博迪投资学Chap021PPT课件
兹维博迪 投资学 中文课件
标准普尔指数
标准普尔500指数
涵盖500家公司的指数
市值加权指数
投资者可以购买指数投资组合:
购买与各种指数相对应的共同基金
购买交易所交易基金 (ETFs)
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其他指数
美国的指数 纽约证交所综合指数 纳斯达克综合指数 威尔希尔5000指数
国外的指数 日本日经指数 英国富时指数 德国综合指数 中国香港恒生指数 加拿大多伦多股市指数
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图 2.6 抵押担保证券余额
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权益证券
普通股:
代表所有权 剩余所有权 有限责任 永续性 固定收益 优先于普通服 税务处理
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优先股:
美国存托凭证
股票市场指数
道琼斯工业平均指数
包括30家大型绩优公司 自1896年被计算出来 价格加权平均数
货币市场基金使得个人投资者可以购买各种货币市场 证券。
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图2.1 货币市场的主要组成
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货币市场证券
短期国库券: 美国政府发行的短期债务。
买方报价与买方报价 银行贴现法
大额存单: 银行定期存单 商业票据: 公司发行的短期无担保债务票据。
Байду номын сангаас
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货币市场证券
银行承兑汇票: 银行客户向银行发出在未来某一日期支付 一笔款项的指令。 欧洲美元: 在美国以外的银行以美元计价的存款。 回购协议与逆回购: 由政府证券支持的短期借款。 联邦基金: 银行之间超短期借款。
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货币市场工具的收益率
除了短期国库券，货币市场证券并不是没有违约风险。 银行大额存单一直以来持续支付高于短期国库券的风险溢 价，且该溢价在经济危机期间更大。 在2008年信用危机中, 在一些基金遭遇严重损失后，联邦 政府为货币市场基金提供了保险。
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Money Market Securities
• Treasury b i l l s : Short-term debt of U.S. government
– Bid and asked price – Bank discount method
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The Bond Market
• I n f l a t i o n - Protected Treasury Bonds
– TIPS: Provide inflation protection
• Federal Agency Debt
• Both t h e premium on bank CDs and the TED spread have often become g r e a t e r during periods of financial crisis
• During the credit c r i s i s of 2008, the fede government offered insurance t o money market mutual funds a f t e r some funds experienced losses
• Money market mutual funds allow i n d i v i d u a l s t o access t h e money market.
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Table 2.1 Major Components of t h e Money Market
兹维博迪-投资学-第九版-中文PPT课件
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系统性风险的上升
银行资产和负债的到期日和流动性之间并不匹配。
负债是短期的、流动的 资产是长期的、非流动的 需要不断再融资改善资产组合
高杠杆比率使得银行几乎没有保证金来确保其安全。
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系统性风险的上升
投资者过分依赖结构化产品，如信用违约掉期来实现信 用升级。
信用违约掉期合约通常是场外交易，缺少公开披露，没 有要求保证金。
为什么会低估信用风险?
没有人会预料到房地产市场的价格会一直下跌 跨地区来分散风险的愿望并未实现 代理问题和评级机构 信用违约掉期并未像预计那样降低风险
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信用违约掉期 (CDS)
信用违约掉期实质上是一种针对借款者违约的保险合同。 投资者购买次级贷款使用信用违约掉期来保证其安全性。
投资银行
• 同意买下新发行的股票 和债券
• 在一级市场上向公众销 售新证券
• 投资者在二级市场上买 卖一级市场发行的证券
商业银行
• 吸收存款、发放贷款
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2008年的金融危机
金融危机的前情:
“大稳健”: 美国经历了一个低利率和经济稳定的时期，只有一 些温和的经济衰退，经济周期似乎已被驯服。
第一章
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投资环境
McGraw-Hill/Irwin
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
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实物资产与金融资产
实物资产 取决于该社会经济的生产能力，为经济创 造净利润。 如土地、建筑物、机器以及可用于生产产 品和提供服务的知识。
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投资者为什么喜欢凸性?
• 曲率大的债券价格在收益下降时的价格上升大于在收益上涨时的价格下降。 • 收益率越不稳定，这种不对称性的吸引力就越大。 • 对于凸性较大的债券而言，投资者必须付出更高的价格并接受更低的到期收
益率。
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可赎回债券
• 当利率下降时，债券的市场价格有一个上限，债券价格不会超过其赎 回价格。
$1,000/1.053.7704 = $831.6717。价格下降 了0.0359%。
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久期法则
法则 1 零息债券的久期等于它的到期时间。 法债则券2 久到期期较时短间。不变时，当息票率较高时， 法限则增3 加票而面增利加率。不变时，债券久期会随期
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久期法则
法收则益4 率保较持低其时他，因息素票都债不券变的，久当期债会券较到长期。
5. 利率风险与债券票面利率成反比。 6. 债券价格对其收益变化的敏感性与当期
出售债券的到期收益率成反比。
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图 16.1 作为到期收益率变化的函数的债券 价格变化
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表 16.1 票面利率为8%的债券价格(半年付 息一次)
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表 16.2 零息债券的价格(半年计一次复利)
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积极债券管理：互换策略
• 替代互换 • 市场价差价互换 • 利率预期互换 • 纯收益获得互换 • 税收互换
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水平分析
• 选择特定的持有期并预测该期末的收益 率曲线。
• 给定持有到期时债券的到期时间， – 它的收益可以从预测的收益率曲线和 计算的期末价格中得出
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感谢下 载
• 负凸性 • 使用有效久期:
有效久期 P / P r
博迪投资学第九版课件
p
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Risk Adjusted Performance: Treynor
2) Treynor Measure
(rP rf )
P
rp = Average return on the portfolio
rf = Average risk free rate ßp = Weighted average beta for portfolio
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Introduction
• Two common ways to measure average portfolio return: 1. Time-weighted returns 2. Dollar-weighted returns • Returns must be adjusted for risk.
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Time-Weighted Return
53 50 2 r1 10% 50 54 53 2 r2 5.66% 53
rG = [ (1.1) (1.0566) ]1/2 – 1 = 7.81% The dollar-weighted average is less than the time-weighted average in this example because more money is invested in year two, when the return was lower.
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M Measure
• Developed by Modigliani and Modigliani • Create an adjusted portfolio (P*)that has the same standard deviation as the market index. • Because the market index and P* have the same standard deviation, their returns are comparable:
2019年-INVESTMENTS 投资学 (博迪BODIE, KANE, MARCUS)Chap021 Option Valuation-PPT精选文档
normal distribution will be less than d
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Black-Scholes Option Valuation
X = Exercise price e = 2.71828, the base of the natural log r = Risk-free interest rate (annualized,
• Time value - the difference between the option price and the intrinsic value
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Figure 21.1 Call Option Value before Expiration
Expanding to Consider Three Intervals
S++
S+++
S+
S++-
S
S+-
SS--
S+-S---
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Possible Outcomes with Three Intervals
Event 3 up 2 up 1 down 1 up 2 down 3 down
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表25.6 综合评分与单项评分
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表 25.7 综合风险与政治风险预测
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表25.7 的解释
该表通过情景分析了解国家风险。 风险稳定度就是最好情况的风险值减去最
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熊市中国际分散化投资还会带来好处吗?
在资本市场动荡时 期，国家组合投资 收益之间的相关性 将增大。
罗尔模型揭示了全 球股价变动背后一 个广泛的因素。
预测：
分散化只能减轻各 国特殊事件的影响。
在1987年发生了什 么? 2008年呢?
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和外币。
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国际化投资的风险因素
风险变化的两 个来源:
1. 以当地货币计量的收益 率
2. 当地货币调整为本国货 币后的收益率
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例 25.1 汇率风险
假设英国无风险利率为10% 现在的汇率是 1英镑兑2美元 。
一个投资者有20,000 美元，即可以购买 10,000 英镑，一年后投资可得11,000英镑。
如果汇率下跌为1英镑兑1.80美元那么最终 只能得到19, 800美元, 损失了200美元。
对美国投资者而言，这项投资并不是无风 险的。
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滋维博迪投资学Chap.ppt
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Rules for Duration
Rule 4 Holding other factors constant, the duration of a coupon bond is higher when the bond’s yield to maturity is lower
Rules 5 The duration of a level perpetuity is equal to: (1+y) / y
• Bonds with greater curvature gain more in price when yields fall than they lose when yields rise.
• The more volatile interest rates, the more attractive this asymmetry.
(y)2
]
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Figure 16.4 Convexity of Two Bonds
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Why do Investors Like Convexity?
• Bonds with greater convexity have more curvature in the price-yield relationship.
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Intrinsic Value and Market Price
• The intrinsic value (IV) is the “true” value, according to a model. • The market value (MV) is the consensus value of all market participants Trading Signal: IV > MV Buy IV < MV Sell or Short Sell IV = MV Hold or Fairly Priced
2013
kg
kg
0.111 0.077
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Honda Example
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Example 18.4 Growth Opportunities
• Firm reinvests 60% of its earnings in projects with ROE of 10%, capitalization rate is 15%. Expected year-end dividend is $2/share, paid out of earnings of $5/share. • g=ROE x b = 10% x .6 = 6%
$5 PVGO $22.22 $11.11 .15
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Life Cycles and Multistage Growth Models • Expected dividends for Honda: 2010 $.50 2012 $ .83 2011 $.66 2013 $1.00 • Since the dividend payout ratio is 30% and ROE is 11%, the “steadystate” growth rate is 7.7%.
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Active or Passive Management
• Active Management
– An expensive strategy – Suitable only for very large portfolios
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Resource Allocation
• I f markets were i n e f f i c i e n t , resources would be s y s t e m a t i c a l l y misallocated.
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Types of Stock Analysis
• Fundamental Analysis - using economic and accounting information to predict stock prices
• How do we explain random stock p r i c e changes?
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E f f i c i e n t Market Hypothesis (EMH)
• EMHsays stock p r i c e s already r e f l e c t a l l available information
投资学Chap021
CHAPTER 21: OPTION VALUATIONPROBLEM SETS1. The value of a put option also increases with the volatility of the stock. We see this from theput-call parity theorem as follows:P = C – S0 + PV(X) + PV(Dividends)Given a value for S and a risk-free interest rate, then, if C increases because of an increase in volatility, P must also increase in order to maintain the equality of the parity relationship.2. A $1 increase in a call option’s exercise price would lead to a decrease in the option’s value ofless than $1. The change in the call price would equal $1 only if: (i) there were a 100%probability that the call would be exercised, and (ii) the interest rate were zero.3. Holding firm-specific risk constant, higher beta implies higher total stock volatility. Therefore,the value of the put option increases as beta increases.4. Holding beta constant, the stock with a lot of firm-specific risk has higher total volatility. Theoption on the stock with higher firm-specific risk is worth more.5. A call option with a high exercise price has a lower hedge ratio. This call option is less in themoney. Both d1 and N(d1) are lower when X is higher.6. a. Put A must be written on the stock with the lower price. Otherwise, given the lowervolatility of Stock A, Put A would sell for less than Put B.b. Put B must be written on the stock with the lower price. This would explain its higherprice.c. Call B must have the lower time to expiration. Despite the higher price of Stock B, CallB is cheaper than Call A. This can be explained by a lower time to expiration.d. Call B must be written on the stock with higher volatility. This would explain its higherprice.e. Call A must be written on the stock with higher volatility. This would explain its higherprice.7.Exercise Price Hedge Ratio 120 0/30 = 0.000 110 10/30 = 0.333 100 20/30 = 0.667 90 30/30 = 1.000As the option becomes more in the money, the hedge ratio increases to a maximum of 1.0.8.S d 1 N(d 1) 45 -0.0268 0.4893 50 0.5000 0.6915 550.97660.83569.a. uS 0 = 130 ⇒ P u = 0 dS 0 = 80 ⇒ P d = 30 The hedge ratio is:5380130300dS uS P P H 00d u -=--=--=b.Riskless Portfolio S = 80 S = 130 Buy 3 shares 240 390 Buy 5 puts 150 0 Total390390Present value = $390/1.10 = $354.545c.The portfolio cost is: 3S + 5P = 300 + 5P The value of the portfolio is: $354.545 Therefore: P = $54.545/5 = $10.9110. The hedge ratio for the call is:5280130020dS uS C C H 00d u =--=--=Riskless Portfolio S = 80 S = 130 Buy 2 shares 160 260 Write 5 calls 0 -100 Total160160Present value = $160/1.10 = $145.455 The portfolio cost is: 2S – 5C = $200 – 5C The value of the portfolio is: $145.455 Therefore: C = $54.545/5 = $10.91 Does P = C + PV(X) – S?10.91 = 10.91 + 110/1.10 – 100 = 10.9111. d 1 = 0.3182 ⇒ N(d 1) = 0.6248d 2 = –0.0354 ⇒ N(d 2) = 0.4859 Xe - r T = 47.56 C = $8.1312. P = $5.69This value is derived from our Black-Scholes spreadsheet, but note that we could have derived the value from put-call parity:P = C + PV(X) – S 0 = $8.13 + $47.56 - $50 = $5.6913. a. C falls to $5.5541b. C falls to $4.7911c. C falls to $6.0778d. C rises to $11.5066e. C rises to $8.718714. According to the Black-Scholes model, the call option should be priced at:[$55 ⨯ N(d1)] – [50 ⨯ N(d 2)] = ($55 ⨯ 0.6) – ($50 ⨯ 0.5) = $8Since the option actually sells for more than $8, implied volatility is greater than 0.30.15. A straddle is a call and a put. The Black-Scholes value would be:C + P = S0 N(d1) - Xe–rT N(d 2) + Xe–rT [1 - N(d 2)] - S0 [1 - N(d1)]= S0 [2N(d1) - 1] + Xe–rT [1 - 2N(d 2)]On the Excel spreadsheet (Spreadsheet 21.1), the valuation formula would be:B5*(2*E4 - 1) + B6*EXP(-B4*B3)*(1 - 2*E5)16. The rate of return of a call option on a long-term Treasury bond should be more sensitive tochanges in interest rates than is the rate of return of the underlying bond. The option elasticityexceeds 1.0. In other words, the option is effectively a levered investment and the rate of return on the option is more sensitive to interest rate swings.17. Implied volatility has increased. If not, the call price would have fallen as a result of thedecrease in stock price.18. Implied volatility has increased. If not, the put price would have fallen as a result of thedecreased time to expiration.19. The hedge ratio approaches one. As S increases, the probability of exercise approaches 1.0.N(d1) approaches 1.0.20. The hedge ratio approaches –1.0. As S decreases, the probability of exercise approaches 1.[N(d1) –1] approaches –1 as N(d1) approaches 0.21. A straddle is a call and a put. The hedge ratio of the straddle is the sum of the hedge ratios ofthe individual options: 0.4 + (–0.6) = –0.222. a. The spreadsheet appears as follows:The standard deviation is: 0.3213b. The spreadsheet below shows the standard deviation has increased to: 0.3568Implied volatility has increased because the value of an option increases with greatervolatility.c. Implied volatility increases to 0.4087 when expiration decreases to four months. Theshorter expiration decreases the value of the option; therefore, in order for the optionprice to remain unchanged at $8, implied volatility must increase.d. Implied volatility decreases to 0.2406 when exercise price decreases to $100. Thedecrease in exercise price increases the value of the call, so that, in order to the optionprice to remain at $8, implied volatility decreases.e. The decrease in stock price decreases the value of the call. In order for the option priceto remain at $8, implied volatility increases.23. a. The delta of the collar is calculated as follows:Position DeltaBuy stock 1.0Buy put, X = $45 N(d1) – 1 = –0.40Write call, X = $55 –N(d1) = –0.35Total 0.25If the stock price increases by $1, then the value of the collar increases by $0.25. Thestock will be worth $1 more, the loss on the purchased put will be $0.40, and the callwritten represents a liability that increases by $0.35.b.If S becomes very large, then the delta of the collar approaches zero. Both N(d1) termsapproach 1. Intuitively, for very large stock prices, the value of the portfolio is simply the(present value of the) exercise price of the call, and is unaffected by small changes in thestock price.As S approaches zero, the delta also approaches zero: both N(d1) terms approach 0.For very small stock prices, the value of the portfolio is simply the (present value of the)exercise price of the put, and is unaffected by small changes in the stock price.24. Put X DeltaA 10 -0.1B 20 -0.5C 30 -0.925. a. Choice A: Calls have higher elasticity than shares. For equal dollar investments, a call’scapital gain potential is greater than that of the underlying stock.b. Choice B: Calls have hedge ratios less than 1.0, so the shares have higher profit potential.For an equal number of shares controlled, the dollar exposure of the shares is greaterthan that of the calls, and the profit potential is therefore greater.26. a. uS 0 = 110 ⇒ P u = 0 dS 0 = 90 ⇒ P d = 10 The hedge ratio is:2190110100dS uS P P H 00d u -=--=--=A portfolio comprised of one share and two puts provides a guaranteed payoff of $110, with present value: $110/1.05 = $104.76 Therefore:S + 2P = $104.76$100 + 2P = $104.76 ⇒ P = $2.38b. Cost of protective put portfolio = $100 + $2.38 = $102.38c.Our goal is a portfolio with the same exposure to the stock as the hypothetical protective put portfolio. Since the put’s hedge ratio is –0.5, the portfolio consists of (1 – 0.5) = 0.5 shares of stock, which costs $50, and the remaining funds ($52.38) invested in T-bills, earning 5% interest. Portfolio S = 90 S = 110 Buy 0.5 shares 45 55 Invest in T-bills 55 55 Total100110This payoff is identical to that of the protective put portfolio. Thus, the stock plus bills strategy replicates both the cost and payoff of the protective put.27. The put values in the second period are:P uu = 0P ud = P du = 110 − 104.50 = 5.50 P dd = 110 − 90.25 = 19.75To compute P u , first compute the hedge ratio:3150.10412150.50udSuuSP P H 0ud uu -=--=--=Form a riskless portfolio by buying one share of stock and buying three puts. The cost of the portfolio is: S + 3P u = $110 + 3P u The payoff for the riskless portfolio equals $121: Riskless Portfolio S = 104.50 S = 121 Buy 1 share 104.50 121.00 Buy 3 puts 16.50 0.00 Total121.00121.00Therefore, find the value of the put by solving:$110 + 3P u = $121/1.05 ⇒ P u = $1.746 To compute P d , compute the hedge ratio:0.125.9050.10475.1950.5ddSduSP P H 0dd du -=--=--=Form a riskless portfolio by buying one share and buying one put. The cost of the portfolio is: S + P d = $95 + P d The payoff for the riskless portfolio equals $110: Riskless Portfolio S = 90.25 S = 104.50 Buy 1 share 90.25 104.50 Buy 1 put 19.75 5.50 Total110.00110.00Therefore, find the value of the put by solving:$95 + P d = $110/1.05 ⇒ P d = $9.762 To compute P, compute the hedge ratio:5344.095110762.9746.1dS uS P P H 00d u -=--=--=Form a riskless portfolio by buying 0.5344 of a share and buying one put. The cost of the portfolio is: 0.5344S + P = $53.44 + PThe payoff for the riskless portfolio equals $60.53:Riskless Portfolio S = 95 S = 110Buy 0.5344 share 50.768 58.784Buy 1 put 9.762 1.746Total 60.530 60.530Therefore, find the value of the put by solving:$53.44 + P = $60.53/1.05 ⇒ P = $4.208Finally, we verify this result using put-call parity. Recall from Example 21.1 that:C = $4.434Put-call parity requires that:P = C + PV(X) – S$4.208 = $4.434 + ($110/1.052) - $100Except for minor rounding error, put-call parity is satisfied.28. If r = 0, then one should never exercise a put early. There is no “time value cost” to waiting toexercise, but there is a “volatility benefit” from waiting. To show this more rigorously, consider the following portfolio: lend $X and short one share of stock. The cost to establish the portfolio is (X – S 0). The payoff at time T (with zero interest earnings on the loan) is (X – S T). Incontrast, a put option has a payoff at time T of (X – S T) if that value is positive, and zerootherwise. The put’s payoff is at least as large as the portfolio’s, and therefore, the put must cost at least as much as the portfolio to purchase. Hence, P ≥ (X – S 0), and the put can be sold for more than the proceeds from immediate exercise. We conclude that it doesn’t pay to exercise early.29. a. Xe-rTb. Xc. 0d. 0e. It is optimal to exercise immediately a put on a stock whose price has fallen to zero. Thevalue of the American put equals the exercise price. Any delay in exercise lowers valueby the time value of money.30. Step 1: Calculate the option values at expiration. The two possible stock prices and thecorresponding call values are:uS 0 = 120 ⇒ C u = 20 dS 0 = 80 ⇒ C d = 0 Step 2: Calculate the hedge ratio.2180120020dS uS C C H 00d u =--=--=Therefore, form a riskless portfolio by buying one share of stock and writing two calls. The cost of the portfolio is: S – 2C = 100 – 2CStep 3: Show that the payoff for the riskless portfolio equals $80: Riskless Portfolio S = 80 S = 120 Buy 1 share 80 120 Write 2 calls 0 -40 Total8080Therefore, find the value of the call by solving:$100 – 2C = $80/1.10 ⇒ C = $13.636Notice that we did not use the probabilities of a stock price increase or decrease. These are not needed to value the call option.31.The two possible stock prices and the corresponding call values are:uS 0 = 130 ⇒ C u = 30 dS 0 = 70 ⇒ C d = 0 The hedge ratio is:2170130030dS uS C C H 00d u =--=--=Form a riskless portfolio by buying one share of stock and writing two calls. The cost of the portfolio is: S – 2C = 100 – 2CThe payoff for the riskless portfolio equals $70: Riskless Portfolio S = 70 S = 130 Buy 1 share 70 130 Write 2 calls 0 -60 Total7070Therefore, find the value of the call by solving:$100 – 2C = $70/1.10 ⇒ C = $18.182Here, the value of the call is greater than the value in the lower-volatility scenario.32. The two possible stock prices and the corresponding put values are:uS 0 = 120 ⇒ P u = 0dS 0 = 80 ⇒ P d = 20 The hedge ratio is:2180120200dS uS P P H 00d u -=--=--=Form a riskless portfolio by buying one share of stock and buying two puts. The cost of the portfolio is: S + 2P = 100 + 2PThe payoff for the riskless portfolio equals $120: Riskless Portfolio S = 80 S = 120 Buy 1 share 80 120 Buy 2 puts 40 0 Total120120Therefore, find the value of the put by solving:$100 + 2P = $120/1.10 ⇒ P = $4.545 According to put-call parity: P + S = C + PV(X) Our estimates of option value satisfy this relationship:$4.545 + $100 = $13.636 + $100/1.10 = $104.54533. If we assume that the only possible exercise date is just prior to the ex-dividend date, then therelevant parameters for the Black-Scholes formula are:S 0 = 60r = 0.5% per month X = 55 σ = 7%T = 2 monthsIn this case: C = $6.04If instead, one commits to foregoing early exercise, then we reduce the stock price by the present value of the dividends. Therefore, we use the following parameters:S 0 = 60 – 2e − (0.005 ⨯ 2) = 58.02 r = 0.5% per month X = 55 σ = 7%T = 3 months In this case, C = $5.05The pseudo-American option value is the higher of these two values: $6.0434. True. The call option has an elasticity greater than 1.0. Therefore, the call’s percentage rate ofreturn is greater than that of the underlying stock. Hence the GM call responds more thanproportionately when the GM stock price changes in response to broad market movements.Therefore, the beta of the GM call is greater than the beta of GM stock.35. True. The elasticity of a call option is higher the more out of the money is the option. (Eventhough the delta of the call is lower, the value of the call is also lower. The proportionalresponse of the call price to the stock price increases. You can confirm this with numerical examples.) Therefore, the rate of return of the call with the higher exercise price respondsmore sensitively to changes in the market index, and therefore it has the higher beta.36. As the stock price increases, conversion becomes increasingly more assured. The hedge ratioapproaches 1.0. The price of the convertible bond will move one-for-one with changes in the price of the underlying stock.37. Salomon believes that the market assessment of volatility is too high. Therefore, Salomonshould sell options because the analysis suggests the options are overpriced with respect to true volatility. The delta of the call is 0.6, while that of the put is 0.6 – 1 = –0.4. Therefore, Salomon should sell puts and calls in the ratio of 0.6 to 0.4. For example, if Salomon sells 2 calls and 3 puts, the position will be delta neutral:Delta = (2 ⨯ 0.6) + [3 ⨯ (–0.4)] = 038. If the stock market index increases 1%, the 1 million shares of stock on which the options arewritten would be expected to increase by:0.75% ⨯ $5 ⨯ 1 million = $37,500The options would increase by:delta ⨯ $37,500 = 0.8 ⨯ $37,500 = $30,000In order to hedge your market exposure, you must sell $3,000,000 of the market indexportfolio so that a 1% change in the index would result in a $30,000 change in the value of the portfolio.39. S = 100; current value of portfolioX = 100; floor promised to clients (0% return)σ = 0.25; volatilityr = 0.05; risk-free rateT = 4 years; horizon of programa. Using the Black-Scholes formula, we find that:d1 = 0.65, N(d1) = 0.7422, d 2 = 0.15, N(d 2) = 0.5596Put value = $10.27Therefore, total funds to be managed equals $110.27 million: $100 million portfolio valueplus the $10.27 million fee for the insurance program.The put delta is: N(d1) – 1 = 0.7422 – 1 = –0.2578Therefore, sell off 25.78% of the equity portfolio, placing the remaining funds in T-bills. Theamount of the portfolio in equity is therefore $74.22 million, while the amount in T-bills is:$110.27 million – $74.22 million = $36.05 millionb.At the new portfolio value, the put delta becomes: –0.2779This means that you must reduce the delta of the portfolio by:0.2779 – 0.2578 = 0.0201You should sell an additional 2.01% of the equity position and use the proceeds to buyT-bills. Since the stock price is now at only 97% of its original value, you need to sell: $97 million ⨯ 0.0201 = $1.950 million of stock40. Using the true volatility (32%) and time to expiration T = 0.25 years, the hedge ratio for Exxonis N(d1) = 0.5567. Because you believe the calls are under-priced (selling at an impliedvolatility that is too low), you will buy calls and short 0.5567 shares for each call you buy.41. The calls are cheap (implied σ = 0.30) and the puts are expensive (impliedσ = 0.34). Therefore, buy calls and sell puts. Using the “true” volatility ofσ = 0.32, the call delta is 0.5567 and the put delta is: 0.5567 – 1.0 = –0.4433Therefore, for each call purchased, buy: 0.5567/0.4433 = 1.256 puts42. a.To calculate the hedge ratio, suppose that the market index increases by 1%. Then the stock portfolio would be expected to increase by:1% ⨯ 1.5 = 1.5% or 0.015 ⨯ $1,250,000 = $18,750Given the option delta of 0.8, the option portfolio would increase by:$18,750 ⨯ 0.8 = $15,000Salomon’s liability from writing these options would increase by the same amount. The market index portfolio would increase in value by 1%. Therefore, SalomonBrothers should purchase $1,500,000 of the market index portfolio in order to hedge its position so that a 1% change in the index would result in a $15,000 change in the value of the portfolio.b.The delta of a put option is:0.8 – 1 = –0.2Therefore, for every 1% the market increases, the index will rise by 10 points and the value of the put option contract will change by:delta ⨯ 10 ⨯ contract multiplier = –0.2 ⨯ 10 ⨯ 100 = –$200 Therefore, Salomon should write: $12,000/$200 = 60 put contractsCFA PROBLEMS1. Statement a: The hedge ratio (determining the number of futures contracts to sell) ought to beadjusted by the beta of the equity portfolio, which is 1.20. The correct hedge ratio would be:400,22.1000,2β2,000β500100$million 100$=⨯=⨯=⨯⨯Statement b: The portfolio will be hedged, and should therefore earn the risk-free rate, not zero, as the consultant claims. Given a futures price of 100 and an equity price of 100, the rate of return over the 3-month period is: (100 - 99)/99 = 1.01% = approximately 4.1% annualized 2.a.The value of the call option is expected to decrease if the volatility of the underlying stock price decreases. The less volatile the underlying stock price, the less the chance of extreme price movements and the lower the probability that the option expires in the money. This makes the participation feature on the upside less valuable.The value of the call option is expected to increase if the time to expiration of the option increases. The longer the time to expiration, the greater the chance that the option will expire in the money resulting in an increase in the time premium component of the option’s value.b. i. When European options are out of the money, investors are essentially saying that theyare willing to pay a premium for the right, but not the obligation, to buy or sell theunderlying asset. The out-of-the-money option has no intrinsic value, but, since optionsrequire little capital (just the premium paid) to obtain a relatively large potential payoff,investors are willing to pay that premium even if the option may expire worthless. TheBlack-Scholes model does not reflect investors’ demand for any premium above thetime value of the option. Hence, if investors are willing to pay a premium for an out-of-the-money option above its time value, the Black-Scholes model does not value thatexcess premium.ii. With American options, investors have the right, but not the obligation, to exercise theoption prior to expiration, even if they exercise for non-economic reasons. This increasedflexibility associated with American options has some value but is not considered in theBlack-Scholes model because the model only values options to their expiration date(European options).3. a. American options should cost more (have a higher premium). American options give theinvestor greater flexibility than European options since the investor can choose whetherto exercise early. When the stock pays a dividend, the option to exercise a call early canbe valuable. But regardless of the dividend, a European option (put or call) never sellsfor more than an otherwise-identical American option.b. C = S0 + P - PV(X) = $43 + $4 - $45/1.055 = $4.346Note: we assume that Abaco does not pay any dividends.c. i) An increase in short-term interest rate ⇒ PV(exercise price) is lower, and call valueincreases.ii) An increase in stock price volatility ⇒ the call value increases.iii) A decrease in time to option expiration ⇒ the call value decreases.4. a. The two possible values of the index in the first period are:uS0 = 1.20 × 50 = 60dS0 = 0.80 × 50 = 40The possible values of the index in the second period are:uuS0 = (1.20)2 × 50 = 72udS0 = 1.20 × 0.80 × 50 = 48duS0 = 0.80 × 1.20 × 50 = 48ddS0 = (0.80)2 × 50 = 32C uu = 72 − 60 = 12 C ud = C du = C dd = 0 Since C ud = C du = 0, then C d = 0.To compute C u , first compute the hedge ratio:214872012udSuuSC C H 0ud uu =--=--=Form a riskless portfolio by buying one share of stock and writing two calls. The cost of the portfolio is: S – 2C u = $60 – 2C u The payoff for the riskless portfolio equals $48: Riskless Portfolio S = 48 S = 72 Buy 1 share 48 72 Write 2 calls 0 -24 Total4848Therefore, find the value of the call by solving:$60 – 2C u = $48/1.06 ⇒ C u = $7.358 To compute C, compute the hedge ratio:3679.040600358.7dS uS C C H 00d u =--=--=Form a riskless portfolio by buying 0.3679 of a share and writing one call. The cost of the portfolio is: 0.3679S – C = $18.395 – C The payoff for the riskless portfolio equals $14.716: Riskless Portfolio S = 40 S = 60 Buy 0.3679 share 14.716 22.074 Write 1 call 0.000 −7.358 Total14.71614.716Therefore, find the value of the call by solving:$18.395 – C = $14.716/1.06 ⇒ C = $4.512P uu = 0P ud = P du = 60 − 48 = 12 P dd = 60 − 32 = 28To compute P u , first compute the hedge ratio:214872120udSuuSP P H 0ud uu -=--=--=Form a riskless portfolio by buying one share of stock and buying two puts. The cost of the portfolio is: S + 2P u = $60 + 2P u The payoff for the riskless portfolio equals $72: Riskless Portfolio S = 48 S = 72 Buy 1 share 48 72 Buy 2 puts 24 0 Total7272Therefore, find the value of the put by solving:$60 + 2P u = $72/1.06 ⇒ P u = $3.962 To compute P d , compute the hedge ratio:0.132482812ddSduSP P H 0dd du -=--=--=Form a riskless portfolio by buying one share and buying one put. The cost of the portfolio is: S + P d = $40 + P d The payoff for the riskless portfolio equals $60: Riskless Portfolio S = 32 S = 48 Buy 1 share 32 48 Buy 1 put 28 12 Total6060Therefore, find the value of the put by solving:$40 + P d = $60/1.06 ⇒ P d = $16.604 To compute P, compute the hedge ratio:6321.04060604.16962.3dS uS P P H 00d u -=--=--=Form a riskless portfolio by buying 0.6321 of a share and buying one put.The cost of the portfolio is: 0.6321S + P = $31.605 + PThe payoff for the riskless portfolio equals $41.888:Riskless Portfolio S = 40 S = 60Buy 0.6321 share 25.284 37.926Buy 1 put 16.604 3.962Total 41.888 41.888Therefore, find the value of the put by solving:$31.605 + P = $41.888/1.06 ⇒ P = $7.912d. According to put-call-parity:C = S0 + P - PV(X) = $50 + $7.912 - $60/(1.062 ) = $4.512This is the value of the call calculated in part (b) above.5. a. (i) Index increases to 1402. The combined portfolio will suffer a loss. The written callsexpire in the money; the protective put purchased expires worthless. Let’s analyze theoutcome on a per-share basis. The payout for each call option is $52, for a total cashoutflow of $104. The stock is worth $1,402. The portfolio will thus be worth: $1,402 -$104 = $1,298The net cost of the portfolio when the option positions are established is:$1,336 + $16.10 (put) - [2 ⨯ $8.60] (calls written) = $1,334.90(ii) Index remains at 1336. Both options expire out of the money. The portfolio will thus beworth $1,336 (per share), compared to an initial cost 30 days earlier of $1,334.90. Theportfolio experiences a very small gain of $1.10.(iii) Index declines to 1270. The calls expire worthless. The portfolio will be worth $1,330,the exercise price of the protective put. This represents a very small loss of $4.90 comparedto the initial cost 30 days earlier of $1,334.90b. (i) Index increases to 1402. The delta of the call approaches 1.0 as the stock goes deep intothe money, while expiration of the call approaches and exercise becomes essentially certain.The put delta approaches zero.(ii) Index remains at 1336. Both options expire out of the money. Delta of eachapproaches zero as expiration approaches and it becomes certain that the options willnot be exercised.(iii) Index declines to 1270. The call is out of the money as expiration approaches. Deltaapproaches zero. Conversely, the delta of the put approaches -1.0 as exercise becomescertain.c. The call sells at an implied volatility (11.00%) that is less than recent historical volatility(12.00%); the put sells at an implied volatility (14.00%) that is greater than historicalvolatility. The call seems relatively cheap; the put seems expensive.。
博迪投资学第九版课件
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9-10
GE Example
• Reward-to-risk ratio for investment in market portfolio:
Market risk premium E (rM ) rf 2 Market variance M
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9-21
Extensions of the CAPM
• Merton’s Multiperiod • Consumption-based Model and hedge CAPM portfolios • Rubinstein, Lucas, • Incorporation of the and Breeden effects of changes in • Investors allocate the real rate of wealth between interest and inflation consumption today and investment for the future
• Information is costless and available to all investors • Investors are rational mean-variance optimizers • There are homogeneous expectations
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E (rM ) rf M E (rM ) rf
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Lower bound = adjusted intrinsic value:
C > S0 - PV (X) - PV (D)
(D=dividend)
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Figure 21.2 Range of Possible Call Option Values
21-9
Early Exercise: Puts
• American puts are worth more than European puts, all else equal.
• The possibility of early exercise has value because:
– The value of the stock cannot fall below zero. – Once the firm is bankrupt, it is optimal to
• Call value cannot be negative. The option payoff is zero at worst, and highly positive at best.
• Call value cannot exceed the stock value.
• Value of the call must be greater than the value of levered equity.
30
30
18.18
3C
0
0
3C = $18.18 C = $6.06
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21-1
CHAPTER 21
Option Valuation
McGraw-Hill/Irwin
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Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
• The right to exercise an American call early is valueless as long as the stock pays no dividends until the option expires.
• The value of American and European calls is therefore identical.
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Binomial Option Pricing: Text Example
120
10
100
Байду номын сангаас
C
90
0
Stock Price
Call Option Value X = 110
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• The call gains value as the stock price rises. Since the price can rise infinitely, the call is “worth more alive than dead.”
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• Time value - the difference between the option price and the intrinsic value
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21-3
Figure 21.1 Call Option Value before Expiration
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21-4
Table 21.1 Determinants of Call Option Values
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21-5
Restrictions on Option Value: Call
Repay loan - 90 - 90
Net Payoff
0 30
30
0 Payoff Structure is exactly 3 times the Call
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Binomial Option Pricing: Text Example
21-2
Option Values
• Intrinsic value - profit that could be made if the option was immediately exercised – Call: stock price - exercise price – Put: exercise price - stock price
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Figure 21.3 Call Option Value as a Function of the Current Stock Price
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Early Exercise: Calls
21-12
Binomial Option Pricing: Text Example
Alternative Portfolio
Buy 1 share of stock at $100
Borrow $81.82 (10% Rate) 18.18
Net outlay $18.18
Payoff
Value of Stock 90 120
exercise the American put immediately because of the time value of money.
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Figure 21.4 Put Option Values as a Function of the Current Stock Price