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前言学弟学妹们,当你们看到这篇复习资料的时候, 学长已经在文档上传的当天上午参加了国际金融的考试, 本复习资料主要针对对象为成都信息工程学院(CUIT)英语系大三学生, 且立足教材也基于托马斯·A ·普格尔(Thomas A. Pugel)先生所著国际金融英文版·第15版, 其他版本或者相似教材也可作为参考, 本资料的整理除了参考维基百科,百度百科以及MBA 智库百科,当然最重要的是我们老师的课件. 为了帮助同学们顺利通过考试, 当然是拿到高分, 希望此资料能够帮助你们节省时间, 达到高效复习的效果.外国语学院2011级,陈爵歌(Louis) 2014年1月6日晚于宿舍 Chapter 2Transnationality Index (跨国化指数)(TNI ) is a means of ranking multinational corporations that is employed by economists and politicians. (反映跨国公司海外经营活动的经济强度，是衡量海外业务在公司整体业务中地位的重要指标) Foreign assets to total assets(外国资产占总资产比)Foreign sales to total sales(海外销售占总销售)Foreign employees to total employees(外籍雇员占总雇员)跨国化指数的构成联合国跨国公司与投资司使用的跨国化指数由三个指标构成：国外资产对公司总资产的百分比；国外销售对公司总销售的百分比；国外雇员人数对公司雇员总人数的百分比关于TNI 的计算公式:International Economic Integration( 国际经济一体化)International economic integration refers to the extent and strength of real -sector and financial -sector linkages among national economies.(国际经济一体化是指两个或两个以上的国家在现有生产力发展水平和国际分工的基础上，由政府间通过协商缔结条约，让渡一定的国家主权，建立两国或多国的经济联盟，从而使经济达到某种程度的结合以提高其在国际经济中的地位)Real Sector(实际经济部门): The sector of the economy engaged in the production and sale of goods and services(指物质的、精神的产品和服务的生产、流通等经济活动。

⾦融市场与机的构Madura第九版题库ch11Chapter 11Stock Valuation and Risk1. The common price-earnings valuation method applied the ______ price-earnings ratio to ________earnings per share in order to value the firm’s stock.A) firm’s; industryB) firm’s; firm’sC) average industry; industryD) average industry; firm’sANSWER: D2. A firm is expected to generate earnings of $2.22 per share next year. The mean ratio of share price toexpected earnings of competitors in the same industry is 15. Based on this information, the valuation of the firm’s shares based on the price-earnings (PE) method isA) $2.22.B) $6.76.C) $33.30.D) none of the aboveANSWER: C3. The PE method to stock valuation may result in an inaccurate valuation for a firm if errors are madein forecasting the firm’s future earnings or in choosing the industry composite used to derive the PE ratio.A) TrueB) FalseANSWER: A4. Bolwork Inc. is expected to pay a dividend of $5 per share next year. Bolwork’s dividends areexpected to grow by 3 percent annually. The required rate of return for Bolwork stock is 15 percent.Based on the dividend discount model, a fair value for Bolwork stock is $_______ per share.A) 33.33B) 166.67C) 41.67D) 60.00ANSWER: C5. Protsky Inc. just paid a divid end of $2.20 per share. The dividend growth rate for Protsky’s dividendsis 3 percent per year. If the required rate of return on Protsky stock is 12 percent, the stock should be valued at $_______ pershare according to the dividend discount model.A) 24.44B) 25.182 Stock Valuation and RiskC) 18.88D) 75.53ANSWER: B6. The limitations of the dividend discount model are more pronounced when valuing stocksA) that pay most of their earnings as dividends.B) that retain most of their earnings.C) that have a long history of dividends.D) that have constant earnings growth.ANSWER: B7. Hancock Inc. retains most of its earnings. The company currently has earnings per share of $11.Hancock expects its earnings to grow at a constant rate of 2 percent per year. Furthermore, theaverage PE ratio of all other firms in Hancock’s industry is 12. Hancock is expected to pay dividends per share of $3.50 during each of the next three years. If investors require a 10 percent rate of return on Hancock stock, a fair price for Hancock stock today is $________.A) 113.95B) 111.32C) 105.25D) none of the aboveANSWER: A8. When evaluating stock performance, ______ measures variability that is systematically related tomarket returns; ______ measures total variabili ty of a stock’s returns.A) beta; standard deviationB) standard deviation; betaC) intercept; betaD) beta; error termANSWER: A9. The ___________ is commonly used as a proxy for the risk-free rate in the Capital Asset PricingModel.A) Treasury bond rateB) prime rateC) discount rateD) federal funds rateANSWER: A10. Arbitrage pricing theory (APT) suggests that a stock’s price is influenced only by a stock’s beta.A) TrueB) FalseANSWER: BStock Valuation and Risk 3 11. Stock prices of U.S. firms primarily involved in exporting are likely to be ________ affected by aweak dollar and __________ affected by a strong dollar.A) favorably; adverselyB) adversely; adverselyC) favorably; favorablyD) adversely; favorablyANSWER: A12. A weak dollar may enhance the value of a U.S. firm whose sales are dependent on the U.S. economy.A) TrueB) FalseANSWER: A13. The January effect refers to the __________ pressure on ______ stocks in January of every year.A) downward; largeB) upward; largeC) downward; smallD) upward; smallANSWER: D14. The expected acquisition of a firm typically results in ____________ in the target’s stock price.A) an increaseB) a decreaseC) no changeD) none of the aboveANSWER: A15. The _______ index can be used to measure risk-adjusted performance of a stock while controlling forthe stock’s volatility.A) SharpeB) TreynorC) arbitrageD) marginANSWER: A16. The _______ index can be used to measure risk-adjusted performance of a stock while controlling for the stock’s beta.A) SharpeB) TreynorC) arbitrageD) marginANSWER: B4 Stock Valuation and Risk17. Stock price volatility increased during the credit crisis.A) TrueB) FalseANSWER: A18. The Sharpe Index measures theA) average return on a stock.B) variability of stock returns per unit of returnC) stock’s beta adjusted for risk.D) excess return above the risk-free rate per unit of risk.ANSWER: D19. A stock’s average return is 11 percent. The average risk-free rate is 9 percent. The stock’s beta is 1and its standard deviation of returns is 10 percent. What is the Sharpe Index?A) .05B) .5C) .1D) .02E) .2ANSWER: E20. A stock’s average return is 10 percent. The average risk-free rate is 7 percent. The standarddeviation of the stock’s return is 4 percent, and the stock’s beta is 1.5. What is the Treynor Index for the stock?A) .03B) .75C) 1.33D) .02E) 50ANSWER: D21. If security prices fully reflect all market-related information (such as historical price patterns) but do not fully reflect all other public information, security markets areA) weak-form efficient.B) semi-strong form efficient.C) strong form efficient.D) B and CE) none of the aboveANSWER: A22. If security markets are semi-strong form efficient, investors cannot solely use ______ to earn excess returns.A) previous price movementsB) insider informationStock Valuation and Risk 5C) publicly available informationD) A and CANSWER: D23. The ______ is commonly used to determine what a stock’s price should have been.A) Capital Asset Pricing ModelB) Treynor IndexC) Sharpe IndexD) B and CANSWER: A24. A stock’s beta is estimated to be 1.3. The risk-free rate is 5 percent, and the market return is expected to be 9 percent. What is the expected return on the stock based on the CAPM?A) 5.2 percentB) 11.7 percentC) 16.7 percentD) 4 percentE) 10.2 percentANSWER: E25. According to the text, other things being equal, stock prices of U.S. firms primarily involved inexporting could be ______ affected by a weak dollar. Stock prices of U.S. importing firms could be ______ affected by a weak dollar.A) adversely; favorablyB) favorably; adverselyC) favorably; favorablyD) adversely; adverselyANSWER: B26. The demand by foreign investors for the stock of a U.S. firm sold on a U.S. exchange may be higherwhen the dollar is expected to ______, other things being equal. (Assume the firm’s operations are unaffected by the value of the dollar.)A) strengthenB) weakenC) stabilizeD) B and CANSWER: A27. A higher beta of an asset reflectsA) lower risk.B) lower covariance between the asset’s returns and market returns.C) higher covariance between the asset’s returns and the market returns.D) none of the above6 Stock Valuation and RiskANSWER: C28. The “January effect” refers to a largeA) rise in the price of small stocks in January.B) decline in the price of small stocks in January.C) decline in the price of large stocks in January.D) rise in the price of large stocks in January.ANSWER: A29. Technical analysis relies on the use of ______ to make investment decisions.A) interest ratesB) inflationary expectationsC) industry conditionsD) recent stock price trendsANSWER: D30. The arbitrage pricing theory (APT) differs from the capital asset pricing model (CAPM) in that it suggests that stock pricesA) are influenced only by the market itself.B) can be influenced by a set of factors in addition to the market.C) are not influenced at all by the market.D) cannot be influenced at all by the industry factors.ANSWER: B31. According to the capital asset pricing model, the required return by investors on a security isA) inversely related with the risk-free rate.B) inversely related with the firm’s beta.C) inversely related with the market return.D) none of the aboveANSWER: D32. Boris stock has an average return of 15 percent. Its beta is 1.5. Its standard deviation of returns is 25 percent. The average risk-free rate is 6 percent. The Sharpe index for Boris stock isA)0.35.B)0.36.C)0.45.D)0.28.E)none of the aboveANSWER: B33. Morgan stock has an average return of 15 percent, a beta of 2.5, and a standard deviation of returns of20 percent. The Treynor index of Morgan stock isA)0.04.B)0.05.Stock Valuation and Risk 7C)0.35.D)0.03.E)none of the above34. Zilo stock has an average return of 15 percent, a beta of 2.5, and a standard deviation of returns of 20percent. The Sharpe index of Zilo stock isA)0.36.B)0.35.C)0.28.D)0.45.E)none of the aboveANSWER: B35. Sorvino Co. is expected to offer a dividend of $3.2 per share per year forever. The required rate ofreturn on Sorvino stock is 13 percent. Thus, the price of a share of Sorvino stock, according to the dividend discount model, is $_________.A) 4.06B) 4.16C)40.63D)24.62E)none of the aboveANSWER: D36. Kudrow stock just paid a dividend of $4.76 per share and plans to pay a dividend of $5 per share nextyear, which is expected to increase by 3 percent per year subsequently. The required rate of return is15 percent. The value of Kudrow stock, according to the dividend discount model, is $__________.A)39.67B)41.67C)33.33D)31.73E)none of the aboveANSWER: B37. LeBlanc Inc. currently has earnings of $10 per share, and investors expect that the earnings per sharewill grow by 3 percent per year. Furthermore, the mean PE ratio of all other firms in the sameindustry as LeBlanc Inc. is 15. LeBlanc is expected to pay a dividend of $3 per share over the next four years, and an investor in LeBlanc requires a return of 12 percent. What is the forecasted stock price of LeBlanc in four years, using the adjusted dividend discount model?A)$150.00B)$163.91C)$45.00D)$168.83E)none of the above8 Stock Valuation and Risk38. Tarzak Inc. has earnings of $10 per share, and investors expect that the earnings per share will growby 3 percent per year. Furthermore, the mean PE ratio of all other firms in the same industry asTarzac is 15. Tarzac is expected to pay a dividend of $3 per share over the next four years, and an investor in Tarzac requires a return of 12 percent. The estimated stock price of Tarzak today should be __________ using the adjusted dividend discount model.A)$116.41B)$104.91C)$161.15D)none of the aboveANSWER: A39. The standard deviation of a stock’s returns is used to measure a stock’sA)volatility.B)beta.C)Treynor Index.D)risk-free rate.ANSWER: A40. The formula for a stock portfolio’s volatility does not contain theA)weight (proportional investment) assigned to each stock.B)variance (standard deviation squared) of returns of each stock.C) correlation coefficients between returns of each stock.D) risk-free rate.ANSWER: D41. If the returns of two stocks are perfectly correlated, thenA) their betas should each equal 1.0.B) the sum of their betas should equal 1.0.C) their correlation coefficient should equal 1.0.D) their portfolio standard deviation should equal 1.0.ANSWER: C42. A stock’s beta can be measured from the estimate of the using regression analysis.A) interceptB) market returnC) risk-free rateD) slope coefficientANSWER: D43. A beta of 1.1 means that for a given 1 percent change in the value of the market, theis expected to change by 1.1 percent in the same direction.A)risk-free rateB)stock’s valueStock Valuation and Risk 9C)s tock’s standard deviationD)correlation coefficientANSWER: B44. Stock X has a lower beta than Stock Y. The market return for next month is expected to be either–1 percent, +1 percent, or +2 percent with an equal probability of each scenario. The probability distribution of Stock X returns for next month isA)the same as that of Stock Y.B)more dispersed than that of Stock Y.C)less dispersed than that of Stock Y.D)zero.ANSWER: C45. The beta of a stock portfolio is equal to a weighted average of theA)betas of stocks in the portfolio.B)betas of stocks in the portfolio, plus their correlation coefficients.C)standard deviations of stocks in the portfolio.D)correlation coefficients between stocks in the portfolio.ANSWER: A46. Value at risk estimates the a particular investment for a specified confidence level.A)beta ofB)risk-free rate ofC)largest expected loss toD)standard deviation ofANSWER: C47. A stock has a standard deviation of daily returns of 1 percent. It wants to determine the lowerboundary of its probability distribution of returns, based on 1.65 standard deviations from theexpected outcome. The stock’s expected daily return is .2 percent. The lower boundary isA)–1.45 percent.B)–1.85 percent.C)0 percent.D)–1.65 percent.ANSWER: A48. A stock has a standard deviation of daily returns of 3 percent. It wants to determine the lower boundary of its probability distribution of returns, based on 1.65 standard deviations from the expected outcome. The stock’s expected daily return is .1 percent. The lower boundary isA)–1.65 percent.B)–3.00 percent.C)–4.85 percent.D)–5.05 percent.10 Stock Valuation and RiskANSWER: C49. Which of the following is not commonly used as the estimate of a stock’s volatility?A)the estimate of its standard deviation of returns over a recent periodB)the trend of historical standard deviations of returns over recent periodsC)the implied volatility derived from an option pricing modelD)the estimate of its option premium derived from an option pricing modelANSWER: D50. The credit crisis only affected the stock performance of stocks in the U.S.A) TrueB) FalseANSWER: B51. When new information suggests that a firm will experience lower cash flows than previously anticipated or lower risk, investors will revalue the corresponding stock downward.A) TrueB) FalseANSWER: B52. A relatively simple method of valuing a stock is to apply the mean price-earnings (PE) ratio of all publicly traded competitors in the respective industry to the firm’s expected earnings for the year.ANSWER: A53. While the previous year’s earnings are often used as a base for forecast ing future earnings, the recent year’s earnings do not always provide an accurate forecast of the future.A) TrueB) FalseANSWER: A54. If investo rs agree on a firm’s forecasted earnings, they will derive the same value for that firm using the PE method to value the firm’s stock.A) TrueB) FalseANSWER: B55. The dividend discount model states that the price of a stock should reflect the present value of the stock’s future dividends.A) TrueB) FalseStock Valuation and Risk 11 ANSWER: A56. The dividend discount model can be adapted to assess the value of any firm, even those that retain most or all of their earnings.A) TrueB) FalseANSWER: A57. For firms that do not pay dividends, a more suitable valuation may be the free cash flow model.A) TrueB) FalseANSWER: A58. The capital asset pricing model (CAPM) is based on the premise that the only important risk of a firm is unsystematic risk.A) TrueB) FalseANSWER: B59. The prime rate is commonly used as a proxy for the risk-free rate in the capital asset pricing model60. A stock with a beta of 2.3 means that for every 1 percent change in the market overall, the stock tends to change by 2.3 percent in the same direction.A) TrueB) FalseANSWER: A61. Stocks that have relatively little trading are normally subject to less price volatility.A) TrueB) FalseANSWER: B62. A firm’s stock price is affected not only by macroeconomic and market conditions but also by firm specific conditions.A) TrueB) FalseANSWER: A12 Stock Valuation and Risk63. Stock repurchases are commonly viewed as an unfavorable signal about the firm.A) TrueB) FalseANSWER: B64. The main source of uncertainty in computing the return of a stock is the dividend to be received next year.A) TrueB) FalseANSWER: B65. A stock portfolio has more volatility when its individual stock returns are uncorrelated.A) TrueB) FalseANSWER: B66. Beta serves as a measure of risk because it can be used to derive a probability distribution of return67. The value-at-risk method is intended to warn investors about the potential maximum loss that couldoccur.A) TrueB) FalseANSWER: A68. Regarding the value-at-risk method, the same methods used to derive the maximum expected loss ofone stock can be applied to derive the maximum expected loss of a stock portfolio for a givenconfidence level.A) TrueB) FalseANSWER: A69. Portfolio managers who monitor systematic risk rather than total risk are more concerned about stockvolatility than about beta.A) TrueB) FalseANSWER: BStock Valuation and Risk 13 70. Regarding the implied standard deviation, by plugging in the actual option premium paid by investorsfor a specific stock in the option-pricing model, it is possible to derive the anticipated volatility level.A) TrueB) FalseANSWER: A71. One way to forecast a portfolio’s beta is to first forecast the betas of the individual stocks in theportfolio and then sum the individual forecasted betas, weighted by the market value of each stock.A) TrueB) FalseANSWER: B72. If beta is thought to be the appropriate measure of risk, a stock’s risk-adjusted returns should bedetermined by the Sharpe index.ANSWER: B73. The Treynor index is similar to the Sharpe index, except that is uses beta rather than standarddeviation to measure the stock’s risk.A) TrueB) FalseANSWER: A74. Fabrizio, Inc. is expected to generate earnings of $1.50 per share this year. If the mean ratio of shareprice to expected earnings of competitors in the same industry is 20, then the stock price per share is $_________.A)13.33B) 3.00C)20.00D)30.00E)none of the aboveANSWER: D75. Which of the following is not a reason the PE ratio method may result in an inaccurate valuation for afirm?A)potential errors in the forecast of the firm’s betaB)potential errors in the forecast of the firm’s future earningsC)potential errors in the choice of the industry composite used to derive the PE ratioD)All of the above are reasons the PE ratio method may result in an inaccurate valuation for a firm.ANSWER: A14 Stock Valuation and Risk76. Sorvino Co. is expected to offer a dividend of $3.2 per share per year forever. The required rate ofreturn on Sorvino stock is 13 percent. Thus, the price of a share of Sorvino stock, according to the dividend discount model, is $_________.A) 4.06B) 4.16C)24.62D)40.63E)none of the aboveANSWER: Csubsequently. The required rate of return is 15 percent. The value of Kudrow stock, according to the dividend discount model, is $__________.A)39.67B)33.33C)31.73D)41.67E)none of the aboveANSWER: D78. The limitations of the dividend discount model are most pronounced for a firm thatA)has a high beta.B)has high expected future earnings.C)distributes most of its earnings as dividends.D)retains all of its earnings.E)none of the aboveANSWER: D79. Which of the following is incorrect regarding the capital asset pricing model (CAPM)?A)It is sometimes used to estimate the required rate of return for any firm with publicly traded stock.B)It is based on the premise that the only important risk of a firm is systematic risk.C)It is concerned with unsystematic risk.D)All of the above are true.ANSWER: C80. The _______________ is not a factor used in the capital asset pricing model (CAPM) to derive thereturn of an asset.A)prevailing risk-free rateB)dividend growth rateC)market returnD)covariance between the asset’s returns and market returnsE)All of the above are factors used in the CAPM.NSWER: BStock Valuation and Risk 15 81. Schwimmer Corp. has a beta of 1.5. The prevailing risk-free rate is 5 percent and the annual marketreturn in recent years has been 11 percent. Based on this information, the required rate of return on Schwimmer Corp. stockB) 6.5C)16.5D)14.0E)none of the aboveANSWER: D82. Which of the following is not a type of factor that drives stock prices, according to your text?A)economic factorsB)market-related factorsC)firm-specific factorsD)All of the above are factors that affect stock prices.ANSWER: D83. ______________ is (are) not a market-related factor(s) that affect(s) stock prices.A)Interest ratesB)Noise tradingC)TrendsD)January effectE)All of the above are market-related factors that affect stock prices.ANSWER: A84. _____________ is (are) not a firm-specific factor(s) that affect(s) stock prices.A)Exchange ratesB)Dividend policy changesC)Stock offerings and repurchasesD)Earnings surprisesE)All of the above are firm-specific factors that affect stock prices.ANSWER: A85. The ____________ is not a measure of a st ock’s risk.A)stock’s price volatilityB)stock’s returnC)stock’s betaD)value-at-risk methodE)All of the above are measures of a stock’s risk.ANSWER: B86. If the standard deviation of a stock’s returns over the last 12 quarters i s 4 percent, and if there is no16 Stock Valuation and RiskB)68; 8C)95; 8D)95; 6E)none of the above ANSWER: A。

Option Pricing: A Simplified Approach†John C. CoxMassachusetts Institute of Technology and Stanford UniversityStephen A. RossYale UniversityMark RubinsteinUniversity of California, BerkeleyMarch 1979 (revised July 1979)(published under the same title in Journal of Financial Economics (September 1979))[1978 winner of the Pomeranze Prize of the Chicago Board Options Exchange][reprinted in Dynamic Hedging: A Guide to Portfolio Insurance, edited by Don Luskin (John Wiley andSons 1988)][reprinted in The Handbook of Financial Engineering, edited by Cliff Smith and Charles Smithson(Harper and Row 1990)][reprinted in Readings in Futures Markets published by the Chicago Board of Trade, Vol. VI (1991)][reprinted in Vasicek and Beyond: Approaches to Building and Applying Interest Rate Models, edited byRisk Publications, Alan Brace (1996)][reprinted in The Debt Market, edited by Stephen Ross and Franco Modigliani (Edward Lear Publishing2000)][reprinted in The International Library of Critical Writings in Financial Economics: Options Marketsedited by G.M. Constantinides and A..G. Malliaris (Edward Lear Publishing 2000)]AbstractThis paper presents a simple discrete-time model for valuing options. The fundamental economic principles of option pricing by arbitrage methods are particularly clear in this setting. Its development requires only elementary mathematics, yet it contains as a special limiting case the celebrated Black-Scholes model, which has previously been derived only by much more difficult methods. The basic model readily lends itself to generalization in many ways. Moreover, by its very construction, it gives rise to a simple and efficient numerical procedure for valuing options for which premature exercise may be optimal.____________________† Our best thanks go to William Sharpe, who first suggested to us the advantages of the discrete-time approach to option pricing developed here. We are also grateful to our students over the past several years. Their favorable reactions to this way of presenting things encouraged us to write this article. We have received support from the National Science Foundation under Grants Nos. SOC-77-18087 and SOC-77-22301.1. IntroductionAn option is a security that gives its owner the right to trade in a fixed number of shares of a specified common stock at a fixed price at any time on or before a given date. The act of making this transaction is referred to as exercising the option. The fixed price is termed the strike price, and the given date, the expiration date. A call option gives the right to buy the shares; a put option gives the right to sell the shares.Options have been traded for centuries, but they remained relatively obscure financial instruments until the introduction of a listed options exchange in 1973. Since then, options trading has enjoyed an expansion unprecedented in American securities markets.Option pricing theory has a long and illustrious history, but it also underwent a revolutionary change in 1973. At that time, Fischer Black and Myron Scholes presented the first completely satisfactory equilibrium option pricing model. In the same year, Robert Merton extended their model in several important ways. These path-breaking articles have formed the basis for many subsequent academic studies.As these studies have shown, 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 assets of the firm.1 Indeed, the theory applies to a very general class of economic problems — the valuation of contracts where the outcome to each party depends on a quantifiable uncertain future event.Unfortunately, the mathematical tools employed in the Black-Scholes and Merton articles are quite advanced and have tended to obscure the underlying economics. However, thanks to a suggestion by William Sharpe, it is possible to derive the same results using only elementary mathematics.2In this article we will present a simple discrete-time option pricing formula. The fundamental economic principles of option valuation by arbitrage methods are particularly clear in this setting. Sections 2 and 3 illustrate and develop this model for a call option on a stock that pays no dividends. Section 4 shows exactly how the model can be used to lock in pure arbitrage profits if the market price of an option differs from the value given by the model. In section 5, we will show that our approach includes the Black-Scholes model as a special limiting case. By taking the limits in a different way, we will also obtain the Cox-Ross (1975) jump process model as another special case.1 To take an elementary case, consider a firm with a single liability of a homogeneous class of pure discount bonds. The stockholders then have a “call” on the assets of the firm which they can choose to exercise at the maturity date of the debt by paying its principal to the bondholders. In turn, the bonds can be interpreted as a portfolio containing a default-free loan with the same face value as the bonds and a short position in a put on the assets of the firm.2Sharpe (1978) has partially developed this approach to option pricing in his excellent new book, Investments. Rendleman and Bartter (1978) have recently independently discovered a similar formulation of the option pricing problem.Other more general option pricing problems often seem immune to reduction to a simple formula. Instead, numerical procedures must be employed to value these more complex options. Michael Brennan and Eduardo Schwartz (1977) have provided many interesting results along these lines. However, their techniques are rather complicated and are not directly related to the economic structure of the problem. Our formulation, by its very construction, leads to an alternative numerical procedure that is both simpler, and for many purposes, computationally more efficient.Section 6 introduces these numerical procedures and extends the model to include puts and calls on stocks that pay dividends. Section 7 concludes the paper by showing how the model can be generalized in other important ways and discussing its essential role in valuation by arbitrage methods.2. The Basic IdeaSuppose the current price of a stock is S = $50, and at the end of a period of time, its price must be either S* = $25 or S* = $100. A call on the stock is available with a strike price of K = $50, expiring at the end of the period.3 It is also possible to borrow and lend at a 25% rate of interest. The one piece of information left unfurnished is the current value of the call, C. However, if riskless profitable arbitrage is not possible, we can deduce from the given information alone what the value of the call must be!Consider the following levered hedge:(1) write 3 calls at C each,(2) buy 2 shares at $50 each, and(3) borrow $40 at 25%, to be paid back atthe end of the period.Table 1 gives the return from this hedge for each possible level of the stock price at expiration. Regardless of the outcome, the hedge exactly breaks even on the expiration date. Therefore, to prevent profitable riskless arbitrage, its current cost must be zero; that is,3C – 100 + 40 = 0The current value of the call must then be C = $20.3 To keep matters simple, assume for now that the stock will pay no cash dividends during the life of the call. We also ignore transaction costs, margin requirements and taxes.Table 1Arbitrage Table Illustrating the Formation of a Riskless Hedgeexpiration datepresent datewrite 3 calls 3C— –150buy 2 shares –100 50 200borrow 40 –50 –50total — —If the call were not priced at $20, a sure profit would be possible. In particular, if C = $25, the above hedge would yield a current cash inflow of $15 and would experience no further gain or loss in the future. On the other hand, if C = $15, then the same thing could be accomplished by buying 3 calls, selling short 2 shares, and lending $40.Table 1 can be interpreted as demonstrating that an appropriately levered position in stock will replicate the future returns of a call. That is, if we buy shares and borrow against them in the right proportion, we can, in effect, duplicate a pure position in calls. In view of this, it should seem less surprising that all we needed to determine the exact value of the call was its strike price, underlying stock price, range of movement in the underlying stock price, and the rate of interest. What may seem more incredible is what we do not need to know: among other things, we do not need to know the probability that the stock price will rise or fall. Bulls and bears must agree on the value of the call, relative to its underlying stock price!This example is very simple, but it shows several essential features of option pricing. And we will soon see that it is not as unrealistic as it seems.3. The Binomial Option Pricing FormulaIn this section, we will develop the framework illustrated in the example into a complete valuation method. We begin by assuming that the stock price follows a multiplicative binomial process over discrete periods. The rate of return on the stock over each period can have two possible values: u – 1 with probability q, or d – 1 with probability 1 – q. Thus, if the current stock price is S, the stock price at the end of the period will be either uS or dS. We can represent this movement with the following diagram:uS with probability qSdS with probability 1 – qWe also assume that the interest rate is constant. Individuals may borrow or lend as much as they wish at this rate. To focus on the basic issues, we will continue to assume that there are notaxes, transaction costs, or margin requirements. Hence, individuals are allowed to sell short any security and receive full use of the proceeds.4Letting r denote one plus the riskless interest rate over one period, we require u > r > d . If these inequalities did not hold, there would be profitable riskless arbitrage opportunities involving only the stock and riskless borrowing and lending.5To see how to value a call on this stock, we start with the simplest situation: the expiration date is just one period away. Let C be the current value of the call, C u be its value at the end of the period if the stock price goes to uS and C d be its value at the end of the period if the stock price goes to dS . Since there is now only one period remaining in the life of the call, we know that the terms of its contract and a rational exercise policy imply that C u = max[0, uS – K ] and C d = max[0, dS – K ]. Therefore,Cu = max[0, uS – K ] with probability qCC d = max[0, dS – K ] with probability 1 – qSuppose we form a portfolio containing Δ shares of stock and the dollar amount B in riskless bonds.6 This will cost ΔS + B . At the end of the period, the value of this portfolio will beΔuS + rB with probability qΔS + BΔdS + rB with probability 1 – qSince we can select Δ and B in any way we wish, suppose we choose them to equate the end-of-period values of the portfolio and the call for each possible outcome. This requires thatΔuS + rB = C uΔdS + rB = C dSolving these equations, we findrd u dC uC B S d u C C u d d u )(,)(!!=!!=" (1)4 Of course, restitution is required for payouts made to securities held short.5 We will ignore the uninteresting special case where q is zero or one and u = d = r .6 Buying bonds is the same as lending; selling them is the same as borrowing.With Δ and B chosen in this way, we will call this the hedging portfolio.If there are to be no riskless arbitrage opportunities, the current value of the call, C , cannot be less than the current value of the hedging portfolio, ΔS + B . If it were, we could make a riskless profit with no net investment by buying the call and selling the portfolio. It is tempting to say that it also cannot be worth more, since then we would have a riskless arbitrage opportunity by reversing our procedure and selling the call and buying the portfolio. But this overlooks the fact that the person who bought the call we sold has the right to exercise it immediately.Suppose that ΔS + B < S – K . If we try to make an arbitrage profit by selling calls for more than ΔS + B , but less than S – K , then we will soon find that we are the source of arbitrage profits rather than the recipient. Anyone could make an arbitrage profit by buying our calls and exercising them immediately.We might hope that we will be spared this embarrassment because everyone will somehow find it advantageous to hold the calls for one more period as an investment rather than take a quick profit by exercising them immediately. But each person will reason in the following way. If I do not exercise now, I will receive the same payoff as a portfolio with ΔS in stock and B in bonds. If I do exercise now, I can take the proceeds, S – K , buy this same portfolio and some extra bonds as well, and have a higher payoff in every possible circumstance. Consequently, no one would be willing to hold the calls for one more period.Summing up all of this, we conclude that if there are to be no riskless arbitrage opportunities, it must be true thatB SC +!=r d u dC uC d u C C u d d u )(!!+!!=r C d u r u C d u d r d u /!"#$%&'()*+,--+'()*+,--= (2)if this value is greater than S – K , and if not, C = S – K .7Equation (2) can be simplified by definingd u d r p !!" and du r u p !!"!1 so that we can writeC = [pC u + (1 – p )C d ]/r (3)It is easy to see that in the present case, with no dividends, this will always be greater than S – K as long as the interest rate is positive. To avoid spending time on the unimportant situations where the interest rate is less than or equal to zero, we will now assume that r is always greater 7 In some applications of the theory to other areas, it is useful to consider options that can be exercised only on the expiration date. These are usually termed European options. Those that can be exercised at any earlier time as well, such as we have been examining here, are then referred to as American options. Our discussion could be easily modified to include European calls. Since immediate exercise is then precluded, their values would always be given by (2), even if this is less than S – K .than one. Hence, (3) is the exact formula for the value of a call one period prior to the expiration in terms of S, K, u, d, and r.To confirm this, note that if uS ≤K, then S < K and C = 0, so C > S – K. Also, if dS ≥K, then C = S – (K/r) > S – K. The remaining possibility is uS > K > dS. In this case, C = p(uS – K)/r. This is greater than S – K if (1 – p)dS > (p – r)K, which is certainly true as long as r > 1.This formula has a number of notable features. First, the probability q does not appear in the formula. This means, surprisingly, that even if different investors have different subjective probabilities about an upward or downward movement in the stock, they could still agree on the relationship of C to S, u, d, and r.Second, the value of the call does not depend on investors’ attitudes toward risk. In constructing the formula, the only assumption we made about an individual’s behavior was that he prefers more wealth to less wealth and therefore has an incentive to take advantage of profitable riskless arbitrage opportunities. We would obtain the same formula whether investors are risk-averse or risk-preferring.Third, the only random variable on which the call value depends is the stock price itself. In particular, it does not depend on the random prices of other securities or portfolios, such as the market portfolio containing all securities in the economy. If another pricing formula involving other variables was submitted as giving equilibrium market prices, we could immediately show that it was incorrect by using our formula to make riskless arbitrage profits while trading at those prices.It is easier to understand these features if it is remembered that the formula is only a relative pricing relationship giving C in terms of S, u, d, and r. Investors’ attitudes toward risk and the characteristics of other assets may indeed influence call values indirectly, through their effect on these variables, but they will not be separate determinants of call value.Finally, observe that p ≡ (r – d)/(u – d) is always greater than zero and less than one, so it has the properties of a probability. In fact, p is the value q would have in equilibrium if investors were risk-neutral. To see this, note that the expected rate of return on the stock would then be the riskless interest rate, soq(uS) + (1 – q)(dS) = rSandq = (r – d)/(u – d) = pHence, the value of the call can be interpreted as the expectation of its discounted future value in a risk-neutral world. In light of our earlier observations, this is not surprising. Since the formula does not involve q or any measure of attitudes toward risk, then it must be the same for any set of preferences, including risk neutrality.It is important to note that this does not imply that the equilibrium expected rate of return on the call is the riskless interest rate. Indeed, our argument has shown that, in equilibrium, holding the call over the period is exactly equivalent to holding the hedging portfolio. Consequently, the riskand expected rate of return of the call must be the same as that of the hedging portfolio. It can be shown that Δ≥ 0 and B ≤ 0, so the hedging portfolio is equivalent to a particular levered long position in the stock. In equilibrium, the same is true for the call. Of course, if the call is currently mispriced, its risk and expected return over the period will differ from that of the hedging portfolio.Now we can consider the next simplest situation: a call with two periods remaining before its expiration date. In keeping with the binomial process, the stock can take on three possible values after two periods,u2SuSS duSdSd2SSimilarly, for the call,C uu = max[0, u2S – K]C uC C du = max[0, duS – K]C dC dd = max[0, d2S – K]C uu stands for the value of a call two periods from the current time if the stock price moves upward each period; C du and C dd have analogous definitions.At the end of the current period there will be one period left in the life of the call, and we will be faced with a problem identical to the one we just solved. Thus, from our previous analysis, we know that when there are two periods left,C u = [pC uu + (1 – p)C ud]/rand (4)C d = [pC du + (1 – p)C dd]/rAgain, we can select a portfolio with ΔS in stock and B in bonds whose end-of-period value will be C u if the stock price goes to uS and C d if the stock price goes to dS. Indeed, thefunctional form of Δ and B remains unchanged. To get the new values of Δ and B, we simply use equation (1) with the new values of C u and C d.Can we now say, as before, that an opportunity for profitable riskless arbitrage will be available if the current price of the call is not equal to the new value of this portfolio or S – K, whichever is greater? Yes, but there is an important difference. With one period to go, we could plan to lock in a riskless profit by selling an overpriced call and using part of the proceeds to buy the hedging portfolio. At the end of the period, we knew that the market price of the call must be equal to the value of the portfolio, so the entire position could be safely liquidated at that point. But this was true only because the end of the period was the expiration date. Now we have no such guarantee. At the end of the current period, when there is still one period left, the market price of the call could still be in disequilibrium and be greater than the value of the hedging portfolio. If we closed out the position then, selling the portfolio and repurchasing the call, we could suffer a loss that would more than offset our original profit. However, we could always avoid this loss by maintaining the portfolio for one more period. The value of the portfolio at the end of the current period will always be exactly sufficient to purchase the portfolio we would want to hold over the last period. In effect, we would have to readjust the proportions in the hedging portfolio, but we would not have to put up any more money.Consequently, we conclude that even with two periods to go, there is a strategy we could follow which would guarantee riskless profits with no net investment if the current market price of a call differs from the maximum of ΔS + B and S – K. Hence, the larger of these is the current value of the call.Since Δ and B have the same functional form in each period, the current value of the call in terms of C u and C d will again be C = [pC u + (1 – p)C d]/r if this is greater than S – K, and C = S – K otherwise. By substituting from equation (4) into the former expression, and noting that C du = C ud, we obtainC = [p2C uu + 2p(1 – p)C ud + (1 – p)2C dd]/r2(5)= [p2max[0, u2S – K] + 2p(1 – p)max[0, duS – K] + (1 – p)2max[0, d2S – K]]/r2A little algebra shows that this is always greater than S – K if, as assumed, r is always greater than one, so this expression gives the exact value of the call.8All of the observations made about formula (3) also apply to formula (5), except that the number of periods remaining until expiration, n, now emerges clearly as an additional determinant of the call value. For formula (5), n = 2. That is, the full list of variables determining C is S, K, n, u, d, and r.8 In the current situation, with no dividends, we can show by a simple direct argument that if there are no arbitrage opportunities, then the call value must always be greater than S – K before the expiration date. Suppose that the call is selling for S –K. Then there would be an easy arbitrage strategy that would require no initial investment and would always have a positive return. All we would have to do is buy the call, short the stock, and invest K dollars in bonds. See Merton (1973). In the general case, with dividends, such an argument is no longer valid, and we must use the procedure of checking every period.We now have a recursive procedure for finding the value of a call with any number of periods to go. By starting at the expiration date and working backwards, we can write down the general valuation formula for any n :n j n j j n j n j r K S d u p p j n j n C /],0max[)1()!(!!0!!"#$$%&''(()*++,-'=''=. (6)This gives us the complete formula, but with a little additional effort we can express it in a more convenient way.Let a stand for the minimum number of upward moves that the stock must make over the next n periods for the call to finish in-the-money. Thus a will be the smallest non-negative integer such that u a d n-a S > K . By taking the natural logarithm of both sides of this inequality, we could write a as the smallest non-negative integer greater than log(K /Sd n )/log(u /d ).For all j < a ,max[0, u j d n-j S – K ] = 0and for all j ≥ a ,max[0, u j d n-j S – K ] = u j d n-j S – K Therefore,n j n j j n j n a j r K S d u p p j n j n C /][)1()!(!!!"#$%&''(()*++,-'=''=.Of course, if a > n , the call will finish out-of-the-money even if the stock moves upward every period, so its current value must be zero.By breaking up C into two terms, we can write!!"#$$%&''()**+,-''()**+,-=--=.n j n j j n j n a j r d u p p j n j n S C )1()!(!!!"#$%&'(()*++,-'''='.j n j n a j n p p j n j n Kr )1()!(!!Now, the latter bracketed expression is the complementary binomial distribution function φ[a ; n , p ]. The first bracketed expression can also be interpreted as a complementary binomial distribution function φ[a ; n , p ′], wherep ′ ≡ (u /r )p and 1 – p ′ ≡ (d /r )(1 – p )p ′ is a probability, since 0 < p ′ < 1. To see this, note that p < (r /u ) andj n j j n j n j n j j n j p p p r d p r u r d u p p !!!!"!"=#$%&'(!=#$%&'(=))*+,,-.!)1()1()1(In summary:Binomial Option Pricing FormulaC = Sφ[a; n, p′] – Kr–nφ[a; n, p]wherep ≡ (r – d)/(u – d) and p′≡ (u/r)pa ≡ the smallest non-negative integergreater than log(K/Sd n)/log(u/d)If a > n, then C = 0.It is now clear that all of the comments we made about the one period valuation formula are valid for any number of periods. In particular, the value of a call should be the expectation, in a risk-neutral world, of the discounted value of the payoff it will receive. In fact, that is exactly what equation (6) says. Why, then, should we waste time with the recursive procedure when we can write down the answer in one direct step? The reason is that while this one-step approach is always technically correct, it is really useful only if we know in advance the circumstances in which a rational individual would prefer to exercise the call before the expiration date. If we do not know this, we have no way to compute the required expectation. In the present example, a call on a stock paying no dividends, it happens that we can determine this information from other sources: the call should never be exercised before the expiration date. As we will see in section 6, with puts or with calls on stocks that pay dividends, we will not be so lucky. Finding the optimal exercise strategy will be an integral part of the valuation problem. The full recursive procedure will then be necessary.For some readers, an alternative “complete markets” interpretation of our binomial approach may be instructive. Suppose that πu and πd represent the state-contingent discount rates to states u and d, respectively. Therefore, πu would be the current price of one dollar received at the end of the period, if and only if state u occurs. Each security — a riskless bond, the stock, and the option — must all have returns discounted to the present by πu and πd if no riskless arbitrage opportunities are available. Therefore,1 = πu r + πd rS = πu(uS) + πd(dS)C = πu C u + πd C dThe first two equations, for the bond and the stock, implyr d u d r u 1!"#$%&''=( and rd u r u d 1!"#$%&''=(Substituting these equalities for the state-contingent prices in the last equation for the option yields equation (3).It is important to realize that we are not assuming that the riskless bond and the stock and the option are the only three securities in the economy, or that other securities must follow a binomial process. Rather, however these securities are priced in relation to others in equilibrium, among themselves they must conform to the above relationships.From either the hedging or complete markets approaches, it should be clear that three-state or trinomial stock price movements will not lead to an option pricing formula based solely on arbitrage considerations. Suppose, for example, that over each period the stock price could move to uS or dS or remain the same at S . A choice of Δ and B that would equate the returns in two states could not in the third. That is, a riskless arbitrage position could not be taken. Under the complete markets interpretation, with three equations in now three unknown state-contingent prices, we would lack the redundant equation necessary to price one security in terms of the other two.4. Riskless Trading StrategiesThe following numerical example illustrates how we could use the formula if the current market price M ever diverged from its formula value C . If M > C , we would hedge, and if M < C , “reverse hedge”, to try and lock in a profit. Suppose the values of the underlying variables areS = 80, n = 3, K = 80, u = 1.5, d = 0.5, r = 1.1In this case, p = (r – d )/(u – d ) = 0.6. The relevant values of the discount factor arer -1 = 0.909, r -2 = 0.826, r -3 = 0.751The paths the stock price may follow and their corresponding probabilities (using probability p ) are, when n = 3, with S – 80,。

Buffett’s Letters To Berkshire Shareholders 1979巴菲特致股东的信 1979年Again, we must lead off with a few words about accounting. Since our last annual report, the accounting profession has decided that equity securities owned by insurance companies must be carried on the balance sheet at market value. We previously have carried such equity securities at the lower of aggregate cost or aggregate market value. Because we have large unrealized gains in our insurance equity holdings, the result of this new policy is to increase substantially both the 1978 and 1979 yearend net worth, even after the appropriate liability is established for taxes on capital gains that would be payable should equities be sold at such market valuations. 首先，还是会计相关的议题，从去年年报开始，会计原则要求保险公司持有的股票投资在资产负债表日的评价方式，从原先的成本与市价孰低法，改按公平市价法列示，由于我们帐上的股票投资拥有大量的未实现利益，因此即便我们已提列了资本利得实现时应该支付的估计所得税负债，我们1978年及1979年的净值依然大幅增加。

美国是世界上公司法、证券法研究最为发达的国家之一，在美国法学期刊（Law Review & Journals）上每年发表400多篇以公司法和证券法为主题的论文。

自1994年开始，美国的公司法学者每年会投票从中遴选出10篇左右重要的论文，重印于Corporate Practice Commentator，至2008年，已经评选了15年，计177篇论文入选。

以下是每年入选的论文列表：2008年（以第一作者姓名音序为序）：1.Anabtawi, Iman and Lynn Stout. Fiduciary duties for activist shareholders. 60 Stan. L. Rev. 1255-1308 (2008).2.Brummer, Chris. Corporate law preemption in an age of global capital markets. 81 S. Cal. L. Rev. 1067-1114 (2008).3.Choi, Stephen and Marcel Kahan. The market penalty for mutual fund scandals. 87 B.U. L. Rev. 1021-1057 (2007).4.Choi, Stephen J. and Jill E. Fisch. On beyond CalPERS: Survey evidence on the developing role of public pension funds in corporate governance. 61 V and. L. Rev. 315-354 (2008).5.Cox, James D., Randall S. Thoma s and Lynn Bai. There are plaintiffs and…there are plaintiffs: An empirical analysis of securities class action settlements. 61 V and. L. Rev. 355-386 (2008).6.Henderson, M. Todd. Paying CEOs in bankruptcy: Executive compensation when agency costs are low. 101 Nw. U. L. Rev. 1543-1618 (2007).7.Hu, Henry T.C. and Bernard Black. Equity and debt decoupling and empty voting II: Importance and extensions. 156 U. Pa. L. Rev. 625-739 (2008).8.Kahan, Marcel and Edward Rock. The hanging chads of corporate voting. 96 Geo. L.J. 1227-1281 (2008).9.Strine, Leo E., Jr. Toward common sense and common ground? Reflections on the shared interests of managers and labor in a more rational system of corporate governance. 33 J. Corp. L. 1-20 (2007).10.Subramanian, Guhan. Go-shops vs. no-shops in private equity deals: Evidence and implications.63 Bus. Law. 729-760 (2008).2007年：1.Baker, Tom and Sean J. Griffith. The Missing Monitor in Corporate Governance: The Directors’ & Officers’ Liability Insurer. 95 Geo. L.J. 1795-1842 (2007).2.Bebchuk, Lucian A. The Myth of the Shareholder Franchise. 93 V a. L. Rev. 675-732 (2007).3.Choi, Stephen J. and Robert B. Thompson. Securities Litigation and Its Lawyers: Changes During the First Decade After the PSLRA. 106 Colum. L. Rev. 1489-1533 (2006).4.Coffee, John C., Jr. Reforming the Securities Class Action: An Essay on Deterrence and Its Implementation. 106 Colum. L. Rev. 1534-1586 (2006).5.Cox, James D. and Randall S. Thomas. Does the Plaintiff Matter? An Empirical Analysis of Lead Plaintiffs in Securities Class Actions. 106 Colum. L. Rev. 1587-1640 (2006).6.Eisenberg, Theodore and Geoffrey Miller. Ex Ante Choice of Law and Forum: An Empirical Analysis of Corporate Merger Agreements. 59 V and. L. Rev. 1975-2013 (2006).7.Gordon, Jeffrey N. The Rise of Independent Directors in the United States, 1950-2005: Of Shareholder V alue and Stock Market Prices. 59 Stan. L. Rev. 1465-1568 (2007).8.Kahan, Marcel and Edward B. Rock. Hedge Funds in Corporate Governance and Corporate Control. 155 U. Pa. L. Rev. 1021-1093 (2007).ngevoort, Donald C. The Social Construction of Sarbanes-Oxley. 105 Mich. L. Rev. 1817-1855 (2007).10.Roe, Mark J. Legal Origins, Politics, and Modern Stock Markets. 120 Harv. L. Rev. 460-527 (2006).11.Subramanian, Guhan. Post-Siliconix Freeze-outs: Theory and Evidence. 36 J. Legal Stud. 1-26 (2007). (NOTE: This is an earlier working draft. The published article is not freely available, and at SLW we generally respect the intellectual property rights of others.)2006年：1.Bainbridge, Stephen M. Director Primacy and Shareholder Disempowerment. 119 Harv. L. Rev. 1735-1758 (2006).2.Bebchuk, Lucian A. Letting Shareholders Set the Rules. 119 Harv. L. Rev. 1784-1813 (2006).3.Black, Bernard, Brian Cheffins and Michael Klausner. Outside Director Liability. 58 Stan. L. Rev. 1055-1159 (2006).4.Choi, Stephen J., Jill E. Fisch and A.C. Pritchard. Do Institutions Matter? The Impact of the Lead Plaintiff Provision of the Private Securities Litigation Reform Act. 835.Cox, James D. and Randall S. Thomas. Letting Billions Slip Through Y our Fingers: Empirical Evidence and Legal Implications of the Failure of Financial Institutions to Participate in Securities Class Action Settlements. 58 Stan. L. Rev. 411-454 (2005).6.Gilson, Ronald J. Controlling Shareholders and Corporate Governance: Complicating the Comparative Taxonomy. 119 Harv. L. Rev. 1641-1679 (2006).7.Goshen , Zohar and Gideon Parchomovsky. The Essential Role of Securities Regulation. 55 Duke L.J. 711-782 (2006).8.Hansmann, Henry, Reinier Kraakman and Richard Squire. Law and the Rise of the Firm. 119 Harv. L. Rev. 1333-1403 (2006).9.Hu, Henry T. C. and Bernard Black. Empty V oting and Hidden (Morphable) Ownership: Taxonomy, Implications, and Reforms. 61 Bus. Law. 1011-1070 (2006).10.Kahan, Marcel. The Demand for Corporate Law: Statutory Flexibility, Judicial Quality, or Takeover Protection? 22 J. L. Econ. & Org. 340-365 (2006).11.Kahan, Marcel and Edward Rock. Symbiotic Federalism and the Structure of Corporate Law.58 V and. L. Rev. 1573-1622 (2005).12.Smith, D. Gordon. The Exit Structure of V enture Capital. 53 UCLA L. Rev. 315-356 (2005).2005年：1.Bebchuk, Lucian Arye. The case for increasing shareholder power. 118 Harv. L. Rev. 833-914 (2005).2.Bratton, William W. The new dividend puzzle. 93 Geo. L.J. 845-895 (2005).3.Elhauge, Einer. Sacrificing corporate profits in the public interest. 80 N.Y.U. L. Rev. 733-869 (2005).4.Johnson, . Corporate officers and the business judgment rule. 60 Bus. Law. 439-469 (2005).haupt, Curtis J. In the shadow of Delaware? The rise of hostile takeovers in Japan. 105 Colum. L. Rev. 2171-2216 (2005).6.Ribstein, Larry E. Are partners fiduciaries? 2005 U. Ill. L. Rev. 209-251.7.Roe, Mark J. Delaware?s politics. 118 Harv. L. Rev. 2491-2543 (2005).8.Romano, Roberta. The Sarbanes-Oxley Act and the making of quack corporate governance. 114 Y ale L.J. 1521-1611 (2005).9.Subramanian, Guhan. Fixing freezeouts. 115 Y ale L.J. 2-70 (2005).10.Thompson, Robert B. and Randall S. Thomas. The public and private faces of derivative lawsuits. 57 V and. L. Rev. 1747-1793 (2004).11.Weiss, Elliott J. and J. White. 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Derivatives, Corporate Hedging, and Shareholder Wealth: Modigliani-Miller Forty Y ears Later. 1998 U. Ill. L. Rev. 1039-1104.ngevoort, Donald C. Rereading Cady, Roberts: The Ideology and Practice of Insider Trading Regulation. 99 Colum. L. Rev. 1319-1343 (1999).ngevoort, Donald C. Half-Truths: Protecting Mistaken Inferences By Investors and Others.52 Stan. L. Rev. 87-125 (1999).11.Talley, Eric. Turning Servile Opportunities to Gold: A Strategic Analysis of the Corporate Opportunities Doctrine. 108 Y ale L.J. 277-375 (1998).12.Williams, Cynthia A. The Securities and Exchange Commission and Corporate Social Transparency. 112 Harv. L. Rev. 1197-1311 (1999).1998年：1.Carney, William J., The Production of Corporate Law, 71 S. Cal. L. Rev. 715-780 (1998).2.Choi, Stephen, Market Lessons for Gatekeepers, 92 Nw. U. L. Rev. 916-966 (1998).3.Coffee, John C., Jr., Brave New World?: The Impact(s) of the Internet on Modern Securities Regulation. 52 Bus. 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Rev. 1018-1094 (1998).11.Skeel, David A., Jr., An Evolutionary Theory of Corporate Law and Corporate Bankruptcy. 51 V and. L. Rev. 1325-1398 (1998).12.Thomas, Randall S. and Martin, Kenneth J., Should Labor Be Allowed to Make Shareholder Proposals? 73 Wash. L. Rev. 41-80 (1998).1997年：1.Alexander, Janet Cooper, Rethinking Damages in Securities Class Actions, 48 Stan. L. Rev. 1487-1537 (1996).2.Arlen, Jennifer and Kraakman, Reinier, Controlling Corporate Misconduct: An Analysis of Corporate Liability Regimes, 72 N.Y.U. L. Rev. 687-779 (1997).3.Brudney, Victor, Contract and Fiduciary Duty in Corporate Law, 38 B.C. L. Rev. 595-665 (1997).4.Carney, William J., The Political Economy of Competition for Corporate Charters, 26 J. Legal Stud. 303-329 (1997).5.Choi, Stephen J., Company Registration: Toward a Status-Based Antifraud Regime, 64 U. Chi. L. Rev. 567-651 (1997).6.Fox, Merritt B., Securities Disclosure in a Globalizing Market: Who Should Regulate Whom. 95 Mich. L. Rev. 2498-2632 (1997).7.Kahan, Marcel and Klausner, Michael, Lockups and the Market for Corporate Control, 48 Stan. L. Rev. 1539-1571 (1996).8.Mahoney, Paul G., The Exchange as Regulator, 83 V a. L. Rev. 1453-1500 (1997).haupt, Curtis J., The Market for Innovation in the United States and Japan: V enture Capital and the Comparative Corporate Governance Debate, 91 Nw. U.L. Rev. 865-898 (1997).10.Skeel, David A., Jr., The Unanimity Norm in Delaware Corporate Law, 83 V a. L. Rev. 127-175 (1997).1996年：1.Black, Bernard and Reinier Kraakman A Self-Enforcing Model of Corporate Law, 109 Harv. L. Rev. 1911 (1996)2.Gilson, Ronald J. Corporate Governance and Economic Efficiency: When Do Institutions Matter?, 74 Wash. U. L.Q. 327 (1996)3. Hu, Henry T.C. Hedging Expectations: "Derivative Reality" and the Law and Finance of the Corporate Objective, 21 J. Corp. 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Static Hedging of Standard Options∗Peter Carr,†Liuren Wu‡First draft:July26,2002;this version:June18,2009I.IntroductionOver the past two decades,the derivatives market has expanded dramatically.Accompanying this ex-pansion is an increased urgency in understanding and managing the risks associated with derivative securities.In an ideal setting under which the price of the underlying security moves continuously (such as in a diffusion with known instantaneous volatility)or withfixed and known size steps(such as in a binomial tree),derivatives pricing theory provides a framework in which the risks inherent in a derivatives position can be eliminated via frequent trading in only a small number of securities.In reality,however,large and random price movements happen much more often than typically assumed in the above ideal settings.The last two decades have repeatedly witnessed turmoil in thefinancial markets such as the1987stock market crash,the1997Asian crisis,the1998Russian default and the ensuing hedge fund crisis,and the tragic event of September11,2001.Juxtaposed between these large crises are many more mini-crises,during which prices move sufficiently fast so as to trigger circuit breakers and trading halts.When these crises occur,a dynamic hedging strategy based on small orfixed size movements often breaks down.Worse yet,strategies that involve dynamic hedging in the underlying asset tend to fail precisely when liquidity dries up or when the market experiences large moves.Unfortunately,it is during thesefinancial crises such as liquidity gaps or market crashes that investors need effective hedging the most dearly.Indeed, several prominent critics have gone further and blamed the emergence of somefinancial crises on the pursuit of dynamic hedging strategies.Perhaps in response to the known dynamic hedging,Breeden and Litzenberger (1978)(henceforth BL)pioneered an alternative approach,which is foreshadowed in the work of Ross(1976)and elaborated on by Green and Jarrow(1987)and Nachman(1988).These authors show that a path-independent payoff can be hedged using a portfolio of standard options maturing with the claim.This strategy is completely robust to model mis-specification and is effective even in the presence of jumps of random size.Its only real drawback is that the class of claims that this strategy can hedge is fairly narrow.First,the BL hedge of a standard option reduces to a tautology. Second,the hedge can neither deal with standard options of different maturities,nor can it deal with path-dependent options.Therefore,the BL strategy is completely robust but has limited range.Incontrast,dynamic hedging works for a wide range of claims,but is not robust.In this paper,we propose a new approach for hedging derivative securities.This approach lies between dynamic hedging and the BL static hedge in terms of both range and robustness.Relative to BL,we the class of allowed stochastic processes of the underlying asset inorder to that can be robustly hedged.In particular,we work in a one-factorthe market price of a security is allowed not only to move diffusively,butto any non-negative value.In this setting,we derive a simple spanning relation between the value of a given European option and the value of a continuum of shorter-term European options.The required position in each of the shorter-term options is proportional to the gamma(second price derivative)that the target option will have at the expiry of the short-term option if the security price at that time is at the strike of this short-term option.As this future gamma does not vary with the passage of time or the change in the underlying price,the weights in the portfolio of shorter-term options are static over the life of these options.Given this static spanning result,no arbitrage implies that the target option and the replicating portfolio have the same value for all times until the shorter term options expire.As a result,we can effectively hedge a long-term option,at least in theory,even in the presence of large random jumps in the asset price movement. Furthermore,given the static nature of the strategy,we do not need to rebalance the hedge portfolio until the shorter-term options mature.Therefore,we do not need to worry about market shutdowns and liquidity gaps in the intervening period.The strategy remains viable and can become even more useful when the market is in stress.As an added advantage,the static hedge only requires a correct specification of the underlying price dynamics between the two option maturities.In contrast,delta hedging only succeeds if the model is correctly specified throughout the life of the target option.As transaction costs and illiquidity render the formation of a portfolio with a continuum of op-tions physically impossible,we develop an approximation for the static hedging strategy using only afinite number of options.This discretization of the ideal trading strategy is analogous to the dis-cretization of a continuous-time dynamic trading strategy(e.g.,delta hedging).To discretize our static hedge,we choose the strike levels and the associated portfolio weights based on a Gauss-Hermite quadrature method.We use Monte Carlo simulation to gauge the magnitude and distribu-tional characteristics of the hedging error introduced by the quadrature approximation.We comparethis hedging error to the hedging error from a delta-hedging strategy based on daily rebalancing with the underlying futures.The simulation results indicate that the two strategies have compara-ble hedging effectiveness in the classic Black and Scholes(1973)environment.The mean absolute hedging errors are comparable when the two strategies involve the same number of transactions. Nevertheless,since typically lower for the underlying asset than it is for the op-tions,these results strategy.The conclusions are similar under other purely continuous asset price dynamics such as the stochastic volatility model of Heston(1993).The conclusion changes when we perform the simulation under the Merton(1976) environment,in which the underlying asset price can exhibit jumps of random size.In the presence of random jumps,the performance of daily delta hedging deteriorates dramatically,but the perfor-mance of the static strategy hardly varies.As a result,under the Merton model,a static strategy with merely three options outperforms delta hedging with daily updating.Further simulations indicate that these results are robust to model misspecification,so long as we perform ad hoc adjustments based on the observed implied volatility.We alsofind that increasing the rebalancing frequency in the delta-hedging strategy does not rescue its performance as long as the underlying asset price can jump by a random amount.In contrast,we can further improve the static-hedging performance by increasing the number of strikes used and by choosing the appropriate maturities for the hedge portfolio.We conclude that the superior performance of static hedging over daily delta hedging in the jump model simulations is not due to model misspecification,nor is it due to the approximation error introduced via discrete rebalancing.Rather,this outperformance is due to the fact that delta hedging is inherently incapable of dealing with jumps of random size in the underlying asset price movement.In contrast,our static spanning relation can handle random jumps and our approximation of this spanning relation performs equally well with and without jumps in the underlying asset price process.To compare the effectiveness of the two types of hedging strategies in practice,we also investi-gate the historical performance of the two strategies in hedging S&P500index options.Based on over six years of data on S&P500index options,wefind that a static hedge using no more thanfive options outperforms daily delta hedging with the underlying futures.The consistency of this result with our jump model simulations lends empirical support for the existence of jumps of random sizein the movement of the S&P500index.We alsofind that our static strategy performs better when the maturity of the options in the hedge portfolio is closer to the maturity of the target option being hedged.As the maturity gap increases, the hedging performance deteriorates moderately,indicating the likely existence of additional ran-dom factors such as stochastic volatility.For clarity of exposition,we focus on hedging a standard European option with a portfolio of shorter-term options;however,the underlying theoretical framework extends readily to the hedging of more exotic,potentially path-dependent options,such as discretely monitored Asian and barrier options,Bermudan options,passport options,cliquets,ratchets,and many other structured notes. We use a globallyfloored,locally capped,compounding cliquet as an example to illustrate how this option contract with intricate path-dependence can be hedged with a portfolio of European options.The hedging strategy is semi-static in the sense that trades only need to occur at the discrete monitoring dates.In related literature,the effective hedging of derivative securities has been applied not only for risk management,but also for option valuation and model verification(Bates(2003)).For example, Bakshi,Cao,and Chen(1997),Bakshi and Kapadia(2003),and Dumas,Fleming,and Whaley (1998)use hedging performance to test different option pricing models.Bakshi and Madan(2000) propose a general option-valuation strategy based on effective spanning using basis characteristic securities.This remainder of the paper is organized as follows.Section II develops the theoretical results underlying our static hedging strategy on a European option.Section III uses Monte Carlo simulation to enact a wide variety of scenarios under which the market not only moves diffusively,but also jumps randomly.Under each scenario,we analyze the hedging performance of our static strategy and compare it with dynamic hedging.Section IV applies both strategies to the S&P500index options data.Section V shows how to extend our theory to the hedging of path-dependent options. Section VI concludes.II.Spanning Options with OptionsWe develop our main theoretical results in this section.Working in a continuous-time one-factor Markovian setting,we show how we can span the risk of a European option by holding a continuum of shorter-term European options.The weights in the portfolio are static as they are invariant to changes in the underlying asset price or the calendar time.We also illustrate how we can use a quadrature rule to approximate the static hedge using afinite number of shorter-term options.A.Assumptions and NotationWe assume frictionless markets and no arbitrage.Tofix notation,we let S t denote the spot price of an asset(say,a stock or stock index)at time t∈[0,T],where T is some arbitrarily distant horizon. For realism,we assume that the owners of this asset enjoy limited liability,and hence S t≥0at alltimes.For notational simplicity,we further assume that the continuously compounded riskfree rate r and dividend yield q are constant.No arbitrage implies that there exists a risk-neutral probability measure Q defined on a probability space(Ω,F,Q)such that the instantaneous expected rate of return on every asset equals the instantaneous riskfree rate r.We also restrict our analysis to the class of models in which the risk-neutral evolution of the stock price is Markov in the stock price S and the calendar time t.Our class of models includes local volatility models,e.g.,Dupire(1994), and models based on L´e vy processes,e.g.,Barndorff-Nielsen(1998),Bates(1991),Carr,Geman, Madan,and Yor(2002),Carr and Wu(2003a),Eberlein,Keller,and Prause(1998),Madan and Seneta(1990),a Merton(1976),and Wu(2006),but does not include stochastic volatility models such as Bates(1996,2000),Bakshi,Cao,and Chen(1997),Carr and Wu(2007),Heston(1993),and Hull and White(1987).We use C t(K,T)to denote the time-t price of a European call with strike K and maturity T.Our assumption that the state is fully described by the stock price and time implies that there exists a call pricing function C(S,t;K,T;Θ)such that(1)C t(K,T)=C(S t,t;K,T;Θ),t∈[0,T],K≥0,T∈[t,T].The call pricing function relates the call price at t to the state variables(S t,t),the contractual param-eters(K,T),and a vector offixed model parametersΘ.We use g(S,t;K,T;Θ)to denote the probability density function of the asset price under the risk-neutral measure Q,evaluated at the future price level K and the future time T and conditional on the stock price starting at level S at some earlier time t.Breeden and Litzenberger(1978)show that this risk-neutral density relates to the second strike derivative of the call pricing function by∂2C(2)g(S,t;K,T;Θ)=e r(T−t)C(K,u;K,T;Θ).∂K2Proof.Under the Markovian assumption in(1),we can compute the initial value of the target call option by discounting the expected value it will have at some future date u ,C (S ,t ;K ,T ;Θ)=e −r (u −t )∞ 0g (S ,t ;K ,u ;Θ)C (K ,u ;K ,T ;Θ)d K =∞0∂2∂KC (S ,t ;K ,u ;Θ) K →∞=0,C (S ,t ;K ,u ;Θ)|K →∞=0,∂A key feature of the spanning relation in (3)is that the weighting function w (K )is independent of S and t .This property characterizes the static nature of the spanning relation.Under no arbitrage,once we form the spanning portfolio,no rebalancing is necessary until the maturity date of the options in the spanning portfolio.The weight w (K )on a call option at maturity u and strike K is proportional to the gamma that the target call option will have at time u ,should the underlying price level be at K then.Since the gamma of a call option typically shows a bell-shaped curve centered near the call option’s strike price,greater weights go to the options with strikes that are closer to that of the target option.Furthermore,as we let the common maturity u of the spanning portfolio approach the target call option’s maturity T ,the gamma becomes more concentrated around K .In the limit when u =T ,all of the weight is on the call option of strike K .Equation (3)reduces to a tautology.The spanning relation in (3)represents a constraint imposed by no-arbitrage and the Markovian assumption on the relation between prices of options at two different maturities.Given that theMarkovian assumption is correct,a violation of equation(3)implies an arbitrage opportunity.For example,if we suppose that at time t,the market price of a call option with strike K and maturity T(left hand side)exceeds the price of a gamma weighted portfolio of call options for some earlier maturity u(right hand side),conditional on the validity of the Markovian assumption(1),the arbi-trage is to sell the call option of strike K and maturity T,and to buy the gamma weighted portfolio of all call options maturing at the earlier date u.The cash received from selling the T maturity call exceeds the cash spent buying the portfolio of nearer dated calls.At time u,the portfolio of expiring calls pays off:∞∂2C.Finite Approximation with Gaussian Quadrature RulesIn practice,investors can neither rebalance a portfolio continuously,nor can they form a static port-folio involving a continuum of securities.Both strategies involve an infinite number of transactions.In the presence of discrete transaction costs,both would lead tofinancial ruin.As a result,dynamic strategies are only rebalanced discretely in practice.The trading times are chosen to balance the costs arising from the hedging error with the cost arising from transacting in the underlying.Simi-larly,to implement our static hedging strategy in practice,we need to approximate it using afinite number of call options.The number of call options used in the portfolio is chosen to balance thecost from the hedging error with the cost from transacting in these options.We approximate the spanning integral in equation(3)by a weighted sum of afinite number(N)of call options at strikes K j,j=1,2,···,N,(7) ∞0w(K)C(S,t;K,u;Θ)d K≈N∑j=1W j C(S,t;K j,u;Θ),where we choose the strike points K j and their corresponding weights based on the Gauss-Hermite quadrature rule.The Gauss-Hermite quadrature rule is designed to approximate an integral of the form ∞−∞f(x)e−x2dx, where f(x)is an arbitrary smooth function.After some rescaling,the integral can be regarded asan expectation of f(x)where x is a normally distributed random variable with zero mean and vari-ance of two.For a given target function f(x),the Gauss-Hermite quadrature rule generates a set of weights w i and nodes x i,i=1,2,···,N,that are defined by(8) ∞−∞f(x)e−x2dx=N∑j=1w j f x j +N!√2N f(2N)(ξ)function between the strikes and the quadrature nodes is given by (9)K(x)=Ke xσ√∂K2=e−q(T−u)n(d1)T−u,where n(·)denotes the probability density of a standard normal and the standardized variable d1is defined asd1≡ln(K/K)+(r−q+σ2/2)(T−u)T−u.We can then obtain the mapping in(9)by replacing d1with√2(T−u)+(q−r−σ2/2)(T−u),with the portfolio weights given by(12)W j=w(K j)K′j(x j)2(T−t)III.Monte Carlo Analysis Based on Popular ModelsConsider the problem faced by the writer of a call option on a certain stock.For concreteness,suppose that the call option matures in one year and is written at-the-money.The writer intends to hold this short position for a month,after which the option position will be closed.During this month,the writer can hedge the risk using various exchange traded liquid assets such as the underlying stock,futures,and/or options on the same stock.We compare the performance of two types of strategies:(i)a static hedging strategy using one-month standard options,and (ii)a dynamic delta hedging strategy using the underlying stock futures.The static strategy is based on the spanning relation in equation (3)and is approximated by a finite number of options,with the portfolio strikes and weights determined by the quadrature method.The dynamic strategy is discretized by rebalancing the futures position daily.The choice of using futures instead of the stock itself for the delta hedge is intended to be consistent with our empirical study in the next section on S&P 500index options.For these options,direct trading in the 500stocks comprising the index is impractical.Practically all delta hedging is done in the very liquid index futures market.Given our assumption of constant interest rates and dividend yields,the simulated performances of the delta hedges based on the stock or its futures are almost identical.Hence,this choice does not affect our results.We compare the performance of the above two strategies based on Monte Carlo simulation.For the simulation,we consider four data generating processes:the benchmark Black-Scholes model (BS),the Merton (1976)jump-diffusion model (MJ),the Heston (1993)stochastic volatility model (HV),and a special case of this model proposed by Heston and Nandi (2000)(HN).Under the objective measure,P ,the stock price dynamics in the two models follows the stochastic differential equations,(13)where W denotes a standard Brownian motion in all three models.Under the MJ model,J(λ)denotes a compound Poisson jump process with constant intensityλ.Conditional on a jump occurring,the MJ model assumes that the log price relative is normally distributed with meanµj and varianceσ2j, with the mean percentage price change induced by a jump given by g=eµj+1calendar time.Therefore,the static spanning relation in (3)nolonger holds.In particular,at time t ,we do not know the variance rate level at time u >t ,v u .Hence,we do not know the gamma of the target call option at time u ,which determines the weighting function of the static hedging portfolio.In our simulation exercise,we replace v u with its time-t risk-neutral expected value to derive the portfolio weight.We investigate the degree to which this violation of the Markovian assumption degenerates the static hedging performance.Finally,neither hedging strategy works perfectly under the Heston with |ρ|=1.The two instruments in the dynamic hedging strategy are not enough to span the two sources of uncertainty under the HV model.The non-Markovian property also invalids the static spanning relation in (3).The presence of stochastic volatility has been documented extensively.Our simulation exercise gauges the degree of performance deterioration for both hedging strategies.We specify the data generating processes in equation (13)under the objective measure P .To price the relevant options and to compute the weights in the hedge portfolios,we also need to specify their respective risk-neutral Q -dynamics,(14)BS:dS t /S t =(r −q )dt +σdW ∗t ,MJ:dS t /S t =(r −q −λ∗g ∗)dt +σdW ∗t +dJ ∗(λ∗),HV:dS t /S t =(r −q )dt +√v t dZ ∗t ,where W ∗denotes a standard Brownian motion under the risk-neutral measure Q ,and (κ∗,θ∗,λ∗,µ∗j ,σ∗j )denote the corresponding parameters under this measure.Option prices under the BS model can be readily computed using the Black-Scholes option pricing formula.Under the MJ model,option prices can be computed as a Poisson-probability weighted sum of the Black-Scholes formulae.Un-der the Heston model and its HN special case,we can price options using either Heston (1993)’s Fourier transform method or the expansion formulae of Lewis (2000)..For the simulation and option pricing exercise,we benchmark the parameter values of the three models to the S&P 500index.We set µ=0.10,r =0.06,and q =0.02for all three models.Wefurther set σ=0.27for the BS model,σ=0.14,λ=λ∗=2.00,µj =µ∗j =−0.10,and σj =σ∗j =0.13for the MJ model,and θ=θ∗=0.272,κ=κ∗=1,and σv =0.1for the HV and HN models.We set ρ=−.5for the HV model.In each simulation,we generate a time series of daily underlying asset prices according to an Euler approximation of the respective data generating process.The starting value for the stock price is set to$100.Under the HV/HN model,we set the starting value of the instantaneous variance rate to its long-run mean:v0=θ.1We consider a hedging horizon of one month and simulate paths over this period.We assume that there are21business days in a month.To be consistent with the empirical study on S&P500index options in the next section,we think of the simulation as starting on a Wednesday and ending on a Thursday four weeks later,spanning a total of21week days and29 actual days.The hedging performance is recorded and,when needed,updated only on week days, but the interest earned on the money market account is computed based on actual/360day count convention.At each week day,we compute the relevant option prices based on the realization of the security price and the specification of the risk-neutral dynamics.For the dynamic delta hedge,we also com-pute the delta each day based on the risk-neutral dynamics and rebalance the portfolio accordingly. For both strategies,we monitor the hedging error(profit and loss)at each week day based on the simulated security price and the option prices.The hedging error at each date t,e t,is defined as the difference between the value of the hedge portfolio and the value of the target call option being hedged,e D t=B t−h e rh+∆t−h(F t−F t−h)−C(S t,t;K,T);(15)e S t=W j C(S t,t;K j,u)+B0e rt−C(S t,t;K,T),where the superscripts D and S denote the dynamic and static strategies,respectively,∆t denotes the delta of the target call option with respect to the futures price at time t,h denotes the time interval between stock trades,and B t denotes the time-t balance in the money market account.The balance includes the receipts from selling the one-year call option,less the cost of initiating and possibly changing the hedge portfolio.In the case of the static hedging strategy,under no arbitrage,the value of the portfolio of the shorter-term options should be equal to the value of the long term target option,and hence B0should be zero.However,since we use afinite number of call options inthe static hedge to approximate the spanning relation,the money market account captures the value difference due to the approximation error,which is normally very small.No rebalancing is needed in the static strategy.Under each model,the delta is given by the partial derivative ∂C (S ,t ;K ,T ;Θ)/∂F ,with F =Se (r −q )(T −t )denoting the forward/futures price.If an investor could trade continuously,this delta hedge removes all of the risk in the BS model and in the HN model.The hedge does not remove all risks in the MJ model because of the random jumps,nor in the HV model because of a second source of diffusion risk.The hedge portfolio for the static strategy is formed based on the weighting function w (K )in equation (4)implied by each model,the Gauss-Hermite quadrature nodes and weights {x i ,w i },and the mapping from the quadrature nodes and weights to the option strikes and weights,as given in equations (11)and (12).Under the HV/HN model,since the is non-Markovian,the static spanning relation in (3)is no longer valid.Furthermore,when we use the spanning relation to form an approximate hedging portfolio,the weighting function in (4)is no longer known at time t because option price at time u >t is also a function of the instantaneous variance rate at time u ,which is not known at time t .To implement the static strategy under these two models,we replace v u by its risk-neutral expected value E Q [v u ]in computing the weighting function w (K )at time t .In computing the strike points for the quadrature approximation of the spanning relation,the an-nualized variance is v =σ2for the BS model,v =θfor the HV/HN model,and v =v .=27%for all models.A.Hedging Comparison under the Diffusive Black-Scholes WorldTable 1reports the summary statistics of the simulated hedging errors,based on 1,000simulations.Panel A in Table 1summarizes the results based on the BS model.Entries are the summary statistics of the hedging errors at the last step (at the end of the 21business days)based on both strategies.For the dynamic strategy (the last column),we perform daily updating.For the static strategy,we consider hedge portfolios with N =3,5,9,15,21one-month options.If the transaction cost for a single option is comparable to the transaction cost for revising aposition in the underlying security,we would expect that the transaction cost induced by buying21 options at one time is close to the cost of rebalancing a position in the underlying stock21times. Hence,it is interesting to compare the performance of daily delta hedging with the performance of the static hedge with21options.The results in panel A of Table1show that the daily updating strategy and the static strategy with21options have comparable hedging performance in terms of the root mean squared error(RMSE).Since the stock market is much more liquid than the stock options market,the simulation results favor the dynamic delta strategy over the static strategy,if indeed stock prices move as in the BS world.The hedging errors from the two strategies show different distributional properties.The kurtosis of the hedging errors from the dynamic strategy is larger than that from all the static strategies.The kurtosis is4.68for the dynamic hedging errors,but is below two for errors from all the static hedges. The maximum profit and loss from the static strategy with21options are also smaller in absolute magnitudes.Therefore,when an investor is particularly concerned about avoiding large losses,the investor may prefer the static strategy.The last row shows the accuracy of the Gauss-Hermite quadrature approximation of the integral in pricing the target options.Under the BS model,the theoretical value of the target call option is $12.35,which we put under the dynamic hedging column.The approximation error is about one cent when applying a21-node quadrature.The approximation error increases as the number of quadrature nodes declines in the approximation.B.Hedging Comparison in the Presence of Random Jumps as in the Merton WorldIn Table1,Panel B shows the hedging performance under the Merton jump-diffusion model.For ease of comparison,we present the results in the same format as in Panel A for the BS model.The performance of all the static strategies are comparable to their corresponding cases under the BS world.If anything,most of the performance measures for the static strategies become slightly better under the Merton jump-diffusion dete-riorates dramatically as we move process of Merton(1976).The root mean the dynamic。

.HEDGING的交际功能浙江财经学院外语系潘晓霞*浙江大学外国语言文化与国际交流学院黄建滨**摘要：HEDGE/HEDGING作为模糊语言学中的特殊语言现象，越来越受到国内外学者的关注。

然而随着对其研究的不断深入，HEDGE/HEDGING的概念也不断地发生变化。

然而对这一概念学者们至今未达成一致，不同的学者提出了不同的定义。

因而，系统地回顾HEDGE/HEDGING的研究很有必要。

本文首先概述HEDGE/HEDGING 的概念演变过程，接着简要回顾HEDGE/HEDGING在不同领域的研究情况，然后分析和探讨其主要交际功能，并在文章的最后提出用“模糊调和”来指称HEDGING可能会比“模糊限制语”更合适。

关键词：模糊限制语交际功能语篇功能人际功能礼貌策略1. HEDGE和HEDGING研究综述1.1 HEDGE和HEDGING的概念演变语言学字典中解释HEDGE和HEDGING这两个概念的词条很少，对它们的定义多半基于LAKOFF最先提出的定义。

HEDGE 作为一个语言学术语是由美国语言学家KOFF（1972）最早提出的，虽然在这之前，ZADEH（1965）和WEINREICH（1966）已注意到了这种语言现象和概念。

根据LAKOFF的定义，HEDGES指的是那些“将事物弄得模模糊糊，或将原本模糊的事物弄得不那么模糊的词语”（LAKOFF，1972，234），诸如sort of, strictly speaking。

ZADEH（1972）按照LAKOFF 的定义从语义学和逻辑学的角度分析了英语中的HEDGES，诸如very，slightly，technically，practically。

HEDGE作为一个语言学术语最初指的是一种用来修饰一述语或名词短语的成员隶属关系的表达，通常为一词语或短语。

由于这类词或短语有着共同的特征，即可以改变某些词的模糊程度，或者说它们在某种程度上起了限制的作用，所以在最早的译文中廖东平（1982）用“模糊限制词”对应英语中的HEDGES。

期货市场套期保值理论述评一、传统套期保值理论传统套期保值是指投资者在期货交易中建立一个与现货交易方向相反、数量相等的交易部位。

由于在某一特定的社会经济系统内，商品的期货价格和现货价格受大体相同的因素影响，两种价格的走势基本一致，在期货合约到期时由于套利行为将使商品的期货价格和现货价格趋于一致，这样就可以用一个市场的利润来弥补另外一个市场的损失。

凯恩斯、希克斯最早从经济学的角度对传统的套期保值理论进行了阐述，认为套期保值者参与期货交易的目的不在于从期货交易中获取高额利润，而是要用期货交易中的获利来补偿在现货市场上可能发生的损失。

二、基差逐利型套期保值理论在完美的市场条件下，即如果期货市场价格和现货市场的价格波动完全一致，不存在交易费用和税收，则可实现完全型的套期保值，即可用一个市场的利润来完全弥补另外一个市场的损失。

但在现实的期货交易中，期货价格和现货价格的变动不完全一致，存在基差风险(Basis risk)，从而期货市场的获利不一定能完全弥补现货市场上的损失。

为克服基差风险，Working(1960)提出了用基差逐利型套期保值来回避基差风险，所谓基差逐利型套期保值是指买卖双方通过协商，由套期保值者确定协议基差的幅度和确定选择期货价格的期限，由现货市场的交易者在这个时期内选择某日的商品期货价格为计价基础，在所确定的计价基础上加上协议基差得到双方交易现货商品的协议价格，双方以协议价格交割现货，而不考虑现货市场上该商品在交割时的实际价格。

基差交易的实质，是套期保值者通过基差交易，将套期保值者面临的基差风险通过协议基差的方式转移给现货交易中的对手，套期保值者通过基差交易可以达到完全的或盈利的保值目的。

Working认为，套期保值的核心不在于能否消除价格风险，而在于能否通过寻找基差方面的变化或预期基差的变化来谋取利润，或者说通过发现期货市场与现货市场之间的价格变动来寻找套期保值的机会。

在这种意义上，套期保值是一种套期图利(Spreading)行为。