09年SOA北美精算师考试第二门FM官方样题第一部分(主要是金融数学)

DERIVATIVES MARKETS

SAMPLE

FOR

QUESTIONS

These questions have been written to assist the student in studying for the Exam FM/2. They are not intended to cover the entire breadth of the syllabus for Financial Economics.

1. Which statement about zero-cost purchased collars is FALSE?

A. A zero-width, zero-cost collar can be created by setting both the put and call strike

prices at the forward price.

B. There are an infinite number of zero-cost collars.

C. The put option can be at-the-money.

D. The call option can be at-the-money.

E. The strike price on the put option must be at or below the forward price.

2. You are given the following information:

?The current price to buy one share of XYZ stock is 500.

?The stock does not pay dividends.

?The risk-free interest rate, compounded continuously, is 6%.

? A European call option on one share of XYZ stock with a strike price of K that expires in one year costs 66.59.

? A European put option on one share of XYZ stock with a strike price of K that expires in one year costs 18.64.

Using put-call parity, determine the strike price, K.

A. 449

B. 452

C. 480

D. 559

E. 582

3. Happy Jalapenos, LLC has an exclusive contract to supply jalapeno peppers to the

organizers of the annual jalapeno eating contest. The contract states that the contest organizers will take delivery of 10,000 jalapenos in one year at the market price. It will cost Happy Jalapenos 1,000 to provide 10,000 jalapenos and today’s market price is

0.12 for one jalapeno. The continuously compounded risk-free interest rate is 6%.

Happy Jalapenos has decided to hedge as follows (both options are one-year,

European):

Buy 10,000 0.12-strike put options for 84.30 and sell 10,000 0.14-stike call options for

74.80.

Happy Jalapenos believes the market price in one year will be somewhere between 0.10 and 0.15 per pepper. Which interval represents the range of possible profit one year from now for Happy Jalapenos?

A. –200 to 100

B. –110 to 190

C. –100 to 200

D. 190 to 390

E. 200 to 400

4. Zero-coupon risk-free bonds are available with the following maturities and yield rates

(effective, annual):

Maturity (years) Yield

1 0.06

2 0.065

3 0.07

You need to buy corn for producing ethanol. You want to purchase 10,000 bushels one year from now, 15,000 bushels two years from now, and 20,000 bushels three years from now. The current forward prices, per bushel, are 3.89, 4.11, and 4.16 for one, two, and three years respectively.

You want to enter into a commodity swap to lock in these prices. Which of the

following sequences of payments at times one, two, and three will NOT be acceptable to you and to the corn supplier?

A. 38,900, 61,650, 83,200

B. 39,083, 61,650, 82,039

C. 40,777, 61,166, 81,554

D. 41,892, 62,340, 78,997

E. 60,184, 60,184, 60,184

5. You are given the following information:

?One share of the PS index currently sells for 1,000.

?The PS index does not pay dividends.

?The effective annual risk-free interest rate is 5%.

You want to lock in the ability to buy this index in one year for a price of 1,025. You can do this by buying or selling European put and call options with a strike price of 1,025. Which of the following will achieve your objective and also gives the cost today of establishing this position.

A. Buy the put and sell the call, receive 23.81

B. Buy the put and sell the call, spend 23.81

C. Buy the put and sell the call, no cost

D. Buy the call and sell the put, receive 23.81

E. Buy the call and sell the put, spend 23.81

6. The current price of one share of XYZ stock is 100. The forward price for delivery of

one share of XYZ stock in one year is 105. Which of the following statements about the expected price of one share of XYZ stock in one year is TRUE?

A. It will be less than 100

B. It will be equal to 100

C. It will be strictly between 100 and 105

D. It will be equal to 105

E. It will be greater than 105.

7. A non-dividend paying stock currently sells for 100. One year from now the stock sells

for 110. The risk-free rate, compounded continuously, is 6%. The stock is purchased in the following manner:

?You pay 100 today

?You take possession of the security in one year

Which of the following describes this arrangement?

A. Outright purchase

B. Fully leveraged purchase

C. Prepaid forward contract

D. Forward contract

E. This arrangement is not possible due to arbitrage opportunities

8. You believe that the volatility of a stock is higher than indicated by market prices for

options on that stock. You want to speculate on that belief by buying or selling at-the-money options. What should you do?

A. Buy a strangle

B. Buy a straddle

C. Sell a straddle

D. Buy a butterfly spread

E. Sell a butterfly spread

9. You are given the following information:

? The current price to buy one share of ABC stock is 100

? The stock does not pay dividends

? The risk-free rate, compounded continuously, is 5%

? European options on one share of ABC stock expiring in one year have the

following prices:

Strike Price Call option price Put option price

90 14.63 0.24 100 6.80 1.93 110 2.17

6.81

A butterfly spread on this stock has the following profit diagram.

Which of the following will NOT produce this profit diagram?

A. Buy a 90 put, buy a 110 put, sell two 100 puts

B. Buy a 90 call, buy a 110 call, sell two 100 calls

C. Buy a 90 put, sell a 100 put, sell a 100 call, buy a 110 call

D. Buy one share of the stock, buy a 90 call, buy a 110 put, sell two 100 puts

E. Buy one share of the stock, buy a 90 put, buy a 110 call, sell two 100 calls.

10.Suppose stock XYZ has a current price of 100. The forward price for delivery of this

stock in 1 year is 110. Also, assume there are no dividends, and that the annual

effective interest rate is 10%, unless otherwise indicated.

Which of the following statements is FALSE?

A. The time-1 profit diagram and the time-1 payoff diagram for long positions in this

forward contract are identical.

B. The time-1 profit for a long position in this forward contract is exactly opposite to

the time-1 profit for the corresponding short forward position.

C. There is no comparative advantage to investing in the stock versus investing in the

forward contract.

D. If the 10% interest rate was continuously compounded instead of annual effective,

then it would be more beneficial to invest in the stock, rather than the forward

contract.

E. If there was a dividend of 3.00 paid 6 months from now, then it would be more

beneficial to invest in the stock, rather than the forward contract.

11.The current stock price is 40, and the effective annual interest rate is 8%.The price of a

35-strike 1-year European call option is 9.12. The price of a 40-strike 1-year European call option is 6.22.The price of a 45-strike 1-year European call option is 4.08.

Assuming that all call positions being compared are long, at what 1-year stock price range does the 45-strike call produce a higher profit than the 40-strike call, but a lower profit than the 35-strike call?

A. S1 < 38.13

B. 38.13 < S1 < 40.44

C. 40.44 < S1 < 42.31

D. S1 > 42.31

E. There is no price for S1 at which this situation occurs.

12.Consider a European put option on a stock index without dividends, with 6 months to

expiration, and a strike price of 1,000. Suppose that the nominal annual risk-free rate is 4% convertible semiannually, and that the put costs 74.20 now.

What price must the index be in 6 months so that being long the put would produce the same profit as being short the put?

A. 922.83

B. 924.32

C. 1,000.00

D. 1,075.68

E. 1,077.17

13.Suppose that you short one share of a stock index for 50, and that you also buy a 60-

strike European call option that expires in 2 years for 10. Assume the effective annual interest rate is 3%.

If the stock index increases to 75 after 2 years, what is the profit on your combined

position, and what is an alternative name for the call in this context?

Profit Name

A. -22.64 Floor

B. -17.56 Floor

C. -22.64 Cap

D. -17.56 Cap

E. -22.64 ‘Written’ Covered Call

14.The current price of a non-dividend paying stock is 40 and the continuously

compounded risk-free rate of return is 8%. You are given that the price of a 35-strike call option is 3.35 higher than the price of a 40-strike call option, where both options expire in 3 months.

How much does the price of an otherwise equivalent 40-strike put option exceed the

price of an otherwise equivalent 35-strike put option?

A. 1.55

B. 1.65

C. 1.75

D. 3.25

E. 3.35

15. The current price of a non-dividend paying stock is 40 and the continuously

compounded risk-free rate of return is 8%. You enter into a short position on 3 call

options, each with 3 months to maturity, a strike price of 35, and an option premium of

6.13. Simultaneously, you enter into a long position on 5 call options, each with 3

months to maturity, a strike price of 40, and an option premium of 2.78.

Assuming all 8 options are held until maturity, what is the maximum possible profit and loss for the entire option portfolio?

Maximum

Loss

Maximum

Profit

A. 3.42 4.58

B. 4.58 10.42

10.42

C. Unlimited

D. 4.58 Unlimited

Unlimited

E. Unlimited

16. The current price of a non-dividend paying stock is 40 and the continuously

compounded risk-free rate of return is 8%. In addition, you are given the following table of call and put option premiums for various exercise prices:

Exercise Price Call Premium Put Premium

35 6.13

0.44

1.99

40 2.78

5.08

45 0.97

You are interested in speculating on volatility in the stock price, and are comparing two investment strategies. The first is a 40-strike ‘straddle’. The second is a ‘strangle’

consisting of a 35-strike put and a 45-strike call. For what range of stock prices in 3 months does the ‘strangle’ outperform the ‘straddle’?

A. The ‘strangle’ never outperforms the ‘straddle.’

B. 33.56 < S T < 46.44

C. 35.13 < S T < 44.87

D. 36.57 < S T < 43.43

E. The ‘strangle’ always outperforms the ‘straddle.’

17. Assume the current price for a stock index is 1,000, and the following premiums exist

for various options to buy or sell the stock index 6 months from now:

Put

Premium

Premium

Call

Strike

Price

120.41 51.78

950

1,000 93.81 74.20

101.21

1,050 71.80

Strategy I is to buy the 1,050-strike call and to sell the 950-strike call.

Strategy II is to buy the 1,050-strike put and to sell the 950-strike put.

Strategy III is to buy the 950-strike call, sell the 1,000-strike call, sell the 950-strike put, and buy the 1,000-strike put.

Assume that the price of the stock index in 6 months will be between 950 and 1,050.

In 6 months, which of the three strategies will have greater payoffs for lower prices of the stock index than for relatively higher prices?

A. I only

B. I and II only

C. I and III only

D. II and III only

E. I, II, and III

18. You are a jeweler who buys gold, which is the primary input needed for your products.

One ounce of gold can be used to produce one unit of jewelry. Assume that the cost of all other inputs is negligible. You are able to sell each unit of jewelry for 700 plus 20% of the market price of gold in one year. In one year, the actual price of gold will be in 1 of 3 possible states, corresponding to the following probability table:

Market Price of Gold in 1-year Probability

.2

750

ounce

per

ounce

.5

per

850

ounce

.3

per

950

You are considering utilizing forward contracts to lock in 1-year gold prices, in which case you would charge the customer (one year from now) 700 plus 20% of the forward price. The 1-year forward price for gold is 850 per ounce. How much does your

expected 1-year profit, per unit of jewelry sold, increase if you buy forward the 1-year price of gold?

A. 0

B. 8

C. 12

D. 20

E. 32

19. You are a producer of gold, and have expenses of 800 per ounce of gold produced.

Assume that the cost of all other production-related expenses is negligible, and that you will be able to sell all gold produced at the market price. In 1 year, the market price of gold will be 1 of 3 possible prices, corresponding to the following probability table:

Gold Price in 1-year Probability

750 per ounce .2

850 per ounce .5

950 per ounce .3

You hedge the price of gold by buying a 1-year put option with an exercise price of 900 per ounce. The option costs 100 per ounce now, and the continuously compounded interest rate is 6%. Which of the following is closest to your expected 1-year profit per ounce of gold produced?

A. 0

B. 3

C. 6

D. 9

E. 12

20. The current price of a stock is 200, and the continuously compounded interest rate is

4%. A dividend will be paid every quarter for the next 3 years, with the first dividend occurring 3 months from now. The amount of the first dividend is 1.50, but each

subsequent dividend will be 1% higher than the one previously paid.

Which of the following is closest to the fair price of a 3-year forward contract on this stock?

A. 200

B. 205

C. 210

D. 215

E. 220

21. You are a market maker in stock index forward contracts. The index spot price is 110,

the continuously compounded interest rate is 5%, and the continuously compounded dividend yield on the index is 2%. If you observe a 6-month forward price of 112,

describe actions you could take to exploit an arbitrage opportunity, and calculate the resulting profit (per index unit).

A. Buy observed forward, sell synthetic forward, Profit = 0.34

B. Buy observed forward, sell synthetic forward, Profit = 0.78

C. Buy observed forward, sell synthetic forward, Profit = 1.35

D. Sell observed forward, buy synthetic forward, Profit = 0.78

E. Sell observed forward, buy synthetic forward, Profit = 0.34

22. A farmer expects to sell 50 tons of pork bellies at the end of each of the next 3 years.

Suppose that the pork bellies forward price for delivery in 1 year is 1,600 per ton. For delivery in 2 years, the forward price is 1,700 per ton. Also, for delivery in 3 years, the forward price is 1,800 per ton. Suppose that interest rates are determined from the

following table:

Years to Maturity Zero-Coupon Bond Yield

1 5.00%

2 5.50%

3 6.00%

If the farmer uses a commodity swap to hedge the price for selling pork bellies, what is the level amount he would receive each year (i.e. – the swap price) for all 50 tons?

A. 84,600

B. 84,800

C. 85,000

D. 85,200

E. 85,400

23. You are given the following spot rates from the latest upward-sloping yield curve:

1 2 3 4 5

Years to Maturity

Spot Rate

5.25%

7.50%

6.25%

4.50%

4.00%

You enter into a 5-year interest rate swap (with a notional amount of 100,000) to pay a

fixed rate and to receive a floating rate based on future 1-year LIBOR rates. If the

swap has annual payments, what is the fixed rate you should pay?

A. 5.20%

B. 5.70%

C. 6.20%

D. 6.70%

E. 7.20%

24.Which of the following statements is NOT a typical reason for why derivative securities

are used to manage financial risk?

A. Derivatives are used as a means of hedging.

B. Derivatives are used to reduce the likelihood of bankruptcy.

C. Derivatives are used to reduce transaction costs.

D. Derivatives are used to satisfy regulatory, tax, and accounting constraints.

E. Derivatives are used as a form of insurance.

25.Which of the following statements concerning risk sharing, in the context of financial

risk management, is LEAST accurate?

A. In an insurance market, individuals that do not incur losses have shared risk with

individuals that do incur losses.

B. Insurance companies can share risk by ceding some of the excess risk from large

claims to reinsurers.

C. Reinsurance companies can further share risk by investing in catastrophic bonds.

D. Risk-sharing reduces diversifiable risk, more so than reducing non-diversifiable

risk.

E. Ideally, any risk-sharing mechanism should benefit all parties sharing the risk.

26. Which of the following positions have an unlimited loss potential from adverse price

movement in the underlying asset, regardless of the initial premium received?

I.Short 1 forward contract

II.Short 1 call option

III.Short 1 put option

A. I only

B. I and II only

C. I and III only

D. II and III only

E. I, II, and III

27.Which of the following positions benefit from falling prices in the underlying asset?

I.Long 1 homeowner’s insurance contract

II.Long 1 equity-linked CD

III.Long 1 synthetic forward contract

A. I only

B. I and II only

C. I and III only

D. II and III only

E. I, II, and III

28. Which of the following is NOT among a firm’s rationales to hedge?

A. To reduce taxes through income shifting

B. To reduce the probability of bankruptcy or distress

C. To reduce the costs associated with external financing

D. To reduce the exposure to exchange rate risk

E. To reduce the debt proportion of external financing

29. The dividend yield on a stock (d) and the interest rate used to discount the stock’s cash

flows (r) are both continuously compounded. There are 4 alternative methods to buy a stock: outright purchase, fully leveraged purchase, prepaid forward contract, and

forward contract. The dividend yield is less than the interest rate, but both are positive.

Which of the following is the correct ranking, from smallest to largest, for the amount of payment needed to acquire the stock?

Note: in establishing your ranking, you should calculate each payment amount at

the exact time of payment, so that for any methods that have payment times in the

future, you do not need to discount these payments back to time-0 when comparing

amounts.

A. Prepaid forward contract, Outright purchase, Forward contract, Fully leveraged

purchase

B. Outright purchase, Prepaid forward contract, Forward contract, Fully leveraged

purchase

C. Outright purchase, Prepaid forward contract, Forward contract, Fully leveraged

purchase

D. Prepaid forward contract, Outright purchase, Fully leveraged purchase, Forward

contract

E. Outright purchase, Prepaid forward contract, Fully leveraged purchase, Forward

contract

30. You are trying to decide whether to use forward contracts or futures contracts when

committing to buy an underlying asset at some date in the future. Which of the

following is NOT a distinguishing characteristic of futures contracts, relative to forward contracts?

A. Contracts are settled daily, and marked-to-market.

B. Contracts are more liquid, as one can offset an obligation by taking the opposite

position.

C. Contracts are more customized to suit the buyer’s needs.

D. Contracts are structured to minimize the effects of credit risk.

E. Contracts have price limits, beyond which trading may be temporarily halted.

金融数学附答案

金融数学附答案文件排版存档编号：[UYTR-OUPT28-KBNTL98-UYNN208]

1、给定股票价格的二项模型，在下述情况下卖出看涨期权 S 0 S u S d X r τ 股数 50 60 40 55 1/2 1000 （1）求看涨期权的公平市场价格。 （2）假设以公平市场价格+美元卖出1000股期权，需要买入多少股股票进行套期保值，无风险利润是多少 答案：（1）d u d r S S S e S q --=τ0=56.040 6040505.005.0=--??e （2）83.2>73.2，τr e S V -?+?='00 83.2> τr e S -?+?'0 40 6005--=--=?d u S S D U =25.0股 104025.00'-=?-=?-=?d S D 753.9975.0105.005.0'-=?-=??-e 美元 则投资者卖空1000份看涨期权，卖空250股股票，借入9753美元 所以无风险利润为1.85835.005.0=?e 美元 2、假定 S 0 = 100，u=，d=，执行价格X=105，利率r=，p=，期权到期时间t=3， 请用连锁法则方法求出在t=0时该期权的价格。(答案见课本46页) 3、一只股票当前价格为30元，六个月期国债的年利率为3%，一投资者购买一份执行价格为35元的六个月后到期的美式看涨期权，假设六个月内股票不派发红利。波动率σ为. 问题：（1）、他要支付多少的期权费【参考N （=；N （）= 】 {提示：考虑判断在不派发红利情况下，利用美式看涨期权和欧式看涨期权的关系}

解析：在不派发红利情况下，美式看涨期权等同于欧式看涨期权！所以利用Ｂ—Ｓ公式，就可轻易解出来这个题！同学们注意啦，N（d1）=N（），N（d2）=N （）。给出最后结果为 4、若股票指数点位是702，其波动率估计值σ=，指数期货合约将在3个月后到期，并在到期时用美元按期货价格计算，期货合约的价格是715美元。关于期货的看涨期权时间与期货相同，执行价是740美元，短期利率位7%，问这一期权的理论价格是多少（N（）=，N）= *= 解：F=715，T-t=,σ=,X=740,r= F/X=715/740=，σ(T-t)=*= d1=ln/+2= d2== G=**740) =美元 5、根据看涨期权bs定价公式证明德尔塔等于N（d1）（答案见课本122页）

金融数学专业职业规划

职业规划书CAREER PLANNING （金融数学） 学生姓名： 学号： 指导教师：

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谈谈我对金融数学专业的认识 一、数学与应用数学（金融数学方向）的介绍 金融数学，又称数理金融学、数学金融学、分析金融学，是利用数学工具研究金融，进行数学建模、理论分析、数值计算等定量分析，以求找到金融学内在规律并用以指导实践。金融数学也可以理解为现代数学与计算技术在金融领域的应用，因此，金融数学是一门新兴的交叉学科，发展很快，是目前十分活跃的前言学科之一。 我们的专业与经济学院的金融学。经济学等专业不同，我们的专业偏重数理金融，强调数学手段研究相关问题。在课程设置上既突出数学基础，也注重金融、证券、保险、经济等基本原理。 二、主要课程 数学分析、解析几何、高等代数、离散数学、常微分方程、概率论、数理统计、计量经济学、数学实验、数学模型、财务会计学、金融学、微观经济学、证券投资学、宏观经济学、公司财务管理、金融时间序列分析。 三、我们的就业前景 我们专业的就业方向比较广。主要有：银行、证券、保险业、基金和一些企事业单位涉及金融的工作岗位。 （1）银行 银行有着比较稳定的收入，较好的福利，受到很多金融数学生的青睐，所以竞争性较强。我国现阶段银行分三类：中央银行、商业银行、政策性银行。四大国有银行：中国工商银行、中国农业银行、中国银行、中国建设银行。三家政策性银行：中国国家开发银行、中国农业发展银行、中国进出口银行。股份制商业银行：中信实业银行。恒丰银行、广东发展银行、深圳发展银行、广大银行、兴业银行、交通银行、民生银行、华夏银行、上海浦东发展银行、浙商银行。 （2）证券公司 证券行业是一个高风险、高压力的行业。特别是前三个月有银高业务要求，竞争非常激烈，并且淘汰率比较高，很难坚持，所以有的时候证券公司招人，但同学们不热情。 （3）保险公司 我国是世界上潜在的保险大国，在寿险、财险、养老保险等方面将有巨大市场，为此需要大量精算师和投资管理专家。精算师是我国最紧缺的尖端人才，目前在我国职业400多名精算从业人员，其中79人取得了国内精算师资格证书，但被世界保险界认可的不足50人。据统计，中国加入WTO以后，大批外资保险公司近日中国，精算师的市场需求量达5000人。因此，精算数学和金融数学的发展必将是大趋势。 朱燕燕

北美精算师考试官方样题2015-12-exam-fm-syllabus

Financial Mathematics Exam—December 2015 The Financial Mathematics exam is three-hour exam that consists of 35 multiple-choice questions and is administered as a computer-based test. For additional details, please refer to Exam Rules The goal of the syllabus for this examination is to provide an understanding of the fundamental concepts of financial mathematics, and how those concepts are applied in calculating present and accumulated values for various streams of cash flows as a basis for future use in: reserving, valuation, pricing, asset/liability management, investment income, capital budgeting, and valuing contingent cash flows. The candidate will also be given an introduction to financial instruments, including derivatives, and the concept of no-arbitrage as it relates to financial mathematics. The Financial Mathematics Exam assumes a basic knowledge of calculus and an introductory knowledge of probability. The following learning objectives are presented with the understanding that candidates are allowed to use specified calculators on the exam. The education and examination of candidates reflects that fact. In particular, such calculators eliminate the need for candidates to learn and be examined on certain mathematical methods of approximation. Please check the Updates section on this exam's home page for any changes to the exam or syllabus. Each multiple-choice problem includes five answer choices identified by the letters A, B, C, D, and E, only one of which is correct. Candidates must indicate responses to each question on the computer. Candidates will be given three hours to complete the exam. As part of the computer-based testing process, a few pilot questions will be randomly placed in the exam (paper and pencil and computer-based forms). These pilot questions are included to judge their effectiveness for future exams, but they will NOT be used in the scoring of this exam. All other questions will be considered in the scoring. All unanswered questions are scored incorrect. Therefore, candidates should answer every question on the exam. There is no set requirement for the distribution of correct answers for the multiple-choice preliminary examinations. It is possible that a particular answer choice could appear many times on an examination or not at all. Candidates are advised to answer each question to the best of their ability, independently from how they have answered other questions on the examination. Since the CBT exam will be offered over a period of a few days, each candidate will receive a test form composed of questions selected from a pool of questions. Statistical scaling methods are used to ensure within reasonable and practical limits that, during the same testing period of a few days, all forms of the test are comparable in content and passing criteria. The methodology that has been adopted is used by many credentialing programs that give multiple forms of an exam. The ranges of weights shown in the Learning Objectives below are intended to apply to the large majority of exams administered. On occasion, the weights of topics on an individual exam may fall outside the published range. Candidates should also recognize that some questions may cover multiple learning objectives.

金融数学附答案

1、给定股票价格的二项模型，在下述情况下卖出看涨期权 S 0 S u S d X r τ 股数 50 60 40 55 1/2 1000 （1）求看涨期权的公平市场价格。 （2）假设以公平市场价格+美元卖出1000股期权，需要买入多少股股票进行套期保值，无风险利润是多少 答案：（1）d u d r S S S e S q --=τ0=56.040 6040505.005.0=--??e （2）83.2>73.2，τr e S V -?+?='00 83.2> τr e S -?+?'0 40 6005--=--=?d u S S D U =25.0股 104025.00'-=?-=?-=?d S D 753.9975.0105.005.0'-=?-=??-e 美元 则投资者卖空1000份看涨期权，卖空250股股票，借入9753美元 所以无风险利润为1.85835.005.0=?e 美元 2、假定 S 0 = 100，u=，d=，执行价格X=105，利率r=，p=，期权到期时间 t=3，请用连锁法则方法求出在t=0时该期权的价格。(答案见课本46页) 3、一只股票当前价格为30元，六个月期国债的年利率为3%，一投资者购买一份执行价格为35元的六个月后到期的美式看涨期权，假设六个月内股票不派发红利。波动率σ为. 问题：（1）、他要支付多少的期权费【参考N （=；N （）= 】 {提示：考虑判断在不派发红利情况下，利用美式看涨期权和欧式看涨期权的关系}

解析：在不派发红利情况下，美式看涨期权等同于欧式看涨期权！所以利用Ｂ—Ｓ公式，就可轻易解出来这个题！同学们注意啦，N（d1）=N（），N（d2）=N（）。给出最后结果为 4、若股票指数点位是702，其波动率估计值σ=，指数期货合约将在3个月后到期，并在到期时用美元按期货价格计算，期货合约的价格是715美元。关于期货的看涨期权时间与期货相同，执行价是740美元，短期利率位7%，问这一期权的理论价格是多少（N（）=，N）= *= 解：F=715，T-t=,σ=,X=740,r= F/X=715/740=，σ(T-t)=*= d1=ln/+2= d2== G=**740) =美元 5、根据看涨期权bs定价公式证明德尔塔等于N（d1）（答案见课本122页）

西交利物浦大学专业解读2020版：金融数学专业

金融数学专业的就业前景、本科毕业后的海外留学前景如何？在高考志愿填报的过程中，广大学生和家长需要对心仪大学的专业特色、课程设置、就业方向等有一个基本了解。下面将为你带来西交利物浦大学专业介绍系列之：金融数学。 西交利物浦大学金融数学专业概览 金融数学专业旨在培养学生具备金融机构所需的专业数学知识和定量技能，重点训练学生将理论知识运用到金融实践中的能力。 为什么选择西浦金融数学专业？ －由来自数学科学系、西浦国际商学院、计算机科学与软件工程系等不同院系的教师授课； －学习专业的知识与技能，有助于你获取国际认可的职业资格证书； －校园距离上海仅80公里，学生可就近获得“世界500强”等知名企业的实习和工作机会； －毕业生可同时获得中国教育部认可的西交利物浦大学学位和国际认可的英国利物浦大学学位。

知识与技能 本专业毕业生将具备以下能力： 1.掌握计算数学和统计学的核心领域知识； 2.拥有较为完备的会计技能、金融及经济学知识； 3.能够定量分析现实中的金融问题。 就业前景 毕业生通常就职于金融、保险、证券、银行等行业。 该专业也为毕业生继续攻读数学或金融领域的硕士学位打下坚实基础。 课程设置 第一学年 在英国，本科阶段学习学制三年，而中国本科阶段学制为四年。因此，对于已获得相应学时、证书的学生，在我校可以直接升入二年级进行专业学习；大多数学

生则是进入一年级学习，包括众多有吸引力的课程，语言课程以及专业学习相关的核心技能学习。 第二学年 金融学财务会计 微观经济学 宏观经济学 金数实分析 概率与统计 金融计算 应用数学方法 第三学年 数值分析计量经济学 抽样和假设检验 运筹学 证券市场 财务管理 金数JAVA编程 EXCEL VBA金融建模 金融数学 统计分布理论 第四学年

金融数学附答案

金融数学附答案 Prepared on 24 November 2020

1、给定股票价格的二项模型，在下述情况下卖出看涨期权 S 0 S u S d X r τ 股数 50 60 40 55 1/2 1000 （1）求看涨期权的公平市场价格。 （2）假设以公平市场价格+美元卖出1000股期权，需要买入多少股股票进行套期保值，无风险利润是多少 答案：（1）d u d r S S S e S q --=τ0=56.040 6040505.005.0=--??e （2）83.2>73.2，τr e S V -?+?='00 83.2> τr e S -?+?'0 40 6005--=--=?d u S S D U =25.0股 104025.00'-=?-=?-=?d S D 753.9975.0105.005.0'-=?-=??-e 美元 则投资者卖空1000份看涨期权，卖空250股股票，借入9753美元 所以无风险利润为1.85835.005.0=?e 美元 2、假定 S 0 = 100，u=，d=，执行价格X=105，利率r=，p=，期权到期时间 t=3，请用连锁法则方法求出在t=0时该期权的价格。(答案见课本46页) 3、一只股票当前价格为30元，六个月期国债的年利率为3%，一投资者购买一份执行价格为35元的六个月后到期的美式看涨期权，假设六个月内股票不派发红利。波动率σ为. 问题：（1）、他要支付多少的期权费【参考N （=；N （）= 】 {提示：考虑判断在不派发红利情况下，利用美式看涨期权和欧式看涨期权的关系}

解析：在不派发红利情况下，美式看涨期权等同于欧式看涨期权！所以利用Ｂ—Ｓ公式，就可轻易解出来这个题！同学们注意啦，N（d1）=N（），N（d2）=N（）。给出最后结果为 4、若股票指数点位是702，其波动率估计值σ=，指数期货合约将在3个月后到期，并在到期时用美元按期货价格计算，期货合约的价格是715美元。关于期货的看涨期权时间与期货相同，执行价是740美元，短期利率位7%，问这一期权的理论价格是多少（N（）=，N）= *= 解：F=715，T-t=,σ=,X=740,r= F/X=715/740=，σ(T-t)=*= d1=ln/+2= d2== G=**740) =美元 5、根据看涨期权bs定价公式证明德尔塔等于N（d1）（答案见课本122页）

北大版金融数学引论第二章答案,DOC

版权所有，翻版必究 第二章习题答案 1．某家庭从子女出生时开始累积大学教育费用5万元。如果它们前十年每年底存 款1000元，后十年每年底存款1000+X 元，年利率7%。计算X 。 解： S=1000s 20 ?p 7%+Xs 10 ?p 7% X= 50000?1000s 20 ?p 7% s 10 ?p7% =651.72 2．价值10,000元的新车。购买者计划分期付款方式：每月底还250元，期限4年。 月结算名利率18%。计算首次付款金额。 解：设首次付款为X ，则有 10000=X+250a 48 ?p1.5% 解得 X=1489.36 3．设有n 年期期末年金，其中年金金额为n ，实利率i=1 。试计算该年金的现值。 解： PV = na?n pi 1?v n n = n 1 n = (n+1)n n 2 ?n n +2 (n+1)n 4．已知：a?n p =X ，a 2 ?n p =Y 。试 用X 和Y 表示d 。 解：a 2 ?n p =a?n p +a?n p (1?d)n 则 Y ?X 1 d=1?( X )n 5．已知：a?7 p =5.58238,a 11 ?p =7.88687,a 18 ?p =10.82760。计算i 。 解： a 18 ?p =a ?7p +a 11 ?p v 7 解得 6.证明： 1 1?v 10 = s 10?p +a ∞?p 。 s 10?p i=6.0% 北京大学数学科学学院金融数学系 第1页

版权所有，翻版必究 证明： s 10 ?p +a ∞?p (1+i)10 ?1+1 1 s 10?p = i (1+i)10 ?1 i i = 1?v 10 7．已知：半年结算名利率6%，计算下面10年期末年金的现值：开始4年每半 年200元，然后减为每次100元。 解： PV =100a?8p3% +100a 20?p 3% =2189.716 8．某人现年40岁，现在开始每年初在退休金帐号上存入1000元，共计25年。然 后，从65岁开始每年初领取一定的退休金，共计15年。设前25年的年利率为8%， 后15年的年利率7%。计算每年的退休金。 解：设每年退休金为X ，选择65岁年初为比较日 1000¨25?p8%=X¨15?p7% 解得 9．已知贴现率为10%，计算¨?8 p 。 X=8101.65 解：d=10%，则 i=1 10.求证： (1)¨?n p =a?n p +1?v n ； 1?d ?1=1 9 ¨?8 p =(1+i) 1?v 8 i =5.6953 (2)¨?n p =s??n p 1+(1+i)n 并给出两等式的实际解释。 证明：(1)¨?n p =1 ? d v n =1 ?i v n =1 ?v n i +1?v n 所以 (2)¨?n p = (1+ i)n ?1 1+i ¨?n p =a?n p +1?v n (1+i )n ?1=(1+i)n ?1 n ?1 d = i 1+i i +(1+i) 所以 ¨?n p =s??n p 1+(1+i) n

金融数学 练习题详解

金融数学第一章练习题详解 第1章利息度量 1.1 现在投资$600，以单利计息，2年后可以获得$150的利息。如果以相同的复利利率投资$2000，试确定在3年后的累积值。 1.2 在第1月末支付314元的现值与第18月末支付271元的现值之和，等于在第T月末支付1004元的现值。年实际利率为5%。求T。 1.3 在零时刻，投资者A在其账户存入X，按每半年复利一次的年名义利率i计息。同时，投资者Ｂ在另一个账户存入2X，按利率i （单利）来计息。假设两人在第八年的后六个月中将得到相等的利息，求i。 1.4 一项投资以δ的利息力累积，27.72年后将翻番。金额为1的投资以每两年复利一次的名义利率δ累积n年，累积值将成为7.04。求n。 1.5 如果年名义贴现率为6%，每四年贴现一次，试确定$100在两年末的累积值。 1.6 如果)(m i=0.1844144，)(m d=0.1802608，试确定m。 1.7 基金A以每月复利一次的名义利率12%累积。基金B以 =t/6 t 的利息力累积。在零时刻，分别存入1到两个基金中。请问何时两个基金的金额将相等。

1.8 基金A 以t δ=a+bt 的利息力累积。基金B 以t δ=g+ht 的利息力 累积。基金A 与基金B 在零时刻和n 时刻相等。已知a>g>0，h>b>0。求n 。 1.9 在零时刻将100存入一个基金。该基金在头两年以每个季度贴现一次的名义贴现率?支付利息。从t=2开始，利息按照t t +=11δ的利息力支付。在t=5时，存款的累积值为260。求δ。 1.10在基金A 中，资金1的累积函数为t+1，t>0；在基金B 中，资金1的累积函数为1+t 2。请问在何时，两笔资金的利息力相等。 1.11已知利息力为t t +=12δ。第三年末支付300元的现值与在第六年末支付600元的现值之和，等于第二年末支付200元的现值与在第五年末支付X 元的现值。求X 。 82 .315))51/(())21(200-)61(600)31(300() 5()2(200)6(600)3(300)1()()1()(22-2211112 12)1ln(212 0=++?+?++?=??+?=?+?+=?+==?=---------++X a X a a a t t a t e e t a t dt t t 1.12已知利息力为1003t t =δ。请求)3(1-a 。 1.13资金A 以10%的单利累积，资金B 以5%的单贴现率累积。请问在何时，两笔资金的利息力相等。 1.14某基金的累积函数为二次多项式，如果向该基金投资1年，在上半年的名义利率为5%（每半年复利一次），全年的实际利率为7%，试确定5.0δ。

金融数学介绍

概述 金融学是现代经济发展的必然产物，是根据经济的发展而兴起的，是研究价值判断和价值规律的学科。主要包括传统金融学理论和演化金融学理论两大领域。而对于金融数学专业更是在金融学和数学的基础上发展起来的，今天我们就讲解一下什么是金融数学专业? 专业介绍 金融数学是新兴综合学科，受到国际金融界和应用数学界的高度重视。该系培养对金融活动进行定量分析和科学预测的复合型金融人才。有金融数学和保险精算学两个方向。除了数学基础课程，该系学生还要学习利息理论及应用、证券投资学、寿险精算等金融数学专业课程，以及经济学和管理学的部分课程。 学系简介 金融数学是近年来蓬勃发展的新学科，在国际金融界和应用数学界受到高度重视。金融数学专业除培养金融数学本科生外，还通过该专业的学习委金融数学与精算学专业输送应用硕士的高级人才。金融数学将培养学生不仅具有扎实的现代数学基础，熟练使用计算机的技能，而且具有深厚的金融专业知识，文理并茂，全面发展。高年级开设概率统计、随机分析、微分方程等数学基础课外，还将开设利息、证券、汇率、保险精算等金融数学的专业课程。金融数学系本科毕业生将能熟练运用数学知识和数据分析方法，从事某些金融保险实际工作，并可继续深造，到高等学校和科研机构应用数学、经济和金融管理等专业攻读硕士学位。 就业方向 金融数学专业考生毕业后就业方向很广泛，可以在(如:中国工商银行、建设银行、农业银行等在内的国有四大银行以及招商银行等股份制商行、城市商业银行、外资银行驻国内分支机构，金融学专业的毕业生常有涉猎，而且往往是广大考生的最佳选择。)、(如:中国人寿保险、平安保险、太平洋保险等)、(如:中央人民银行、银行业监督管理委员会、证券业监督管理委员会、保险业监督管理委员会等)、(国家开发银行、中国农业发展银行等)、(含基金管理公司、上交所、深交所、期交所等)、(如:社保基金管理中心或社保局等)、(如信托投资公司、金融投资控股公司、投资咨询顾问公司、大型企业财务公司等)、和 就业前景 金融学做为商学中显学的地位在近年来的中国研究生教育中日益提高，无论是了解亦或是不了解这一行的朋友，一听到“金融”二字都会兴奋不已，因为在许多人看来，这是与财富、声誉最为靠近的一门学科，各式各样金融评论员在媒体上的狂轰乱炸更是将这种看法带入极致。 同时由于金融学涉及的范围比较广泛，所以就业的方向也就很多，也就使得我们的就业前景十分明朗。

【SOA】关于北美精算师,你必须知道的入门级知识——Exam P

关于北美精算师，你必须知道的入门级知识——Exam P 成为一名北美准精算师（ASA）必须要经历五门SOA的准精算师考试，而其中最简单也是大部分人最先开始学习准备的就是Exam P，即probability。顾名思义，Exam P考察的就是最基本的数理统计与概率问题。下面我们就来了解一下Exam P的考试形式与内容。 考试目的 考生可以掌握用于定量评估风险的基本的概率方法，并着重于用这些方法应用解决精算学中遇到的问题。参加这门考试的考生应具有一定的微积分基础，并了解基本的概率、保险和风险管理的概念。 考试形式 Exam P采用机考的形式，总共30道单项选择题，考试时间为3个小时。每道选择题共有5个选项，其中只有一个正确选项。 与SAT考试不同的是，Exam P考试答错并不会额外扣分，也就是说考生一定不要空任何一道题。Exam P中会随机分布几道“pilot question”，这些题目是主办方用来分析从而改进将来的考试而出现的，它们的正确与否并不会影响到考生的实际分数。但是由于考生并无法分辨出这些题目，所以对每一道题目，考生都要同样认真地对待。 考试内容

概率（占总分10%－20%） 最基本的事件概率计算问题。包括集合方程与表示（sat functions）、互斥事件（mutually exclusive events）、事件独立性（independence of events）、组合概率（Combinatorial probability）、条件概率（Conditional probability）以及贝叶斯定理（Bayes theorem）等。 拥有单因素概率分布的随机变量（占总分35%－45%） 连续分布或离散分布的单因素随机变量的研究。包括PDF&CDF（Probability density functions and Cumulative distribution functions）、独立随机事件的和的分布、众数（Mode）、中位数（Median）、百分位数（Percentile）、动差（Moment）、方差（Variance）以及变形等问题。 拥有多因素概率分布的随机变量（占总分35%－45%） 包括联合PDF&CDF、中心极限定理（central limit theorem）、条件与边缘概率分布与动差（Conditional and marginal probability distributions and moments）、条件与边缘概率分布的方差、协方差与概率系数（Covariance and correlation coefficients）以及变换与顺序统计（Transformation and order statistics）等。 提醒：众所周知，2018年7月1日起，SOA新课程体系将正式生效，其中Exam P科目不变，考试大纲有变动，具体有那些变化？？？后台回复“Exam P”免费获取Exam P最新考试大纲。 考试时间