中引入一个惩罚项，使其成为一个偏微分方程。此方法避免了数值解的震荡，但是收敛性会随着离散的优化而降低。[Ikonen & Toivanen(2005)][Ikonen & Toivanen(2004)]应用了另外一种基于算子分裂的方法来解决美式期权的约束条件。通过离散将空间算子分解为几个简单的算子，并将每个时间层分解成与这些分解后的算子等数目。[Ikonen & Toivanen(2005)]中，他们使用这种方法来解决随机波动率下的美式期权定价问题，将差分算子及可提前执行约束分解成一系列一维线性互补问题(LCPs)，然后应用Brennan&Schwartz格式[Brennan & Schwartz(1977)]近似一维线性互补问题。这种方法的优点是将复杂的高维问题化为多个简单的一维问题，使得计算的复杂性大大减少。
但是考虑到期权定价中满足极值原理的重要性，本文针对描述美式期权定价的二维问题提出了一类新的有限体积九点格式和相应的算子分裂格式，该格式结合价格漂移方向近似二阶混合导数，使得提出的格式满足极大值原理和一致误差估计。
亚式期权是奇异期权的一种，其到期收益函数依赖于标的资产有效期至少某一段时间内的某种形式的平均[约翰·赫尔(1997)]。通常取标的资产价格按预定时间内的算术平均值或几何平均值作为其平均价格。目前亚式期权是OTC(柜台交易)市场上广受交易者青睐的金融工具，但即使在标的遵循几何布朗运动的假设下，也只有几何平均亚式期权的定价可以得到显式表达式。而在OTC市场上交易的绝大部分亚式期权都是标的算术平均，对标的算术平均亚式期权进行定价更多的是采用数值方法如二叉树方法[Klassen(2001)]、特征线差分方法[Jiang & Dai(2002)]或以标准几何平均亚式期权[Seydel(2004)]来近似逼近。
本文对亚式期权定价问题给出了恰当的边界条件，并提出了一类加权迎风有限体积格式和相应的交替方向格式。对价格漂移占优问题，采用加权迎风技术以避免数值解的非物理震荡；同时，结合亚式期权定价问题特性提出相应的交替方向。从理论上严格证明了提出的格式满足极大值原理，得到了一致误差估计。
本文最后提出了高维偏微分方程的迎风有限体积数值格式，因在BSDE模型中我们讨论的问题都是高维问题，而金融市场中变动因素的多样性和复杂性也决定了在金融定价问题中我们实际上面对的都是高维问题．所以本文结合问题特性提出的格式满足了极大值原理，解决了通常的有限体积格式不满足极值原理的问题并得到了一致误差估计。
论文组织和具体安排如下：
本文第1章主要介绍了金融衍生产品的历史与现状，及金融衍生产品定价的由来，随后从BSDE的角度给出了金融衍生产品定价的数学模型。
本文第2章介绍了倒向随机微分方程(BSDE)与一类拟线性二阶偏微分方程(PDE)存在的对应关系，这就是[Peng(1991)]中获得的非线性Feynman-Kac公式。于是BSDE中可问题可以转化到相对比较成熟的PDE中，使用PDE数值方法方法来解BSDE。同时可以反过来使用BSDE的随机算法来解决PDE中的问题。
本文第3章则简单回顾了一下已有的金融衍生产品的定价方法，如二叉树方法，Monte Carlo随机模拟方法以及有限差分方法、特征方法等PDE中的各种数值方法。
本文第4章主要研究了美式期权定价问题的有限体积数值模拟方法。对随机波动率下美式期权定价问题，我们用新的方法近似二阶混合交叉导数项，用迎风技术近似一阶对流项[Liang & Zhao(1997)]，提出一类新的有限体积九点格式。
其中算子A_x、A_y、A_(xy)和A_(yx)分别由自四个不同方向(x-方向，xy-方向，yx-方向以及y-方向)的差分方程的系数构成：
其中
相应的θ格式：
同时，我们对所提出的九点格式提出有效的算子分裂格式[Wang & Zhao(2003)]。其中k=0，…，l-1，
该分裂格式按照x，y，xy和yx四个方向分解，使得问题的求解变为四个方向的一维求解。对提出的格式，我们有极大值原理和一致误差估计：
若记
定理0．1．1(极值原理)假定V_(ij)~k为G_h上的一网函数，满足以下不等式：
其中L_h中的A_(i，j)满足假设(4．3．1)，则V_(ij)~k不可能在内点取正的极大，除非V_(ij)~k为常数。
定理0．1．2(稳定性估计)若L_hV_(ij)~k=0，则
定理0．1．3(误差估计)在定理0．1．1的条件下，离散格式(0．1．5)具有误差估计：‖E~k‖≤K sup_k‖R~k‖，其中E~k为分量为e_(ij)~k的向量。
为说明所提格式的有效性，我们给出几个数值算例。通过与投影超松弛(PSOR)方法[Ikonen & Toivanen(2005)]计算出的结果进行比较，我们发现，结果与其吻合。更加复杂的一般的多因素期权定价问题将在以后讨论。
本文第5章主要研究算术平均亚式期权定价问题的交替方向迎风有限体积方法。对算术平均亚式期权定价问题
我们通过方程(0．1．8)给出了适宜的边界条件，令
则亚式期权问题(0．1．8)的初边值条件为：
并在此基础上对价格漂移占优问题，采用加权迎风技术以避免数值解的非物理震荡；同时结合亚式期权定价问题特性，我们提出相应的交替方向。最终得到了一类新的交替方向加权迎风有限体积(ADFV)格式：(0．1．11)中V_(ij)~(n-1／2)的边值和V_(ij)的初边值条件为：
其中
我们从理论上严格证明了提出的格式满足极大值原理，也得到了一致误差估计：
定理0．1．4(极值原理)假定V_(ij)~n满足以下条件：
如果V_(ij)~n在Θ上不是常数，则V_(ij)~n的正最大值只能在(?)Θ上达到。
定理0．1．5(稳定性估计)因定价问题中f=0，我们有以下稳定性估计：
其中L~∞(Ω)为标准Banach空间，
定理0．1．6(误差估计)若v∈L~∞(0，T；L~(4，∞)(Ω))∩W_∞~1(0，T；W_∞(Ω))∩W_∞~2(0，T；L_∞(Ω))，在条件(5．3．3)及定理1的条件下，我们有误差估计：
其中
需要指出的是这一类加权迎风有限体积格式和相应的交替方向格式对于高维问题同样适用。
本文第6章主要研究提出了高维偏微分方程的迎风有限体积数值格式，该格式同样具有极大值原理和一致误差估计。
在BSDE模型中我们讨论的问题都是高维问题，而金融市场中变动因素的复杂性也决定了在实际金融定价问题中我们面对的都是高维问题。以下我们考虑如下二阶抛物型偏微分方程的数值解，
其中f=f((x，y)，t)，r=r((x，y)，t)，v=(v_1((x，y)，t)，v_2((x，y)，t)，…，v_d((x，y)，t))~T，D=(d_(ij)((x，y)，t))_(d×d)，c=c((x，y)，t)。Ω是定义在R~d上的求解区域，终端时刻为T。
基于第6．4节中的分析，我们得到了(0．1．16)的中心有限体积格式(6．5．8)、迎风有限体积格式Ⅰ(6．5．9)和全离散迎风有限体积格式Ⅱ(6．5．10)。需要特别指出的是通常的中心有限体积格式(6．5．8)和迎风有限体积格式Ⅰ(6．5．9)是不满足极值原理的，而对本文提出的全离散迎风有限体积格式Ⅱ(6．5．10)，我们有如下定理：
注意到格式(6．5．10)可以表示成
其中
所以用前面类似的方法可以得到如下极值原理、稳定性估计和误差估计：
定理0．1．7(极值原理)假定V_(ij)~k为G_h上的一网函数，满足以下不等式：
则V_(ij)~k不可能在内点取正的极大，除非V_(ij)~k为常数。
定理0．1．8(稳定性估计)若L_hV_(ij)~k=0，则
定理0．1．9(误差估计)在定理1的条件下，离散格式(6．5．8)具有误差估计：
其中E~k为分量为e_(ij)~k的向量。C为是不依赖于解u，V，剖分h和Δt的一个常数。
注0．1．1对格式(6．5．10)同样可以提出相应的分裂格式，可证明其满足极值原理，具有以上误差估计。详细细节和数值模拟正在进行和验证中。
In early financial market there are only four kinds of financial tools: bank deposit, bank draft (bank acceptance), bond and stock. The earliest bank deposit appeared in the 13th centuries. It is known that Ricciardi bank of Luca is the earliest one. During the year from 1272 to 1310, this bank provided a loan of 400,000 pound to the royalty of England. Although the default of this loan caused this company's bankruptcy, it did give an example of the financial risk of early days. Bill of Exchange, which is the creditor's rights with the goal to pay and can be circulated and transferred, arises almost at the same time with bank deposit. However, all kinds of bond, in which the issuer owes the holders a debt and is obliged to repay the principal and interest (the coupon) at a later date (termed maturity), didn't appeared until the 16th century. The first real government bond is Grand Parti of Francis I in 1955. It was issued not just for a handful of banks, but faced all the investors. Stock originated from the British East India company in 1600. The earliest stock is a certificate of pooling capital to finance the building of ships. The first permanent stock company to issue shares of stock was the Dutch East India Company, in 1602. From 1613, this company started to distribute the profit every four voyages, which was commonly regarded as a predecessor of "stockholder" and "distributed dividends".
Thus, we can see that the early financial investment is relatively simple. Investors followed some plain philosophy principles such as "lowly buy - highly sell" , "don' t put all the eggs into one basket" , etc. It was not until the latter half of 20th century has the financial market grown up rapidly and mathematical tools took more and more part in it. These changes were mainly due to the two financial revolutions in the Wall street.
The first financial revolution started from the paper[Markowitz( 1952)] "Portfolio Selection" , which is Harry Markowitz' s doctoral thesis. The early version of this thesis discussed how to get the portfolio which could maximize the anticipated income through the combination of risk assets (dispersal investment) and at the same time it could keep the risk of single security at an acceptable level. The mathematical tool in his thesis was called Markowitz mean-variance analysis) [Markowitz(1959)]. Before his portfolio theory, investors also discussed risk and profit, but as they couldn' t quantify some important indices, their portfolios were usually very subjective and hardly to make clear why they could get such anticipated profit. Markowitz' s theory solved these problems. Later, many people, such as Willam Sharp, made further research on the problem of profit and risk when markets reached the "balance" (supply and demand are equal) and gave capital asset pricing model - CAPM [Sharpe(1964)]. This model indicates that when markets achieve balance, the factors which determine the asset profit (i.e. pricing assets) are the ,β-measure system risk, whilst non-system risk plays no role in pricing assets, and it is a kind of linear relationship between expecting profit andβ. The standard CAPM gives a complete answer to the problem of the determining mechanism of asset profits when the markets achieve balance. Because of their excellent work, Markowitz and Sharp shared the Nobel prize in economics with Merton Miller in 1990.
Before those years we mentioned above, we should point out especially that in 1900, Louis Bachelier published his thesis "Theorie de la Speculation" [Bachelier(1900)], which is a milestone of modern financial study. In his thesis, he used random walk for the first time to describe stock prices and also he mentioned the option pricing problem.
At the beginning of the 70's, Black and Scholes obtained a significant breakthrough, which raised the second financial revolution. They deduced a differential equation which should be satisfied for any price of derivative securities based on any non-dividend payment stock.
In their breakthrough paper, Black and Scholes solved their differential equation successfully and got the precise formula for European call and put options. But in the real market, the precise formula can' t be obtained for pricing many derivative securities. But we can use numerical methods to solve it.
The linear form of backward stochastic differential equations (BSDEs) was first introduced in [Bismut(1978)] in 1978. Later, [Pardoux & Peng(1990)] studied the existence and uniqueness of a kind of nonlinear backward stochastic differential equations under Lipschitz condition. In [Duffie & Epstein(1992b)], they introduced a special case of backward stochastic differential equations independently during their study in stochastic differential utility. They found it could be used to describe the consumable preference under uncertain economic environment (i.e. econometrics foundation- utility function theory).
Subsequently, El Karoui and Quenez found that the theoretical price of many important derivative securities (e.g. futures and options) could be solved by backward stochastic differential equations, especially Black-Scholes formula, which is a special linear form of BSDEs. See [El Karoui & Quenez(1995)],[El Karoui, Peng & Quenez(1997)], [Duffie & Lions(1992)], [Duffie, Geoffard, & Skiadas(1994)].
In [Peng(1991)], Peng obtained a probabilistic interpretation for system of second order quasilinear parabolic partial differential equation, i.e. nonlinear Feynman-Kac formula, so that it connected the solution of BSDE associated with a kind of SDE with PDE. Therefore we can use the ready-made methods of PDE to solve BSDE, and inversely, some PDE problems can be solved by BSDEs' stochastic algorithm. See[Ma, Protter & Yong(1994)], [Ma & Yong(1995)], [Duffie, Ma, & Yong (1995)], [Ma, Protter, San Martin & Torres(2002)] etc. In [Hu & Peng(1995)], Peng also obtained the existence and uniqueness of a kind of completely coupling SDE and BSDE.
The European option price can be obtained by solving an initial and border value problem of a highly dimensional parabolic partial differential equation(s) with mixed derivatives. Usually we use numerical methods, such as difference method, finite element method, limited volume method etc. Different from European option, American option can be executed at any time during the period of validity. It can be changed into a linear complementary problem (LCP) depending on time, [Oxsterlee(2003)].
There are many numerical methods to solve LCP, such as [Clarke & Parrott(1996)], [Clarke & Parrott(1999)], which gave a method of PFAS for an approximately linear complementary problem. The advantage is its iterative times are independent with the grid number, and get an effective numerical solution, but it is hard to perform it. PFAS was used and improved in [Oxsterlee(2003)] and the vibration of numerical solutions was avoided effectively. In [Zvan, Forsyth & Vetzal(1998)], they used penalization method to solve the American option which can be executed in advance, i.e. a penalty term was introduced in the previous partial differential inequality so that it became into a partial differential equation. This method avoids the vibration of numerical solutions, but the convergence property decreases with the optimization of discretizing. In [Ikonen & Toivanen(2005)] [Ikonen & Toivanen(2004)], they used another method based on operator splitting to solve the American option problem with constraint. They discreted space operators into several simple ones, and separated every time layer into the same amount with operators. In [Ikonen & Toivanen(2005)], they used this method to solve the pricing of American option under stochastic fluctuating ratio, i.e. they decomposed the difference operators and the constraint of executing in advance into a series of one dimensional LCPs and ap- plied Brennan&Schwartz format [Brennan & Schwartz(1977)] to approximate LCPs. The advantage of this method is to transform a complicated highly dimensional problem into several simple one dimensional ones so that decreasing much complex calculating.
At the beginning of 90' s, with diversification of market requirements, it is hard to satisfy the special demand of clients only using the standard options (e.g. European option, American option). So some options with more dealing manners and dealing price appear, which is called exotic options [Hull(2000)].
Asian option is a kind of exotic option. The income function in the due day depends on the average of some form at some period at least in the valid period of target assets. The arithmetic or geometric average of target assets in the anticipated period is usually used as it' s average price. At present, Asian option is a financial tool widely used in the OTC (over-the counter) market, but even in the assumption that the target follows geometric Brownian motion, only pricing of geometric average Asian option can be expressed explicitly. However, the target of most Asian option trading in the OTC market is arithmetic average. Pricing of this kind of Asian option, we often use numerical method such as binomial method[Klassen(2001)], characteristic difference method [Jiang & Dai(2002)]or approaching with standard geometric-average Asian options[Seydel(2004)]. This paper is organized as follows:
In Chapter 1, a brief history review of financial derivatives is given. Since BSDEs play a important role in mathematical finance, some pricing model are given in BSDEs.
In Chapter 2, The correspondence between BSDE and a kind of quasi-linear 2-order parabolic PDE , i.e. the "nonlinear Feynman-Kac formula" is introduced . this formula is given in [Peng(1991)], thus the BSDE problem and PDE problem can be transformed into each other.
In Chapter 3, Some pricing method for financial derivatives are given, such as binomial method, Monte Carlo method and some PDE method.
In Chapter 4 of my doctoral thesis, I studied finite volume numerical simulation method of pricing for American option. For American option under stochastic volatility, a new kind of 9-point finite volume scheme is proposed, in which using a new technique for the 2-order hybrid cross derivatives, and upwind method for the convection item [Liang & Zhao(1997)]where operators A_x, A_y A_(xy) and A_(yx) are consisted of coefficiencies of difference equation from four different direction (the x-direction, the xy-direction, the yx-direction and the y-direction) .whereThe correspondingθ-scheme:
Meanwhile, the operator splitting scheme can be proposed for this 9-point scheme [Wang & Zhao(2003)].where k = 0,…,l-1,
This operator splitting scheme splits according to x, y, xy, and yx four directions, thus the problem turns into four 1-dim problem in different directions.
We have maximum principle and error estimate for the scheme proposed:Let
Theorem 0.2.1 (Maximum Principle) Let V_(ij)~k be a net function on G_h satisfies following inequality:where the A_(i,j) in L_h satisfies assumptions(4.3.1), Then V_(ij)~k will not achieve its maximum at the inner points, unless V_(ij)~k is constant.
Theorem 0.2.2 (Stability) Let L_hV_(ij)~k = 0, then
Theorem 0.2.3 (Error Estimate) Under the condition of theorem 0.2.1, the discrete scheme(0.2.5)has the following error estimate:where E~k is a vector with components of e_(ij)~k.
To show the validity of this scheme, some numerical examples are given, and the result tallies with the PSOR method [Ikonen & Toivanen(2005)]. More general multi-factor American option pricing problem will be considered later.
In Chapter 5 of my doctoral thesis, I studied an alternating-direction implicit upwind finite volume method for pricing Asian options. [Seydel(2004)]:Proper boundary conditions are given through equation (0.2.8) , letThen the initial and boundary condition for pricing Asian option(0.2.8) is: For convection-dominated problems, using upwind method to avoid non-physical shock, a new kind of alternating-direction implicit finite volume method according to the Asian option can be proposed:the boundary condition for V_(ij)~(n-1/2) and the initial and boundary condition for V_(ij)~n in (0.2.11) are:
Maximum principle for this scheme is theoretically proved and the error estimates also derived:
Theorem 0.2.4 (Maximum Principle) Let V_(ij)~n satisfies the following condition:Let V_(ij)~n is not constant on 6, then the positive maximum of V_(ij)~n can only be achieved onde.
Theorem 0.2.5 (Stability) For f=0 in pricing model, we have the following stability estimate:where L~∞(Ω) is standard Banach space,
Theorem 0.2.6 (Error Estimate) Let , Under the condition (5.3.3) and the assumptions of theorem (0.2.4), we have error estimate:
It should be pointed out that this alternating-direction implicit finite volume method is also valid for high-dimensional problems.
In Chapter 6, a kind of upwind control volume method are proposed, which is designated for the high dimensional problems arises in financial markets. Maximum principle and error estimates also derived:
Based on the analysis in section 6.4, we propose three scheme for problem(6.2.1): the Central Control Volume Scheme(6.5.8), the Upwind Control Volume Scheme I (6.5.9) and the Upwind Control Volume Scheme II (6.5.10) . It should be noted that the traditional Central Control Volume Scheme(6.5.8) and Upwind Control Volume Scheme I (6.5.9) don't satisfy maximum principle, but for the Upwind Control Volume Scheme II we proposed, the answer is yes.
Scheme(6.5.10)can be expressed as:By analogy to the forenamed procedure, we can get the following maximum principle, stability analysis and the error estimate.
Theorem 0.2.7 (Maximum Principle) LetV_(ij)~kbe a net function on G_h, which satisfies inequality:Then V_(ij)~k will not achieve its maximum at the inner points, unless V_(ij)~k is constant.
Theorem 0.2.8 (Stability)
Theorem 0.2.9 (Error Estimate) Under the condition of theorem 0.2.1, the discrete scheme (6.5.8)has the following error estimate:where E~k is a vector with components of e_(ij)~k, C is a constant, doesn't depend on the solution u,V, partition h adn△t.Remark 0.2.1 Splitting scheme for (6.5.10) can also be proposed, together with the maximum principle and error estimates, while the details and the numerical simulation are under verifying.
由此可以看出，早期的金融投资相对简单，投资者遵循的是“低买高卖”，“不要把所有的鸡蛋放在一个篮子里”之类的朴素的投资哲学。直到20世纪后半叶，金融市场的发展才出现急剧上升的趋势，数量工具也在金融市场中崭露头脚。
1900年，Louis Bachelier发表了他的学位论文“Théorie de la Spéculation”(投机交易理论)[Bachelier(1900)]。它被公认为现代金融学的里程碑。他在论文中首次利用随机游走的思想给出了股票价格运行的随机模型，在这篇论文中，他就提到了期权的定价问题。
七十年代初，Black和Scholes取得了一个重大的突破[Black(1989)]，掀起了第二次金融变革。他们推导出了基于无红利支付股票的任何衍生证券的价格必须满足的微分方程
Fisher Black和Myron Scholes在他们那篇突破性的论文[Black & Scholes(1973)]中成功求解了他们的微分方程，得到了欧式看涨期权和看跌期权的精确公式。RobertMerton随即将模型推广到了更一般的范畴[Merton(1971)][Merton(1973)]。因为他们的的杰出工作，Robert Merton和Myron Scholes一起分享了1997年的诺贝尔经济学奖。
金融市场的迅速发展也推动了相关的数学工具的迅猛发展[巴克斯特、伦尼(2006)]。近年来在金融数学方面应用越来越广泛的倒向随机微分方程(BSDE)的线性形式首先由(Bismut(1978)]在1978年引入，1990年[Pardoux & Peng(1990)]研究了Lipschitz条件下非线性倒向随机微分方程解的存在唯一性定理。[Duffie & Epstein(1992b)]在研究随机微分效用过程中也独立地引进了倒向随机微分方程的一个特别典型的情况。他们发现可以用它来描述不确定经济环境下的消费偏好(即计量经济学的基础—效用函数理论)随后El Karoui和Quenez发现金融市场的许多重要的衍生证券(如期货期权等)的理论价格都可以用倒向随机微分方程解出，特别是上面提到的Black-Scholes公式正可以被归纳为BSDE线性情况下的一种特殊形式。可参见[Duffie & Lions(1992)]，[Duffie，Geoffard，& Skiadas(1994)]，[El Karoui & Quenez(1995)]，[El Karoui，Peng & Quenez(1997)]。
求解BSDEs一般有两种途经：一种途径是用随机方法直接对BSDEs求解，如Monte Carlo方法；另一种途径则是通过求解PDE来求解BSDEs。第二种途径的理论基础是[Peng(1991)]获得的一类与正向随机微分方程(FBSDEs)的解耦合的倒向随机微分方程(BSDE)的解与一类拟线性二阶偏微分方程(PDE)的解的对应关系，即非线性Feynman-Kac公式。Feynman-Kac公式将与SDE耦合的BSDE的解和PDE的解联系起来。这样一方面可以使用相对比较成熟的PDE方法来解BSDE，另一方面可以反过来使用BSDE的随机算法来解决PDE中的问题。见[Ma，Protter & Yong(1994)]，[Ma & Yong(1995)]，[Duffie，Ma，& Yong(1995)]，[Ma，Protter，San Martín & Torres(2002)]，[Peng & Xu(2006a)]等。随后[Hu & Peng(1995)]又通过概率方法获得了一类完全耦合的SDE和BSDE解的存在唯一性。
在实际市场中，更多的衍生证券定价是得不到确切的解析公式的，这时候就可以使用数值计算方法。已有的金融衍生产品的定价方法种类非常丰富，如二叉树方法，Monte Carlo随机模拟方法以及有限差分方法、特征方法等各种PDE中的数值方法(可参见[Karatzas & Shreve(1998)]、[Seydel(2004)]、[姜礼尚(2003)])。其中经典的二叉树方法作为一种简便易行的期权估价方法由[Cox，Ross & Rubinstein(1979)]在著名的CRR模型中引入，随后扩展出三叉树方法以及切片法(可参见[Parkinson(1977)]和[Boyle(1988)])，并不断改进(见[Hull & White(1988)][Levy，Avellaneda & Paras(1994)][Tian(1993)]等)。二叉树方法和切片法的收敛性证明分别由[Lamberton(1998)]和[Amin & Khanna(1994)]给出。至于Monte Carlo方法，[Douglas，Ma and Protter 1996]，[Ma，Protter and Yong 1994]针对正倒向随机微分方程(FBSDE)情形。[Bouchard，Ekeland and Touzi 2002]考虑了非线性情形下的蒙特卡洛(Monte-Carlo)方法。最近，[zhao，Chen & Peng(1998)]通过蒙特卡洛(Monte-Carlo)方法研究提出了一类解倒向随机微分方程的高精度数值格式。
通过BSDE与PDE的联系，[Ma，Protter & Yong(1994)]提出了一种用PDE来解一类正倒向随机微分方程的“四步法”，随后[Douglas，Jr．，Ma & Protter et al．(1996)]提供了一种特征线差分方法来解上述问题，本论文就是基于“四步法”这种途径分别考虑了美式期权和亚式期权的定价问题：
欧式期权价格可以通过解一个混合导数的高维抛物形偏微分方程(组)的初边值问题得到，通常用数值方法求解，常用的方法有差分方法、有限元方法、有限体积方法等，可参见[李荣华&冯果忱(1996)][李德元&陈光南(1995)]。与欧式期权不同，由于美式期权可在期权有效期的任何一个时刻执行，美式期权实际上是一个非线性问题。该问题可转化为一个自由边界问题[姜礼尚(2003)]或者依赖于时间的线性互补问题(linear complementarity problem，LCP)[Oxsterlee(2003)]。解决线性互补问题现有很多数值方法，如[Clarke & Parrott(1996)]提出了近似线性互补方程的PFAS方法，其优点是该多层格式的迭代次数与网格数是不相关的，并且得到了有效的数值解，但是其执行难度是非常大的。[Oxsterlee(2003)]应用并改进了PFAS格式，有效的避免了数值解的震荡。[Zvan，Forsyth & Vetzal(1998)]对于美式期权的可提前执行约束采用惩罚方法来解决，在原偏微分不等式中引入一个惩罚项，使其成为一个偏微分方程。此方法避免了数值解的震荡，但是收敛性会随着离散的优化而降低。[Ikonen & Toivanen(2005)][Ikonen & Toivanen(2004)]应用了另外一种基于算子分裂的方法来解决美式期权的约束条件。通过离散将空间算子分解为几个简单的算子，并将每个时间层分解成与这些分解后的算子等数目。[Ikonen & Toivanen(2005)]中，他们使用这种方法来解决随机波动率下的美式期权定价问题，将差分算子及可提前执行约束分解成一系列一维线性互补问题(LCPs)，然后应用Brennan&Schwartz格式[Brennan & Schwartz(1977)]近似一维线性互补问题。这种方法的优点是将复杂的高维问题化为多个简单的一维问题，使得计算的复杂性大大减少。
但是考虑到期权定价中满足极值原理的重要性，本文针对描述美式期权定价的二维问题提出了一类新的有限体积九点格式和相应的算子分裂格式，该格式结合价格漂移方向近似二阶混合导数，使得提出的格式满足极大值原理和一致误差估计。
亚式期权是奇异期权的一种，其到期收益函数依赖于标的资产有效期至少某一段时间内的某种形式的平均[约翰·赫尔(1997)]。通常取标的资产价格按预定时间内的算术平均值或几何平均值作为其平均价格。目前亚式期权是OTC(柜台交易)市场上广受交易者青睐的金融工具，但即使在标的遵循几何布朗运动的假设下，也只有几何平均亚式期权的定价可以得到显式表达式。而在OTC市场上交易的绝大部分亚式期权都是标的算术平均，对标的算术平均亚式期权进行定价更多的是采用数值方法如二叉树方法[Klassen(2001)]、特征线差分方法[Jiang & Dai(2002)]或以标准几何平均亚式期权[Seydel(2004)]来近似逼近。
本文对亚式期权定价问题给出了恰当的边界条件，并提出了一类加权迎风有限体积格式和相应的交替方向格式。对价格漂移占优问题，采用加权迎风技术以避免数值解的非物理震荡；同时，结合亚式期权定价问题特性提出相应的交替方向。从理论上严格证明了提出的格式满足极大值原理，得到了一致误差估计。
本文最后提出了高维偏微分方程的迎风有限体积数值格式，因在BSDE模型中我们讨论的问题都是高维问题，而金融市场中变动因素的多样性和复杂性也决定了在金融定价问题中我们实际上面对的都是高维问题．所以本文结合问题特性提出的格式满足了极大值原理，解决了通常的有限体积格式不满足极值原理的问题并得到了一致误差估计。
论文组织和具体安排如下：
本文第1章主要介绍了金融衍生产品的历史与现状，及金融衍生产品定价的由来，随后从BSDE的角度给出了金融衍生产品定价的数学模型。
本文第2章介绍了倒向随机微分方程(BSDE)与一类拟线性二阶偏微分方程(PDE)存在的对应关系，这就是[Peng(1991)]中获得的非线性Feynman-Kac公式。于是BSDE中可问题可以转化到相对比较成熟的PDE中，使用PDE数值方法方法来解BSDE。同时可以反过来使用BSDE的随机算法来解决PDE中的问题。
本文第3章则简单回顾了一下已有的金融衍生产品的定价方法，如二叉树方法，Monte Carlo随机模拟方法以及有限差分方法、特征方法等PDE中的各种数值方法。
本文第4章主要研究了美式期权定价问题的有限体积数值模拟方法。对随机波动率下美式期权定价问题，我们用新的方法近似二阶混合交叉导数项，用迎风技术近似一阶对流项[Liang & Zhao(1997)]，提出一类新的有限体积九点格式。
其中算子A_x、A_y、A_(xy)和A_(yx)分别由自四个不同方向(x-方向，xy-方向，yx-方向以及y-方向)的差分方程的系数构成：
其中
相应的θ格式：
同时，我们对所提出的九点格式提出有效的算子分裂格式[Wang & Zhao(2003)]。其中k=0，…，l-1，
该分裂格式按照x，y，xy和yx四个方向分解，使得问题的求解变为四个方向的一维求解。对提出的格式，我们有极大值原理和一致误差估计：
若记
定理0．1．1(极值原理)假定V_(ij)~k为G_h上的一网函数，满足以下不等式：
其中L_h中的A_(i，j)满足假设(4．3．1)，则V_(ij)~k不可能在内点取正的极大，除非V_(ij)~k为常数。
定理0．1．2(稳定性估计)若L_hV_(ij)~k=0，则
定理0．1．3(误差估计)在定理0．1．1的条件下，离散格式(0．1．5)具有误差估计：‖E~k‖≤K sup_k‖R~k‖，其中E~k为分量为e_(ij)~k的向量。
为说明所提格式的有效性，我们给出几个数值算例。通过与投影超松弛(PSOR)方法[Ikonen & Toivanen(2005)]计算出的结果进行比较，我们发现，结果与其吻合。更加复杂的一般的多因素期权定价问题将在以后讨论。
本文第5章主要研究算术平均亚式期权定价问题的交替方向迎风有限体积方法。对算术平均亚式期权定价问题
我们通过方程(0．1．8)给出了适宜的边界条件，令
则亚式期权问题(0．1．8)的初边值条件为：
并在此基础上对价格漂移占优问题，采用加权迎风技术以避免数值解的非物理震荡；同时结合亚式期权定价问题特性，我们提出相应的交替方向。最终得到了一类新的交替方向加权迎风有限体积(ADFV)格式：(0．1．11)中V_(ij)~(n-1／2)的边值和V_(ij)的初边值条件为：
其中
我们从理论上严格证明了提出的格式满足极大值原理，也得到了一致误差估计：
定理0．1．4(极值原理)假定V_(ij)~n满足以下条件：
如果V_(ij)~n在Θ上不是常数，则V_(ij)~n的正最大值只能在(?)Θ上达到。
定理0．1．5(稳定性估计)因定价问题中f=0，我们有以下稳定性估计：
其中L~∞(Ω)为标准Banach空间，
定理0．1．6(误差估计)若v∈L~∞(0，T；L~(4，∞)(Ω))∩W_∞~1(0，T；W_∞(Ω))∩W_∞~2(0，T；L_∞(Ω))，在条件(5．3．3)及定理1的条件下，我们有误差估计：
其中
需要指出的是这一类加权迎风有限体积格式和相应的交替方向格式对于高维问题同样适用。
本文第6章主要研究提出了高维偏微分方程的迎风有限体积数值格式，该格式同样具有极大值原理和一致误差估计。
在BSDE模型中我们讨论的问题都是高维问题，而金融市场中变动因素的复杂性也决定了在实际金融定价问题中我们面对的都是高维问题。以下我们考虑如下二阶抛物型偏微分方程的数值解，
其中f=f((x，y)，t)，r=r((x，y)，t)，v=(v_1((x，y)，t)，v_2((x，y)，t)，…，v_d((x，y)，t))~T，D=(d_(ij)((x，y)，t))_(d×d)，c=c((x，y)，t)。Ω是定义在R~d上的求解区域，终端时刻为T。
基于第6．4节中的分析，我们得到了(0．1．16)的中心有限体积格式(6．5．8)、迎风有限体积格式Ⅰ(6．5．9)和全离散迎风有限体积格式Ⅱ(6．5．10)。需要特别指出的是通常的中心有限体积格式(6．5．8)和迎风有限体积格式Ⅰ(6．5．9)是不满足极值原理的，而对本文提出的全离散迎风有限体积格式Ⅱ(6．5．10)，我们有如下定理：
注意到格式(6．5．10)可以表示成
其中
所以用前面类似的方法可以得到如下极值原理、稳定性估计和误差估计：
定理0．1．7(极值原理)假定V_(ij)~k为G_h上的一网函数，满足以下不等式：
则V_(ij)~k不可能在内点取正的极大，除非V_(ij)~k为常数。
定理0．1．8(稳定性估计)若L_hV_(ij)~k=0，则
定理0．1．9(误差估计)在定理1的条件下，离散格式(6．5．8)具有误差估计：
其中E~k为分量为e_(ij)~k的向量。C为是不依赖于解u，V，剖分h和Δt的一个常数。
注0．1．1对格式(6．5．10)同样可以提出相应的分裂格式，可证明其满足极值原理，具有以上误差估计。详细细节和数值模拟正在进行和验证中。
In early financial market there are only four kinds of financial tools: bank deposit, bank draft (bank acceptance), bond and stock. The earliest bank deposit appeared in the 13th centuries. It is known that Ricciardi bank of Luca is the earliest one. During the year from 1272 to 1310, this bank provided a loan of 400,000 pound to the royalty of England. Although the default of this loan caused this company's bankruptcy, it did give an example of the financial risk of early days. Bill of Exchange, which is the creditor's rights with the goal to pay and can be circulated and transferred, arises almost at the same time with bank deposit. However, all kinds of bond, in which the issuer owes the holders a debt and is obliged to repay the principal and interest (the coupon) at a later date (termed maturity), didn't appeared until the 16th century. The first real government bond is Grand Parti of Francis I in 1955. It was issued not just for a handful of banks, but faced all the investors. Stock originated from the British East India company in 1600. The earliest stock is a certificate of pooling capital to finance the building of ships. The first permanent stock company to issue shares of stock was the Dutch East India Company, in 1602. From 1613, this company started to distribute the profit every four voyages, which was commonly regarded as a predecessor of "stockholder" and "distributed dividends".
Thus, we can see that the early financial investment is relatively simple. Investors followed some plain philosophy principles such as "lowly buy - highly sell" , "don' t put all the eggs into one basket" , etc. It was not until the latter half of 20th century has the financial market grown up rapidly and mathematical tools took more and more part in it. These changes were mainly due to the two financial revolutions in the Wall street.
The first financial revolution started from the paper[Markowitz( 1952)] "Portfolio Selection" , which is Harry Markowitz' s doctoral thesis. The early version of this thesis discussed how to get the portfolio which could maximize the anticipated income through the combination of risk assets (dispersal investment) and at the same time it could keep the risk of single security at an acceptable level. The mathematical tool in his thesis was called Markowitz mean-variance analysis) [Markowitz(1959)]. Before his portfolio theory, investors also discussed risk and profit, but as they couldn' t quantify some important indices, their portfolios were usually very subjective and hardly to make clear why they could get such anticipated profit. Markowitz' s theory solved these problems. Later, many people, such as Willam Sharp, made further research on the problem of profit and risk when markets reached the "balance" (supply and demand are equal) and gave capital asset pricing model - CAPM [Sharpe(1964)]. This model indicates that when markets achieve balance, the factors which determine the asset profit (i.e. pricing assets) are the ,β-measure system risk, whilst non-system risk plays no role in pricing assets, and it is a kind of linear relationship between expecting profit andβ. The standard CAPM gives a complete answer to the problem of the determining mechanism of asset profits when the markets achieve balance. Because of their excellent work, Markowitz and Sharp shared the Nobel prize in economics with Merton Miller in 1990.
Before those years we mentioned above, we should point out especially that in 1900, Louis Bachelier published his thesis "Theorie de la Speculation" [Bachelier(1900)], which is a milestone of modern financial study. In his thesis, he used random walk for the first time to describe stock prices and also he mentioned the option pricing problem.
At the beginning of the 70's, Black and Scholes obtained a significant breakthrough, which raised the second financial revolution. They deduced a differential equation which should be satisfied for any price of derivative securities based on any non-dividend payment stock.
In their breakthrough paper, Black and Scholes solved their differential equation successfully and got the precise formula for European call and put options. But in the real market, the precise formula can' t be obtained for pricing many derivative securities. But we can use numerical methods to solve it.
The linear form of backward stochastic differential equations (BSDEs) was first introduced in [Bismut(1978)] in 1978. Later, [Pardoux & Peng(1990)] studied the existence and uniqueness of a kind of nonlinear backward stochastic differential equations under Lipschitz condition. In [Duffie & Epstein(1992b)], they introduced a special case of backward stochastic differential equations independently during their study in stochastic differential utility. They found it could be used to describe the consumable preference under uncertain economic environment (i.e. econometrics foundation- utility function theory).
Subsequently, El Karoui and Quenez found that the theoretical price of many important derivative securities (e.g. futures and options) could be solved by backward stochastic differential equations, especially Black-Scholes formula, which is a special linear form of BSDEs. See [El Karoui & Quenez(1995)],[El Karoui, Peng & Quenez(1997)], [Duffie & Lions(1992)], [Duffie, Geoffard, & Skiadas(1994)].
In [Peng(1991)], Peng obtained a probabilistic interpretation for system of second order quasilinear parabolic partial differential equation, i.e. nonlinear Feynman-Kac formula, so that it connected the solution of BSDE associated with a kind of SDE with PDE. Therefore we can use the ready-made methods of PDE to solve BSDE, and inversely, some PDE problems can be solved by BSDEs' stochastic algorithm. See[Ma, Protter & Yong(1994)], [Ma & Yong(1995)], [Duffie, Ma, & Yong (1995)], [Ma, Protter, San Martin & Torres(2002)] etc. In [Hu & Peng(1995)], Peng also obtained the existence and uniqueness of a kind of completely coupling SDE and BSDE.
The European option price can be obtained by solving an initial and border value problem of a highly dimensional parabolic partial differential equation(s) with mixed derivatives. Usually we use numerical methods, such as difference method, finite element method, limited volume method etc. Different from European option, American option can be executed at any time during the period of validity. It can be changed into a linear complementary problem (LCP) depending on time, [Oxsterlee(2003)].
There are many numerical methods to solve LCP, such as [Clarke & Parrott(1996)], [Clarke & Parrott(1999)], which gave a method of PFAS for an approximately linear complementary problem. The advantage is its iterative times are independent with the grid number, and get an effective numerical solution, but it is hard to perform it. PFAS was used and improved in [Oxsterlee(2003)] and the vibration of numerical solutions was avoided effectively. In [Zvan, Forsyth & Vetzal(1998)], they used penalization method to solve the American option which can be executed in advance, i.e. a penalty term was introduced in the previous partial differential inequality so that it became into a partial differential equation. This method avoids the vibration of numerical solutions, but the convergence property decreases with the optimization of discretizing. In [Ikonen & Toivanen(2005)] [Ikonen & Toivanen(2004)], they used another method based on operator splitting to solve the American option problem with constraint. They discreted space operators into several simple ones, and separated every time layer into the same amount with operators. In [Ikonen & Toivanen(2005)], they used this method to solve the pricing of American option under stochastic fluctuating ratio, i.e. they decomposed the difference operators and the constraint of executing in advance into a series of one dimensional LCPs and ap- plied Brennan&Schwartz format [Brennan & Schwartz(1977)] to approximate LCPs. The advantage of this method is to transform a complicated highly dimensional problem into several simple one dimensional ones so that decreasing much complex calculating.
At the beginning of 90' s, with diversification of market requirements, it is hard to satisfy the special demand of clients only using the standard options (e.g. European option, American option). So some options with more dealing manners and dealing price appear, which is called exotic options [Hull(2000)].
Asian option is a kind of exotic option. The income function in the due day depends on the average of some form at some period at least in the valid period of target assets. The arithmetic or geometric average of target assets in the anticipated period is usually used as it' s average price. At present, Asian option is a financial tool widely used in the OTC (over-the counter) market, but even in the assumption that the target follows geometric Brownian motion, only pricing of geometric average Asian option can be expressed explicitly. However, the target of most Asian option trading in the OTC market is arithmetic average. Pricing of this kind of Asian option, we often use numerical method such as binomial method[Klassen(2001)], characteristic difference method [Jiang & Dai(2002)]or approaching with standard geometric-average Asian options[Seydel(2004)]. This paper is organized as follows:
In Chapter 1, a brief history review of financial derivatives is given. Since BSDEs play a important role in mathematical finance, some pricing model are given in BSDEs.
In Chapter 2, The correspondence between BSDE and a kind of quasi-linear 2-order parabolic PDE , i.e. the "nonlinear Feynman-Kac formula" is introduced . this formula is given in [Peng(1991)], thus the BSDE problem and PDE problem can be transformed into each other.
In Chapter 3, Some pricing method for financial derivatives are given, such as binomial method, Monte Carlo method and some PDE method.
In Chapter 4 of my doctoral thesis, I studied finite volume numerical simulation method of pricing for American option. For American option under stochastic volatility, a new kind of 9-point finite volume scheme is proposed, in which using a new technique for the 2-order hybrid cross derivatives, and upwind method for the convection item [Liang & Zhao(1997)]where operators A_x, A_y A_(xy) and A_(yx) are consisted of coefficiencies of difference equation from four different direction (the x-direction, the xy-direction, the yx-direction and the y-direction) .whereThe correspondingθ-scheme:
Meanwhile, the operator splitting scheme can be proposed for this 9-point scheme [Wang & Zhao(2003)].where k = 0,…,l-1,
This operator splitting scheme splits according to x, y, xy, and yx four directions, thus the problem turns into four 1-dim problem in different directions.
We have maximum principle and error estimate for the scheme proposed:Let
Theorem 0.2.1 (Maximum Principle) Let V_(ij)~k be a net function on G_h satisfies following inequality:where the A_(i,j) in L_h satisfies assumptions(4.3.1), Then V_(ij)~k will not achieve its maximum at the inner points, unless V_(ij)~k is constant.
Theorem 0.2.2 (Stability) Let L_hV_(ij)~k = 0, then
Theorem 0.2.3 (Error Estimate) Under the condition of theorem 0.2.1, the discrete scheme(0.2.5)has the following error estimate:where E~k is a vector with components of e_(ij)~k.
To show the validity of this scheme, some numerical examples are given, and the result tallies with the PSOR method [Ikonen & Toivanen(2005)]. More general multi-factor American option pricing problem will be considered later.
In Chapter 5 of my doctoral thesis, I studied an alternating-direction implicit upwind finite volume method for pricing Asian options. [Seydel(2004)]:Proper boundary conditions are given through equation (0.2.8) , letThen the initial and boundary condition for pricing Asian option(0.2.8) is: For convection-dominated problems, using upwind method to avoid non-physical shock, a new kind of alternating-direction implicit finite volume method according to the Asian option can be proposed:the boundary condition for V_(ij)~(n-1/2) and the initial and boundary condition for V_(ij)~n in (0.2.11) are:
Maximum principle for this scheme is theoretically proved and the error estimates also derived:
Theorem 0.2.4 (Maximum Principle) Let V_(ij)~n satisfies the following condition:Let V_(ij)~n is not constant on 6, then the positive maximum of V_(ij)~n can only be achieved onde.
Theorem 0.2.5 (Stability) For f=0 in pricing model, we have the following stability estimate:where L~∞(Ω) is standard Banach space,
Theorem 0.2.6 (Error Estimate) Let , Under the condition (5.3.3) and the assumptions of theorem (0.2.4), we have error estimate:
It should be pointed out that this alternating-direction implicit finite volume method is also valid for high-dimensional problems.
In Chapter 6, a kind of upwind control volume method are proposed, which is designated for the high dimensional problems arises in financial markets. Maximum principle and error estimates also derived:
Based on the analysis in section 6.4, we propose three scheme for problem(6.2.1): the Central Control Volume Scheme(6.5.8), the Upwind Control Volume Scheme I (6.5.9) and the Upwind Control Volume Scheme II (6.5.10) . It should be noted that the traditional Central Control Volume Scheme(6.5.8) and Upwind Control Volume Scheme I (6.5.9) don't satisfy maximum principle, but for the Upwind Control Volume Scheme II we proposed, the answer is yes.
Scheme(6.5.10)can be expressed as:By analogy to the forenamed procedure, we can get the following maximum principle, stability analysis and the error estimate.
Theorem 0.2.7 (Maximum Principle) LetV_(ij)~kbe a net function on G_h, which satisfies inequality:Then V_(ij)~k will not achieve its maximum at the inner points, unless V_(ij)~k is constant.
Theorem 0.2.8 (Stability)
Theorem 0.2.9 (Error Estimate) Under the condition of theorem 0.2.1, the discrete scheme (6.5.8)has the following error estimate:where E~k is a vector with components of e_(ij)~k, C is a constant, doesn't depend on the solution u,V, partition h adn△t.Remark 0.2.1 Splitting scheme for (6.5.10) can also be proposed, together with the maximum principle and error estimates, while the details and the numerical simulation are under verifying.
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[巴克斯特、伦尼(2006))]巴克斯特、伦尼(2006)．金融数学：衍生产品定价引论．人民邮电出版社，叶中行、王桂兰、林建忠译．
[嵇少林(1999)]嵇少林(1999)．算术亚式期权的无套利定价问题．山东大学学报 (自然科学版)．34(2)：144-148．
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