金融数学课件(英文版)第5讲
金融数学ppt课件
考虑T时刻到期的欧式期权,假定到期时,期 权的内在价值为V(T)=g(P(T));
设V(t,x)表示在t时刻股票价格为x时,期权的价值, 利用Ito公式可得到如下Black-Scholes方程
终V t端(t,条x 件) r V(T x x( ,tx,)x V ) g(1 2 x)2 x 2 V x(t x ,x ) r( V t,x () 5.2)
解上述联立方程可得
0 V S 1 1 ( ( H H ) ) V S 1 1 ( ( T T ) ) ,V 0 1 1 r 1 u r d d V 1 ( H ) u u ( 1 d r ) V 1 ( T ) *
注
0 称为套期保值比。 注意若取
向量自回归模型及其应用 14
1.投资组合理论简介
在投资活动中,人们发现,投资者手中持有多种 不同风险的证券,可以减轻风险带来的损失,对于投 资若干种不同风险与收益的证券形成的证券组称为证 券投资组合。
证券投资组合的原则是,组合期望收益愈大愈好, 组合标准差愈小愈好,但在同一证券市场中,一般情 形是一种证券的平均收益越大,风险也越大,因而最 优投资组合应为一个条件极值问题的解,即对一定的 期望收益率,选择资产组合使其总风险最小。
15
Markowitz 提出的证券组合均值方差问题,是证券 组合理论的基本问题,可描述为有约束的线性规划问
题
mi
n2p
mi w
nwTw
s.t. 1Tw1
E(Xp) E(X)Tw
解上述问题可得最优资产组合w*的表达式,且最 优资产组合的方差为
p 2 a 2 2 b c
诺贝尔经济奖简介(3)
2003年度诺贝尔经济学奖授予 Robert F.Engle和 Clive Granger。
金融数学完整课件全辑
风险管理政策
制定明确的风险管理政策和流程,确保业务 操作的合规性。
危机应对计划
制定应对重大风险的应急预案,确保在危机 发生时能够迅速、有效地应对。
05
投资组合优化
马科维茨投资组合理论
总结词
该理论是现代投资组合理论的基石,它通过 数学模型和优化技术,为投资者提供了构建 最优投资组合的方法。
详细描述
债券是一种常见的固定收益证券,其价格与利率之间存在密切关系。债券定价模型用于确定债券的理 论价格,通常基于现值计算方法。不同类型的债券(如国债、企业债等)具有不同的风险和收益特征 ,因此需要采用不同的定价模型。
复杂衍生品定价
总结词
概述了复杂衍生品定价的难点和方法, 包括信用衍生品、利率衍生品和商品衍 生品等。
数据清洗
对数据进行预处理,去除异常值、缺 失值和重复值,提高数据质量。
数据存储
采用分布式存储系统,高效地存储和 管理大规模金融数据。
数据可视化
通过图表、图像等形式直观地展示数 据分析结果,帮助用户更好地理解数 据。
机器学习在金融中的应用
风险评估
信贷审批
利用机器学习算法对历史金融数据进行分 析,预测未来市场走势和风险状况。
微积分
微积分是研究函数、极限、导数和积 分的数学分支。在金融领域,微积分 用于计算金融衍生品的价格和风险度 量。
线性代数
线性代数是研究线性方程组、矩阵和 向量空间的数学分支。在金融领域, 线性代数用于数据处理、模型建立和 优化问题求解等方面。
03
金融衍生品定价
期权定价模型
总结词
详细描述了期权定价模型的基本原理、应用场景和优缺点。
通过机器学习模型对借款人的信用状况进 行评估,提高信贷审批的效率和准确性。
北大金融学课件(英文版)第5章(4课时)
© 2008 , All Copy Rights Reserved.
5-2
The Variability of Exchange Rate
Three types of variability after early 1970s. • Long-term trends • Medium-term trends (which can be counter to the long-term trends) • Short-term trends
A country with a relatively low inflation rate will have an appreciating currency (an increasing nominal-exchange-rate value of its currency).
© 2008 , All Copy Rights Reserved.
© 2008 , All Copy Rights Reserved.
5-3
Figure 5.1, Panel A
© 2008 , All Copy Rights Reserved.
5-4
Figure 5.1, Panel B
© 2008 , All Copy Rights Reserved.
5-5
© 2008 , All Copy Rights Reserved.
5-12
In the Long run: Purchasing Power Parity
· Changes over time (relative PPP):
From time 0 to time t:
金融工程讲义(ppt 30页)(英文版)
19.18
Forward Prices
In a world that is FRN wrt P(0,T), the expected value of a security at time T is its forward price
19.19
Interest Rates
In a world that is FRN wrt P(0,T2) the expected value of an interest rate lasting between times T1 and T2 is the forward interest rate
If Eg denotes a world that is FRN wrt g
f0 g0
E
g
fT gT
19.14
Aleternative Choices for the Numeraire Security g
• Money Market Account • Zero-coupon bond price • Annuity factor
d m dt s dz Imagine two derivatives dependent on
with prices ?1 and ?2. Suppose
d ?1 ƒ1
1
dt
1
dz
d ƒ2 ƒ2
2
dt 2
dz
19.4
Forming a Riskless Portfolio
We can set up a riskless portfolio , consisting of + 2 ƒ2 of the 1st derivative and 1ƒ1 of the 2nd derivative
金融数学课件资料PPT课件
期次(半年)
票息
0
1
40.00
2
40.00
3
40.00
4
40.00
合计
160.00
利息收入 折价累计额
48.23 48.64 49.07 49.52 195.46
8.23 8.64 9.07 9.52 35.46
账面值
964.54 972.77 981.41 990.48 1000.00
例题5-3
例:债券的面值为1000元,年息票率为6% ,期限为3年,到期按面值偿还。市场利率 为8%,试计算债券在购买6个月后的价格 和帐面值。
解:已知: C = F= 1000 r = g = 6% n=3 i= 8% 所以债券在购买日的价格为
在购买6个月后的价格为
在购买6个月后的帐面值等于价格扣除 应计息票收入: 按理论方法计算
P Nr(1 t)a Cvn n
Nr(1 t)a K n
该公式称为计算债券价格的基本公式,债券价格 的计算还有另外两种变型公式:
(1)溢价/折价公式: P C [Nr(1 t) Ci]a n
(2)Makeham公式: P K g(1 t) (C K )
例:
面值1000元的五年期债券,票息率为每年 计息两次的年名义利率10%,可以面值赎 回,现以每年计息两次的年名义利率12% 的收益率购买,求分期偿债表中的总利息 收入。
SUCCESS
THANK YOU
•
5.1.3票息支付周期内债券的估价
债券的平价:债券购买日的实际交付款项 债券的市价:扣除应计票息后的买价 计算方法: 理论法 实务法 混合法
债券的面值N=1000 债券的收益率i=0.05
金融市场和净现值讲义(英文版)(ppt 23页)
• This graph will show intertemporal consumption opportunities.
Ms. Patience
to be.
Ms. Impatience
$20,000
$40,000
$60,000
$80,000
$100,000
$120,000
Consumption today
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
$80,000
$100,000
$120,000
Consumption today
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
3-8
Intertemporal Consumption Opportunity Set
McGraw-Hill/Irwin
Copyright © 2002 by The McGraw-Hill Companies, Inc. All rights reserved.
3-7
Intertemporal Consumption Opportunity Set
Consumption at t+1
3-4
The Financial Market Economy: Example
• Financial intermediation can take three forms:
金融数学 第 (5)节51页PPT
39、勿问成功的秘诀为何,且尽全力做你应该做的事吧。——美华纳
40、学而不思则罔,思而不学则殆。——孔子
•
30、风俗可以造就法律,也可以废除 法律。 ——塞·约翰逊
金融数学 第 (5)节
谢谢!
36、自己的鞋子,自己知道紧在哪里。——西班牙
37、我们唯一不会改正的缺点是软弱。——拉罗什福科
xiexie! 38、我这个人走得很慢,但是我从不后退。——亚伯拉罕·林肯
•
26、我们像鹰一样,生来就是自由的 ,但是 为了生 存,我 们不得 不为自 己编织 一个笼 子,然 后把自 己关在 里面。 ——博 莱索
•
27、法律如果不讲道理,即使延续时 间再长 ,也还 是没有 制约力 的。— —爱·科 克
Hale Waihona Puke •28、好法律是由坏风俗创造出来的。 ——马 克罗维 乌斯
•
29、在一切能够接受法律支配的人类 的状态 中,哪 里没有 法律, 那里就 没有自 由。— —洛克
金融数学课件(英文版)第1讲
L1.11
Arbitrage
Prices which admit arbitrage are, in some sense, incorrect. Existence of arbitrage is a severe form of the inconsistency and mispricing that we want to avoid. Assume no arbitrage, unless otherwise indicated. Thus, when we try to price some security, we are looking for an arbitrage-free price. Some authors define arbitrage without “type 2”. The distinction between our definition and their definition is essentially harmless, because: If there exists an asset whose price is always nonnegative and not always zero, then type 1 arb exists whenever type 2 arb exists. Bj¨rk’s term for arbitrage is “arbitrage possibility.” o
Introduction General properties of arbitrage-free prices One-period binomial model
《金融数学》(双语) 课程教学大纲
《金融数学》(双语) 课程教学大纲课程编号:07140080课程类别:专业课程授课对象:数学科学学院学生开课学期:第五学期学分:3学分主讲教师:严力指定教材:Petr Zima,Robert L.Brown, 《Mathematics of Finance》(Fifth Edition) ,McGraw-Hill Ryerson Limited 出版社,2001年教学目的:通过学习本课程,主要达到以下两个目的:1.学会国际金融界中的基本利息的概念与理论, 会计算单利, 复利, 各种年金. 学会用计算机软件Excel 制作抵押贷款(按揭)的分期还贷付款表, 偿债基金表. 会分析一般金融问题. 培养理财观念及国际意识. 为今后就业,个人理财打下基础.2.创造英语环境, 以英语授课为主, 辅以适当中文解释. 学会用英语阅读, 理解专业书籍. 提高英语听力及书面表达力, 增强英语解题能力, 培养英语思维及交流能力. 为今后就业,考研,出国培养扎实的英文应用能力.第一章Simple Interest and Simple Discount课时:2周,共6课时教学内容第一节Simple Interest一、Simple Interest, principal, present / accumulated value介绍单利,本金,现时值,将来值及计算方法-二、Demand loan介绍需求贷款及计算方法三、Invoice cash discount介绍发票中的现金折扣及计算方法思考题:1、什么是单利,本金,现时值,将来值及计算方法?2、如何制作需求贷款表?3.如何利用现金折扣节省开支?第二节Discounted value at simple interest一、Simple discount, discount factor介绍单利下的贴现因子及现时值计算方法二、Promissory Note介绍短期期票及计算方法思考题:1、什么是单贴现, 贴现因子及现时值,将来值计算方法?2、如何计算短期期票?第三节Equations of value一、Dated Value, focal date, time diagram介绍时间价值,参考日, 时间图二、Equations of value介绍时间价值计算方法三、Property of transitivity介绍等价概念思考题:1、什么是时间价值及计算方法?2、如何判定等价的时间价值?第四节Partial payment一、Merchant’s Method思考题:1、如何用商人法则计算余额?2、如何用递减法则计算余额?第五节Simple discount at a discount rate一、Simple Interest, principal, present / accumulated value介绍单贴现, 贴现因子及现时值,将来值计算方法- 二、Equivalent relationship介绍等价的贴现率与利率的计算方法思考题:1、什么是单贴现,本金,现时值,将来值及计算方法?2、什么是单贴现率与利率的等价关系及计算方法?第二章Compound Interest and Compound Discount 课时:5周,共15课时第一节Fundamental compound interest formula一、Fundamental compound interest formula介绍基本复利计算公式二、Compound Interest, principal, present / accumulated value介绍复利,本金,现时值,将来值及计算方法-思考题:1、什么是复利,本金,现时值,将来值及计算方法?2、什么是复利和单利的区别?第二节Equivalent compound interest rate一、Annual effective rate, nominal rate介绍年有效利率及名义利率计算方法二、Equivalent relation介绍两等价名义利率的计算方法思考题:1、什么是有效利率及名义利率?2、如何转换两等价名义利率?第三节Discounted Value at Compound Interest一,Present value, compound discount, discount factor 介绍现时值,复贴现,贴现因子及计算方法二、Long-term promissory note介绍长期票据及计算方法思考题:1、什么是现时值,复贴现,贴现因子及计算方法?2、如何计算长期票据?3.什么是长期票据和短期票据的区别?第四节Accumulated and Discounted Value for a Fractional Period of Time一、Exact / Theoretical method介绍理论计算方法-二、Approximate / Practical method介绍实用计算方法思考题:1、什么是理论计算方法?2、什么是实用计算方法?3.什么是理论和实用计算方法的区别?第五节Finding the Rate and the Time一、Finding the rate介绍利率计算方法-二、Finding the time介绍期限计算方法思考题:1、什么是利率计算方法?2. 什么是期限计算方法?第六节Equations of value一、Dated Value, focal date, time diagram介绍时间价值,参考日, 时间图二、Equations of value介绍时间价值计算方法三、Property of transitivity介绍复利等价概念思考题:1、什么是复利时间价值及计算方法?2、如何判定复利等价的时间价值?第七节Compound Interest at Changing Interest Rate一、Finding the interest at Changing Interest Rate介绍利率改变后利息计算方法-二、Finding the present value at Changing Interest Rate介绍利率改变后现时值计算方法思考题:1、如何计算不同利率阶段的利息?2、如何计算不同利率阶段的本金?第八节Other Applications of Compound Interest Theory, Inflation and the “Real” Rate of Interest一、Economics application介绍经济应用-二、“Real” Rate of Interest介绍“实值”利率及计算方法三,Inflation / Deflation介绍通货膨胀和通货紧缩思考题:1、什么通货膨胀和通货紧缩?2,什么是“实值”利率及计算方法?第九节Continuous Compounding一、Finding the rate of continuous Compounding介绍连续利率计算方法-二、Finding the present value and future value介绍连续利率时现时值与将来值计算方法思考题:1、什么是连续利率计算方法?2、什么是连续利率与其他利率的关系?第十节Compound Discount at a Discount Rate and Equivalent Discount Rates 一、Finding the Compound Discount Rate介绍复贴现率计算方法-二、Finding the present value and future value介绍复贴现率时现时值与将来值计算方法三T-bill介绍债劵思考题:1、什么是复贴现率计算方法?2、如何计算复贴现率时现时值与将来值?3,什么是债劵及计算方法?第三章Simple Annuities课时:3周,共9课时第一节Definitions一、Simple, general, ordinary, due, deferred annuities介绍简单,普通,初付,延付及延期年金二、Notations介绍惯用符号-思考题:1、什么是简单,普通,初付,延付及延期年金?2、什么是不同年金的区别?第二节Accumulated Value of an Ordinary Simple一、Accumulated Value介绍简单延付年金累积值计算方法二、Payment介绍简单延付年金复费计算方法思考题:1、什么是简单延付年金累积值计算方法?2、什么是简单延付年金复费计算方法?第三节Discounted Value of an Ordinary Simple Annuity 一,Discounted Value介绍简单延付年金现时值计算方法二、Decision making决策制定思考题:1、什么是简单延付年金现时值计算方法?2、如何根据净现时值制定决策?第四节Other Simple Annuities一、Annuity due介绍初付年金计算方法-二、Deferred annuity介绍延期年金计算方法思考题:1、什么是初付年金计算方法?2、什么是延期年金计算方法?第五节Finding the Term of an Annuity一、Finding the term介绍期限计算方法-二、Concluding payment介绍最后付费计算方法思考题:1、什么是期限计算方法?2,什么是最后付费计算方法?第六节Finding the Interest Rate一、Finding the Interest Rate介绍利率计算方法二、Linear interpolation介绍线性插值计算方法思考题:1、什么是利率计算方法?2、什么是线性插值计算方法?第四章General and Other Annuities课时:3周,共9课时第一节General Annuities一、General annuities介绍普通初付,延付及延期年金二、Application综合应用思考题:1、什么是普通,初付,延付及延期年金?2、如何计算不同年金?第二节Mortgages in Canada二、Mortgages介绍抵押贷款计算方法二、Small payment介绍小付费计算方法思考题:1、什么是抵押贷款计算方法?2、什么是小付费计算方法?第三节Perpetuities一,Perpetuities介绍永久年金现时值计算方法二、Payment介绍永久年金付费计算方法思考题:1、什么是永久年金计算方法?2、如何区别永久年金和普通年金?第四节Annuities where Payment Vary一、Geometric progression介绍几何级数变化计算方法-二、Arithmetic progression介绍算术级数变化计算方法思考题:1、什么是几何级数变化计算方法?2、什么是算术级数变化计算方法?第五章Amortization Method and Sinking Funds 课时:5周,共15课时第一节Amortization of a Deb t一、Amortization schedule介绍贷款清偿表的制作二、Outstanding principle balance介绍本金余额计算方法思考题:1、什么是贷款清偿表计算方法?2、什么是本金余额计算方法?第二节Outstanding Balance一, Retrospective method介绍过去计算方法二、Prospective method介绍将来计算方法思考题:1、什么是过去计算方法?2、什么是将来计算方法?第三节Refinance a Loan-the Amortization Method 一,Amortization Method介绍重新贷款计算方法1二、Decision making决策制定思考题:1、什么是重新贷款计算方法1?2、如何根据罚金制定决策?第四节Refinance a Loan the Sum-of-Digits Method一、Sum-of-Digits Method介绍重新贷款计算方法2-二、Deferred annuity介绍延期年金计算方法思考题:1、什么是重新贷款计算方法2?2、什么是计算方法1和计算方法2的区别?第五节Sinking Fund一、Sinking Fund介绍偿债基金计算方法-二、Schedule介绍偿债基金表的制作思考题:1、什么是偿债基金?2,如何制作偿债基金表?第六节The Sinking-Fund Method of Retiring a DebtRetiring a Debt介绍偿债计算方法第七节Comparison of Amortization and Sinking-Fund Methods 比较不同偿债计算方法执笔人:严力。
金融英语听说课件unit5insurance
the process of applying for an insurance policy, which includes providing information about the insured person or property and the type of insurance coverage desired.
02
an insurance policy that covers the life of an individual.
Life Insurance
Property Insurance
Liability Insurance
Health Insurance
an insurance policy that covers property against losses due to specific perils.
Avoid Adverse Actions
Do not attempt to repair the damage yourself or use someone who is not authorized by the insurance company.
Storm Damage Claim
A policyholder successfully claimed for damage caused by a storm, providing evidence of the damage and the cost to repair.
Financial English Listening and Speaking Coursewar
目录
contents
Insurance Introductioninsurance contractInsurance productsInsurance claimsinsurance industry
Part Five金融英语课件
3. Settlement----Payments and Funds Transfer Settlement is the striking of a balance between two or more parties having mutual dealing(s). The provision of a mechanism of payment and funds transfer is an important function of banks.
Intermediary Services in China: 1.Settlement 3.Agency mitment 7.Trust of Funds 9.others 2.Bank Cards 4.Guarantee 6.Trading 8.consulting
I. Intermediary Services
5. Factoring A.Definition Factoring is a short-term financing from the nonrecourse sale of accounts receivable to a third party, known as a factor, usually a bank-holding company or the bank.
Phrases & Words of I.1
bailor valuables bailee safekeeping fiduciary authority compensate loyalty indemnity custody
privacy pledge collateral
2. Intermediary Services A.Settlement;★ B.Trust Services; C. Lease; D. Factoring.
金融学英文课件 (5)
3-9
Characteristics of Financial Instruments
• Contracts are very complex. • Complexity is costly and people do not want to bear these costs. • Standardization of financial instruments overcomes potential costs of complexity.
3-11
Characteristics of Financial Instruments…cont
• The solution to high cost of obtaining information is to standardize both the instrument and the information about the issuer. • Financial instruments are designed to handle the problem of asymmetric information.
【PPT精品课件】金融工程-05_Options_Funda
期权
13
股票期权——执行与交割
1. 股票期权交易实行“自动执行”原则。如果标的股票 在交割月的第三个星期五的收盘价格高出执行价0.75 美元(含0.75美元)以上,那么,除非买权多头方通 知清算机构放弃执行买权,否则买权将自动执行。卖 权的自动执行原则是类似的
2. 如果股票期权不符合自动执行的条件,但是多头方希 望执行期权,那么期权持有人必须通过经纪公司通知 期权清算公司
期权
4
期权简史——郁金香事件
1. 17世纪,在荷兰的贵族阶层中,郁金香是身份的象征。 当时,郁金香批发商普遍采用远期合约出售郁金香。 但是,固定价格的远期合约蕴含了很高的风险,因为 他们不知道将来郁金香的成本是多少
2. 为了控制风险,许多批发商从花农手里购买期权,这 些期权实际上就是买权——保证批发商在未来某一段 时间内有权按照既定的价格从花农手中购买郁金香。 当时郁金香现货价格急剧波动,但许多批发商凭借期 权得以维持经营。那些没有采用期权管理价格风险的 批发商大多在价格的暴涨暴跌中破产
期权
5
期权简史——郁金香事件
3. 随着郁金香热在其它国家升温,郁金香期权的二级市 场发展起来。除了批发商以外,公众也开始交易郁金 香期权合约,企图从郁金香市场的繁荣中获利。随着 郁金香价格的直线上升,许多人在缺乏监管的期权市 场上暴富起来
4. 当荷兰在1638年经济开始衰退时,郁金香热走到了 尽头,价格开始高台跳水。许多出售郁金香卖权的投 机者无力履行期权合约规定的义务。在荷兰乃至欧洲, 郁金香热极大地损害了期权的声誉
3. 在期权多头方提交执行通知以后的第三个工作日进行 交割
4. 股票期权采用实物交割
期权
14
股票期权——执行价
1. 上市交易的股票期权的执行价都是按照一定的规则生 成的,尽管各个交易所制定的规则可能存在细微的差 别
〈金融学〉Chapter Five.ppt
金融系 刘玲
4
光票托收和跟单托收
托收
光票托收
跟单托收
金融单据(汇票)
金融单据和商业单据 商业单据(运输单据)
金融系 刘玲
5
3) Direct collection
Direct collection is an arrangement whereby the seller obtains his bank’s pre-numbered direct collection letter, thus enabling him to send his documents directly to his bank’s correspondent bank for collection. This kind of collection accelerates the paperwork process.(直接托收是这样一种安 排,委托人从他的银行即托收行那里获得托收指示的空白格式, 由他自己填写,连同托收单据直接寄给买方银行,即代收行。)
1) Documentary collections
Documentary collections may be described as collections on financial instruments being accompanied by commercial documents or collections on commercial documents without being accompanied by financial instruments, that is, commercial documents without a bill of exchange.跟单托收意指金融单据附带商业票据 的托收,或者商业单据不附带金融单据的托收,即并 无汇票,单是商业票据的托收。跟单托收一定有商业 单据;不一定有金融单据。
金融管理基础英文课件 (5)
DFL = Percent change in EPS Percent change in EBIT
• For the purpose of computation, it can be restated as:
DFL = EBIT .
EBIT – I
• DFL for two plans can be calculated using values from Table 5-5
Break-Even Chart: Conservative Firm
5-11
Volume-Cost-Profit Analysis: Conservative Firm
Table 5-3
5-12
The Risk Factor
• Factors influencing decision on maintaining a conservative or leveraged position include:
– May not hold good in real world
• Price weakening to capture increasing market • Cost overruns when moved beyond optimum-size
operation
– Relationships are not so fixed as assumed
150,000 (24,000 shares at $6.25)
$200,000
5-22
Impact of Financing Plan on Earnings per Share
Table 5-5
5-23
Financing Plans and Earnings per Share
金融理论英文课件 (5)
2
(1) Factors that Affect Prices of European Options
o o
European Options: Options can be exercised only on their maturity date American Options: Options can be exercised at any date up to maturity
o o o
Default-free bond face value D = $1,000 Coupon of coup = $100
33
34
35
36
(7) The Black-Scholes Option Pricing Model
o o o
T: the life of the option expressed as a fraction of a year j: the rate that is compounded n times in interval T r : rate of return for one year
Pricing Contingent Claims: Option Pricing Theory and Evidence
Option pricing theory is relevant to almost every area of finance. For example, virtually all corporate securities can be interpreted as portfolios of puts and calls on the firm. ---- J. C. Cox, S. A. Ross, and M. Rubinstein, “Option Pricing: A Simplified Approach,” Journal of Financial Economics, 1979
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is a martingale under P (meaning, each component is a martingale). For any self-financing trading strategy Θt with value Vt , we’ll show that V /B is a martingale, where Vt := Θt · Xt . Once we show this, we are done, because the familiar argument applies: If V0 = 0, then V0 /B0 = 0, hence E(VT /BT ) = 0. If also VT ≥ 0, then VT /BT ≥ 0, so VT /BT = 0, hence VT = 0 Conclusion: Θ is not an arbitrage.
Fundamental theorem in continuous time Black-Scholes
L5.6
Fundamental theorem: Proof
No arb ⇒ ∃ equivalent martingale measure P: Intuition of proof: Same as L2.21-L2.24. Let’s simplify the problem by assuming only J possible time-T outcomes Ω = {ω1 , . . . , ωJ }. For each j = 1, . . . , J, define the jth Arrow-Debreu asset to be a contract paying: 1 unit of BT if and when ωj occurs, else 0. Let pj be the time-0 value of the jth Arrow-Debreu asset, where value is expressed in units of the asset B. So pj is how many units of B it costs today, to buy 1 unit of B in the event that ωj occurs. (But what’s “the value” of an A-D asset that does not trade? If market is complete, it’s the value of a replicating portfolio. If market is incomplete, then proof becomes more difficult.)
L5.1
Financial Mathematics 330 Lecture 5
Roger Lee
2010 November 3
Fundamental theorem in continuous time
Black-Scholes
L5.2
Fundamental theorem in continuous time
Fundamental theorem in continuous time
then dYt = ρt dMt = ρt σt dWt , which is driftless.
Black-Schheorem: Proof
∃ equivalent martingale measure P ⇒ No arb: We are given that the vector of discounted asset prices 1 Xt := Bt
Fundamental theorem in continuous time
change measure, then we can find out what happens to S.
Black-Scholes
L5.10
Girsanov’s theorem
In this class we will need only one fact from the Girsanov theory of measure change. We will need only what Bj¨rk (2nd/3rd) calls the o “converse of Girsanov”. Theorem: If W is a Brownian motion under P,
j
pj = 1.
To check the MG property, let X be the price (in dollars) of any asset. Then X0 = B0
J
j=1
XT (ωj ) XT · pj = E BT (ωj ) BT
The first step simply says that the time-0 value of X equals the value of the portfolio of Arrow-Debreu assets that replicates XT (where all values are expressed in units of B). That portfolio holds XT (ωj )/BT (ωj ) units of the jth A-D asset, and the portfolio’s total value is the sum of quantity times price. The concluding step is by definition of expectation wrt P.
Black-Scholes
Fundamental theorem in continuous time
Black-Scholes
L5.3
More about martingales in Brownian settings
[Under integrability conditions. . . ]
N
= Θt · d(At Xt ) =
n=1
n n θt d(At Xt )
n Since each At Xt is a martingale, AV = V /B is a martingale also.
Idea: Martingales are wealth processes arising from zero-expectation games. Varying your bet size across games and across time still produces, collectively, a zero-expectation game. Can’t risklessly make something from nothing by playing zero-expectation games.
Fundamental theorem in continuous time Black-Scholes
L5.7
Fundamental theorem: Proof
Define P({ωj }) := pj . Can check this really is an equivalent measure: pj > 0 and
Fundamental theorem in continuous time Black-Scholes
L5.8
Fundamental theorem: Comments
Idea: The P probability of an event is simply the price (in units of B) of a asset that pays 1 unit of B iff that event occurs. Note: In this entire proof, we never assumed that B is the bank account, and never assumed that it is riskless. It is enough to assume that B is some asset with positive price process. Later, in some applications, we will prefer to normalize using some such asset (some numeraire) that is not the bank account. By default, if we say risk-neutral or martingale measure without specifying the numeraire, it is understood to be the bank account.
Fundamental theorem in continuous time Black-Scholes
L5.5
Fundamental theorem: Proof
To see that V /B is a martingale, let At := 1/Bt . Then d(At Vt ) = Vt dAt + At dVt + dAt dVt = Θt · Xt dAt + At (Θ · dXt ) + dAt Θt · dXt = Θt · (Xt dAt + At dXt + dAt dXt )
W M is a martingale wrt {Ft }t≥0 if and only if there exists an
adapted process σ such that dMt = σt dWt This is a form of the martingale representation theorem. We knew that every Itˆ integral is a martingale. This says that o conversely every martingale is an Itˆ integral plus a constant. o If M is a martingale, then stochastic integrals with respect to M are martingales. That’s because dMt = σt dWt for some σt , so if dYt = ρt dMt