[Applied Mathematical Finance, West] Calibration of the SABR Model in Illiquid Markets

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西悉尼大学应用金融学硕士

Theory
Economics
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西悉尼大学 University of Western Sydney

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所在地：澳大利亚新南威尔士州悉尼

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《数理金融》课程大纲

《数理金融》课程大纲课程名称（中文）：数理金融课程名称（英文）：Mathematical Finance课程代码：s开课单位：XXXX学院授课对象：XXXX学科学术学位硕士研究生学时：32学分：2开课学期：2 考核方式：开卷、完成作业或论文先修课程：测度论、随机过程教师信息教师电子邮件学位职称张三统计学博士教授李四金融学博士副教授课程简介数理金融是金融学自身发展而衍生出来的一个新的分支，是数学与金融学相结合的产物，是金融学由定性分析向定性分析与定量分析相结合、由规范研究向实证研究转变。

由理论阐述向理论研究与实用研究并重，金融模糊决策向精确化决策发展的结果。

教学目标与基本要求通过本课程的学习使学生理解数理金融的基本概念与基本理论，掌握概率统计方法在金融中的应用方法，能用所学的基本理论进行实证分析，为进一步的数理金融的学习与研究奠定基础，提高学生解决实际问题的能力。

课程内容与学时分配课程内容本课程主要学习金融的基本概念，无套利原理，完全市场模型；偏好与期望效用理论；金融市场的风险与风险厌恶的投资者行为的静态分析；随机占优；投资组合的选择理论；两基金分离，资本资产定价与套利定价模型；离散时间参数下的期权定价方法；连续时间参数下的期权定价方法；Black-Scholes期权的定价公式；概率定价方法、二叉树定价方法和状态价格定价模型几个方面全面地介绍期权定价的理论和实践。

课程具体安排实验、实践等其他环节内容与要求要求学生课下用至少20小时自行动手，利用实际数据模拟和分析本课程内容（不计学时）。

教材及主要参考资料[1]叶中行. 数理金融. 科学出版社, 2000.[2]姜礼尚. 期权定价的数学模型和方法. 高等教育出版社, 2012.[3]Tomas Bjork. Arbitrage Theory in Continuous Time. Oxford University Press, 1998.[4]H.Follmer and A.Schied. Stochastic Finance. Walter de Gruyter, 2002.撰写人：XXXX。

美国金融学专业必读教材

美国金融学专业必读教材去美国读金融研究生推荐看的书：1. Real Analysis(实分析)Calculus/ Michael Spivak2. Ordinary Differential Equations(常微分方程)Ordinary Differential Equations With Applications/ Larry C. Andrews本书涵盖了几乎所有基本的常微分知识：一阶、二阶、拉普拉斯变换、方程系统等。

更为可贵的是，有两个章节详细介绍了ODE 在物理中的应用，这对Finance的学习会大有裨益，毕竟Finance和Physics有千丝万缕的联系。

3. Partial Differential Equations(偏微分方程)Partial Differential Equations for Scientists & Engineers/ S. J. Farlow4. Numerical Analysis(数值分析)Numerical Analysis/ Burden, Faires/ 8th Edition5. Intro to Topology(拓扑学引论)Topology：A First Class/ James R. MunkresIntroduction to Topology and Mordern Analysis/ George Simmons6. Applied Linear Algebra(线代)Numerical Linear Algebra/ Lloyd N. Trefethen, David Bau, III.Matrix Computations/Golub and Van Loan.7. Probability & Statistics (概率与统计)Mathematical Statistics with Applications/ Dennis Wackerly, William Mendenhall, Richard L. Scheaffer.Statistical Inference/ George Casella, Roger Berger8. Complex Variables(复变函数)9. Wavelet(小波分析)Introduction to Wavelets and Wavelets Transforms/ Sidney Burrus, Ramesh GopinathAn introduction to Wavelets/ David WalnutAn introcution to Wavelets and other Filtering Method in Finance and Economics/ Ramazan Gencay需要提醒的是，这些原版的书不容易在国内买到，但是，有志向去美国读金融或经济的同学可以选择内容相近的中文书进行学习，毕竟一个扎实的数学功底能够使将来的美国金融专业的学习更有收获。

Mathematical Techniques in Finance (17)

• Find capital gains from holding the option, using the Itô formula
19
Black-Scholes PDE as a no-arbitrage condition, take #1 (11.4.2) • Rearrange: • LHS represents capital gains of a portfolio long 1 option and short stocks • This portfolio costs • Capital gain in a risk-free bank account would be • The two capital gains are risk-free, => must be the same:
3
Risk-neutral pricing in continuous time (11.2)
• Under the risk-neutral measure
• The new process BtQ has to be a martingale under Q because the above implies • The Girsanov Theorem claims that BtQ is not only a martingale under Q, it is a Brownian motion under measure Q!
Mathematical Techniques in Finance 2nd Edition
Lecture 17 Black-Scholes formula in continuous time
1
Summary of important results (11.1) • Itô process X, given by SDE , is a martingale if and only if • • Risk-neutral pricing (no dividends): if X is an asset price then EtQ[dXt/Xt] = r dt

SCI收录的数学类期刊杂志

数学全文电子期刊导航以刊物名首词字母排序AAdvances in Applied Mathematics 《应用数学进展》美国ISSN:0196-8858，1980年创刊，全年8期，Elsevier Science 出版社出版，SCI收录期刊，SCI 2003年影响因子0.558。

发表原始论文和评述文章，反映应用数学（包括实验数学、统计学、数学生物学和理论计算机科学等）的最新研究进展。

Advances in Mathematics《数学进展》美国ISSN:0001-8708，1965年创刊，全年16期，Elsevier Science 出版社出版，SCI收录期刊，SCI 2003年影响因子0.930。

侧重发表纯数学研究论述与评论，反映该学科各个方面的重要进展。

Applied Mathematical Modelling《应用数学模型》美国ISSN:0307-904X，1976年创刊，全年12期，Elsevier Science 出版社出版，SCI收录期刊，SCI 2002年影响因子0.500。

数学模型及相关数字技术和软件在工程和环境系统设计和分析中起到越来越重要的作用。

本刊为这一领域开辟论坛，发表论文、评论、札记、会议报告和书评等。

Annales de l'Institut Henri Poincare (B) Probability and Statistics《亨利·彭加莱研究所纪事：概率计算与统计学》法国ISSN:0246-0203，1964年创刊，全年6期，Elsevier Science 出版社出版，SCI收录期刊，SCI 2003年影响因子0.789。

刊载概率计算和统计学包括经济计算和运筹学等方面的研究论文和评论。

Annales de l'Institut Henri Poincare © Non Linear Analysis《亨利·彭加莱研究所纪事：非线性分析》法国ISSN:0294-1449，1984年创刊，全年6期，Elsevier Science 出版社出版，SCI收录期刊，SCI 2003年影响因子0.881。

数理金融(资产定价的原理与模型)

PV
Cn C1 C2 1 r 1 r 2 1 r n
2.1.4 年金
1.普通年金的终值 2.年金的现值 3.永续年金
2.2债券及其期限结构
2.2.1债券的定义和要素 1.面值 2.期限 3.附息债券与票面利率 4.付息频率 5.分期偿还特征 6附加选择权 2.2.2债券的风险 1.利率风险 2.违约风险
2.2债券及其期限结构
1.2 期望效用函数
1.2.1 彩票（lottery）及其运算 1.2.2 彩票集合上的偏好关系 1.2.3 基数效用函数存在定理
定理 1.2 (基数效用函数存在定理) 设 B 具有性质 1—性质 3 的偏好关系” ” 则存在效用函数 U : B R 满足: (1) (2) (3)
~
P Q 当且仅当 U ( x) U ( y) P ～ Q 当且仅当 U ( x) U ( y)
任意最小方差投资组合都可以唯一的表示为全局最小方差投资组合33具有无风险资产的均值方差分析331具有无风险资产的有效投资组合本节假定市场存在n种风险资产无风险资产的收益率是一常数设为是投资于无风险资产的权系数表示投资于种资产的投资组合的期望收益则若投资者投资于无风险资产的权系数为正则表示储蓄若权系数为负表示借贷为购买风险资产筹集资金此时最小方差资产组合问题可表示为如下的优化问题332具有无风险资产的均值方差分析333两基金分离定理334切点组合的含义335具有无风险资产情况下的超额收益率定义33资产称为资产组合的超额收益率将3312a写成分量形式就得到下面的定理定理32当市场存在无风险利率时任意资产的超额收益率可以用如下公式表示336市场仅存在风险资产情况下的超额收益率定理33
1.5.2 一阶随机占优

大学328门专业课程标准英文翻译模板(经济,财会,金融,管理等专业)

泽稷网校-财务金融证书在线教育领导品牌
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金融工程课件

➢交易方案的制定者（市场参与者）——为满足客户需要而构造出方案并将其出售给客户
➢新观念的创造者（创新者）——创造新的金融产品与手段
➢钻法律空子者（在法律边缘活动的人）——精熟会计和税法。
4、金融工程与金融理论关系
• 西方金融理论的含义（微观金融的三个层次：Investment、Corporate Finance、 Asset Pricing）
Carnegie Mellon University Graduate School of Business Master of Science in Computational
Finance /mscf/
建立在工程学院的有 Princeton University Department of Operations Research & Financial
应用性
金融工程发展的动力源泉
金融工程的目的性
解决金融财务问题盈利风险管理合理避税逃避管制
金融工程的5个步骤
1、诊断：识别金融问题的实质和根源 2、分析：根据当前的体制、技术和金融
理论找出解决问题的最佳方案 3、生产：生产一种新工具 4、定价：确定生产成本和边际收益 5、修正：为满足每个客户的特殊需求，
➢公司治理问题，它讨论的是公司组织结构和激励机制等问题。公司金融在国内往往被译为公司财务，实际上其内容远远超出了公司的财务问题。
金融工程的主要内容
• 固定收入证券 • 衍生证券 • 风险管理 • 金融产品的开发与创新
5、金融工程的工具
• 概念性工具：金融科学的思想和概念，包括估值理论、投资组合理论、套期保值理论、会计关系以及税收知识.
Engineering M.S.E. in Operations Research and Financial

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Handling these market skews and smiles correctly is critical for hedging. One would like to have a coherent estimate of volatility risk, across all the different strikes and maturities of the positions in the book.
dF ~wF dtzsF dW1
ð1Þ
ds2~ms2 dtzjs2 dW2
ð2Þ
dW1 dW2~r dt
ð3Þ
where w, m and j are time and state dependent functions, and dW1 and dW2 are correlated Brownian motions.
Correspondence Address: Graeme West, Financial Modelling Agency, 19 First Ave East, Parktown North, 2193, South Africa. Email: graeme@finmod.co.za 1350-486X Print/1466-4313 Online/05/040371–15 # 2005 Taylor & Francis DOI: 10.1080/13504860500148672
Local volatility models are self-consistent, arbitrage-free, and can be calibrated to precisely match observed market smiles and skews. Currently these models are the most popular way of managing smile and skew risk. Possibly they are often preferred to stochastic volatility models for computational reasons: the local volatility models are tree models; to price with stochastic volatility models typically means Monte Carlo. However, it has recently been observed (Hagan et al., 2002) that the dynamic behaviour of smiles and skews predicted by local volatility models is exactly opposite the behaviour observed in the marketplace: local volatility models predict that the skew moves in the opposite direction to the market level, in reality, it moves in the same direction. This leads to extremely poor hedging results within these models, and the hedges are often worse than the naive Black model hedges, because these naive hedges are in fact consistent with the smile moving in the same direction as the market.
372 G. West
stochastic volatility approach, for which the reader is referred to Hull & White (1987), Heston (1993), Lewis (2000), Fouque et al. (2000), Lipton (2003), and finally Hagan et al. (2002), which is the model we will consider here.
KEY WORDS: SABR model, equity derivatives, volatility skew calibration, illiquid markets
Why another Skew Model? Vanilla OTC European options or European futures options are priced and often hedged using respectively the Black–Scholes or Black model. In these models there is a one-to-one relation between the price of the option and the volatility parameter s, and option prices are often quoted by stating the implied volatility simp, the unique value of the volatility which yields the option price when used in the formula. In the classical Black–Scholes–Merton world, volatility is a constant. But in reality, options with different strikes require different volatilities to match their market prices. This is the market skew or smile. Typically, although not always, the word skew is reserved for the slope of the volatility/strike function, and smile for its curvature.
Applied Mathematical Finance, Vol. 12, No. 4, 371–385, December 2005
Calibration of the SABR Model in Illiquid Markets
GRAEME WEST
Financial Modelling Agency, South Africa, and Programme in Advanced Mathematics of Finance, School of Computational & Applied Mathematics, University of the Witwatersrand, South Africa
To resolve this problem, Hagan et al. (2002) derived the SABR model. The model allows the market price and the market risks, including vanna and volga risks, to be obtained immediately from Black’s formula. It also provides good, and sometimes spectacular, fits to the implied volatility curves observed in the marketplace. More importantly, the SABR model captures the correct dynamics of the smile, and thus yields stable hedges.
Calibration of the SABR Model in Illiquid Markets 373
As another example, the models of Fouque et al. (2000) are variations on the following initial set-up:
(Received 14 July 2004; in revised form 9 November 2004) ABSTRACT Recently the SABR model has been developed to manage the option smile which is observed in derivatives markets. Typically, calibration of such models is straightforward as there is adequate data available for robust extraction of the parameters required asinputs to the model. The paper considers calibration of the model in situations where input data is very sparse. Although this will require some creative decision making, the algorithms developed here are remarkably robust and can be used confidently for mark to market and hedging of option portfolios.