Why Large Financial Institutions Buy LongTerm Put Options from Companies


 Anabel White
 2 years ago
 Views:
Transcription
1 Why Large Financial Institutions Buy LongTerm Put Options from Companies Vladimir Atanasov Department of Finance Penn State University 609 Business Administration Building University Park, PA Stanley B. Gyoshev Finance Department Drexel University Academic Building Philadelphia, PA (215) Current Draft: March, 2001
2 Why Large Financial Institutions Buy LongTerm Put Options from Companies Abstract This study explores the strategic interaction between large institutional investors and firms that issue put options written on their own stock. The firms experience large positive abnormal annual returns after they sell put options. The vast majority of issued put options expire without being exercised, and the buyers of these options, which are predominantly investment banks, lose money. We propose a model that gives a rationale why an uniformed party, an investment bank will trade in put options with an informed party, the issuing firm, although the expected profits from this trade are negative. The model shows how trading with an informed party can be profitable because the bank can acquire valuable information and afterwards earn abnormal returns on trades in other securities of the same firm. Finally, we outline several predictions from the model, and propose empirical tests to establish our proposition that an investment bank can legally acquire private information and trade profitably on it.
3 I. Introduction The seminal work of Akerlof (1970) showed how asymmetric information might lead to the collapse of a market because uninformed participants are not willing to incur losses while trading with informed parties. Asymmetric information and the associated with it winner s curse and adverse selection lead to inefficient market outcomes like larger bidask spreads and IPO underpricing because, in general, the uninformed parties have to receive an outside payment before they trade with informed parties. Our work focuses on the interaction between investment banks and corporations when corporations sell to banks longterm put options written on the stock of the corporation. The analysis of this setting documents a rare example where an uninformed party, an investment bank willingly accepts to trade with an informed party, a corporation, although the trade has a negative expected profit for the investment bank 1. The reason for this, at first glance irrational behavior, is that the investment bank can infer from the behavior of the corporation valuable information about the distribution of stock prices, and trade on this information on the stock market. In other words, by accepting an expected loss trade with an informed party the investment bank invests in a lawful acquisition of inside information, and then earns abnormal return on this investment. Our model is similar to the screening models used in the insurance, banking and industrial organization literature where an uninformed firm proposes different contracts 1 Other large institutional investors may also participate in the put option sale. For brevity we will denote all these investors by the term investment bank. 2
4 to various types of informed customers, and by choosing the optimal contracts manages to separate the customers based on their type 2. In our case, the investment bank offers to buy longterm put options from corporations that have recently announced stock repurchases. By setting optimally the premium of the options, the bank learns which corporations have positive and which corporations have negative private information about their future prospects. Later, based on this information the investment bank can take long positions in publicly traded put options written on the stock of the companies with bad prospects, or take short positions in put options and long positions in call options written on the stocks of companies with good prospects. In order to implement this scheme, the investment bank has to use a surprisingly simple strategy. The only thing that the investment banks needs to do is to offer a put option premium to the issuing corporations that is equal to the fair premium computed using public information. As a consequence, only companies that have positive private information about the distribution of their stock price in the future will accept to sell put options. The companies with bad prospects will refuse participating in the trade because it has negative expected value for them. This assures the existence and uniqueness of the separating equilibrium. Note that through this screening scheme the investment bank has acquired private information in a perfectly legal way. Gibson and Singh (2000) propose a theoretical model that also analyses the use of put options by firms. In their model corporations that need to raise new capital use put options to signal their quality and reduce their financing costs by fetching a better price for their newly offered securities. The model is not applicable to firms that repurchase 2 See for example the pioneering works of Rotschild and Stiglitz (1976) and Wilson (1977) 3
5 stocks because they are not in need of new capital. In addition, in the model of Gibson and Singh (2000), as in any signaling model, the informed party moves first. There is plenty of anecdotal evidence that the investment banks are usually the first to contact the firms with ongoing stock buybacks, and promote the put options scheme that will enhance the stock repurchase. The outline of the paper is as follows. Section II describes the institutional details of the logterm put options sold by companies. The theoretical model is outlined in Section III. Section IV describes the predictions of the model and proposes ways to test them. Section V concludes. II. Institutional details of the sale of put options by firms The remarkable spread in the use of derivatives, combined with wellknown cases of large losses associated with their use, has tremendously increased the interest in derivatives usage by firms. Derivatives can be used to hedge or to speculate. Hedging attempts to lessen or avoid unexpected revenue loss or gain from activities not related to the core firm operations by making counterbalancing investments. In contrast, speculation is the divesture of financial derivatives or real assets that increases the variability of the firm cash flows or earnings. Selling put options on firm s own stock is a good example of speculation. When a firm has good financial performance there is an additional positive cash flow from the collected option premiums. When a firm experiences weak financial performance, there is an additional negative cash flow because its stock price is likely to fall and the put options sold by the firm are going to be exercised. 4
6 Because of the possibility of losing a substantial amount of money, the managers will sell put options only if they have strong expectations that their firms are going to have a good performance during the life of the options. This notion is supported by Gyoshev and Tsetsekos (2000). The authors find in a sample of 45 firms that have sold put options an average of 14.1% riskadjusted abnormal return in the year following the announcement of the put sale in the 10Q or 10K statements. They also report that most of the put options expire out of the money. This evidence of good firm performance after the put option sale implies that the buyers of the put options, which are mainly investment banks and other large financial institutions, lost money on these trades. It is interesting to note that the investment banks actively solicit firms to buy put options from them. In the Investment Dealers' Digest from December 5, 1994, Paul Mazzilli, a principal in equity capital markets at Morgan Stanley & Co., is cited to say that "A large portion of the companies that do [share repurchase] programs with us have been introduced to it, and use the strategy from time to time." Tyler Dickson, a VP in equity capital markets at Salomon Brothers is cited to say that: "This year, put warrants have come of age," and also that: "Companies are much more familiar with them as an enhancing vehicle to share repurchases." He added that Salomon has purchased put warrants from three Fortune 500 companies in the last two weeks alone. The put option sale is not publicly known at the time when it was done. The earliest time it may become public information is after the release of the next 10Q or 10 K statement, but a large subset of the firms investigated by Gyoshev and Tsetsekos (2000) report the put option trades a lot later. On average the time from the date of the trade to the time the public learned about it is more than four months. Moreover, SEC 5
7 states that if the put option sale affects the financial situation of the firm in a nonmaterial way, there is no legal requirement to disclose it. This implies that potentially some firms have chosen not to disclose their put option trades at all. III. Theoretical model III.A. Outline of the game between the investment bank and the firms Players Consider one risk neutral investment bank denoted by I, and two types of firms that have recently initiated market stock repurchase programs 3. Type A firms have a positive signal about their future prospects, while type B firms have a negative or neutral signal about their future prospects 4. Timeline The order of moves is as follows. At time t = 0, Nature picks firm type A or B. At time t = 1, the Investment bank, I offers each of the firms to buy from them at a prespecified premium put options written on the firms stock. At time t = 2, based on their 3 The assumption that the firms are facing a single investment bank for that deal is reasonable for two reasons. First, there is usually a longterm relationship between the investment bank and the firm. It is costly for both of them to build a new relationship. Second, as evident in Gyoshev and Tsetsekos (2000), the option contracts are privately negotiated with nonstandard features like long maturity and European style exercise. These types of options are not traded on organized exchanges. 4 The results of this paper can be easily extended for a continuum of firm types. For expositional simplicity we focus only on two firm types. 6
8 type and the size of the premium, the firms agree or disagree to sell put options to the investment bank. Last, at time t = 3, the Investment bank infers firm type from the actions of the firms at time t = 2, and trades on this legally acquired private information. The extensive form of the game between the investment bank I and the two types of firms A and B is shown in Figure 1. Information At time t = 0, the investment bank has no private information about the future stock price of the firms. In other words, the investment bank cannot distinguish firm type. At time t = 0, type A firms have private information that their stock price will go up. In general, type A firms are more optimistic than the other market participants, including the investment bank, about their future performance. Or, the firms know that they are less risky than the market expects given the available public information. Type B firms, on the other hand, have a private signal that their future price will go down or that they are more risky than the market expectations. At time t = 2, if the bank has proposed the optimal premium offer and has ensured the existence of a separating equilibrium, the bank infers firm type and learns which firms will have a stock increase (or are less risky) and which will not. Payoffs The payoffs for the firms from participating in the game are denoted by P A for firm A, and P B for firm B, where the following is true: P P [ 0, Pr emium ( Value Type A) ] A = put = max (1) [ 0, Pr emium ( Value Type B) ] B = put = max (2) 7
9 Both firms will get 0 if they don t agree to sell put options to the investment bank. Therefore, the firms will agree to the terms of the investment bank only if the premium they will get is larger than the value of the put option computed given their private information. P I The payoff for the bank is denoted by P I, where the following is true: [ 0, Pr emium E( Value )] E( V separating _ equilibriu m) = min + (3) put I The first term of the investment bank payoff is the negative of the payoffs for the firms, because the sale of put options is a zero sum game. The nature of the second term underlines the main contribution of this paper. This is the value of information that the bank can infer about firm type if there is a unique separating equilibrium in the game. If there is a pooling equilibrium in the game, the bank cannot infer firm type. The second term then is equal to 0, and the model reduces to the classical adverse selection model of Akerlof (1970), where the bank as an uninformed party will be facing negative expected profits from participating in the trade. The monetary gains V I from acquiring private information can potentially be very large. If the bank knows what firms have a positive signal about their future prospects, the bank can purchase call options on these firms stock, sell put options, or buy their stock that is currently undervalued. In Section III.C., we illustrate the potential value of private information V I with two numerical examples. III.B. Resulting equilibrium The investment bank will engage in the put option sale only if it assures the existence and uniqueness of a separating equilibrium, where firms of type A accept the conditions of the sale, and firms of type B reject the contract. If the separating 8
10 equilibrium exists and it is unique, then the bank acquires private information about firm types and the second term, V I in (3) is positive. If the derivatives or stock markets of the firms are liquid enough, the value of V I will dominate over the negative adverse selection term, and the bank will earn positive profits from the transaction. Below we construct a feasible strategy for the investment bank that ensures a unique separating equilibrium. Let s assume that the following condition about firm type is true for any price > 0: price price ( price) dprice > f ( price) dprice f ( price) f A P > 0 0 price 0 B dprice (4) Where f A (price) and f B (price) are the probability distribution functions (p.d.f) of the prices of firms type A and firms type B, and f P (price) is the unconditional p.d.f. of the price of the average firm, given that the public cannot distinguish firm types. The interpretation of this assumption is that the firms of type A are with better than average prospects and it is more likely for them to have higher stock prices in the future than firms of type B. The type of the firm is private information. The rest of the market has an unconditional cumulative distribution of the future stock price of the average firm that in a stochastic sense is dominated by the distribution of the firm type A, and dominates the distribution of firm type B. Consider the following strategy for I at time t = 1: Propose to every firm that has a highly liquid market in derivatives to buy Europeanstyle outofthe money put options with a long maturity for a put premium that is equal to: Strike Put _ premium = ( Strike price) * f ( price) dprice (5) 0 P 9
11 The interpretation of equation (5) is that investment bank offers a fair price for the put options given the public information that all market participants have about the future distribution of stock prices. Necessary conditions for a separating equilibrium We have to show that the above proposed strategy of the bank leads to a unique separating equilibrium. In order for a separating equilibrium to exist and be unique, the following sets of individual rationality and incentive compatibility constraints have to be satisfied for both firm types: IR(A): P A 0 IR(B): P B 0 IC(A): P A The payoff for a type A firm if it pretends to be a type B firm IC(B): P B The payoff for a type B firm if it pretends to be a type A firm Given the stochastic dominance condition (4), it turns out that: A) The individual rationality constraint for firm type A coincides with the incentive compatibility constraint for firm type A, and both reduce to the following inequality: Strike Put _ premium ( Strike price)* f ( price) dprice (6) 0 A This condition directly follows from the description of I s strategy (5), and condition (4). B) The individual rationality constraint for firm type B coincides with the incentive compatibility constraint for firm type B, and both reduce to the following inequality: 10
12 Strike Put _ premium < ( Strike price)* f ( price) dprice (7) 0 B Similar to A) this condition directly follows from the description of I s strategy (5), and condition (4). The strategy of the bank to propose a take it or leave it offer to buy put options for a premium equal to the expression in (5) assures that only firms of type A will agree to sell options to the bank. Firms of type A have positive private information about their future performance. The true value of the put options computed using their private information is lower than the premium proposed by the bank. Therefore, firms of type A will accept the proposal by the bank and earn positive profits. On the other hand, firms of type B have private information that their performance will be less than average. The value of the put option computed using their private information will be higher than the premium proposed by the bank and all firms of type B will not accept the terms of the investment bank. As a consequence, the separating equilibrium of the game exists and it is unique. When the bank sees that a firm accepts its terms, the bank can immediately update its beliefs that this firm is a firm of type A, and later use this information to earn profits trading in other options of the same firm. III.C. Numerical Examples In this subsection we present two numerical examples that illustrate how the investment bank can earn informational rents after losing money trading in put options with the issuing firms. 11
13 Example 1. The difference between firm types is based on future stock returns 5 This example uses a binomial optionpricing model to show the value of information about future price changes of the stocks of the two firms. Suppose there are only two future states of nature, a good and a bad state. Let type A firms have a payoff of 120 in the good state and a payoff of 60 in the bad state. Type B firms have a payoff of 100 in the good state and a payoff of 40 in the bad state. If we assume for simplicity that there is an equal number of firms of both types, then the expected payoff of a firm of unknown type is 110 in the good state and 50 in the bad state. Let the stock price of the average firm to be 80, and the rate on Tbills (the riskfree security) to be 5%. Now, the investment bank offers to each firm to buy put options with a strike of 65. The payoff of this put option given the public information is 0 is the good state and 15 in the bad state. The put option price computed using only public information is then $ Both firms know their type and therefore they know for sure the true value of the put option for them. The value of the put for firm type A is $2.86, while the value of the put for firm type B is $6.35. As a result, only firms of type A will agree to sell put options and the bank will lose on this trade 3.49 per option. After the losing trade, the bank learns what firms are type A, and what firms are type B. After acquiring this private information that the rest of the market does not have, the bank buys a call option with a strike of 100, 5 See Lo and Wang (1995) for a sophisticated option pricing model that incorporates information about future returns. 6 See, for example, page 662 of Bodie, Kane, and Marcus (1999) for an exposition how to price options using the binomial pricing model 12
14 written on the stock of a type A firm. The true value of this call option is $7.62, while the bank can buy it from an uninformed investor for only $5.40 (the fair price given public information). The bank makes a profit of $2.22 per option. In order to make positive profits from the whole transaction, the bank needs to make sure that it buys at least 1.57 times more call options from the market than the number of put options that it bought from the issuing firms. For example if the bank proposes to buy 100 put option contracts (10,000 options) from one firms, and the firm agrees. Then, the bank can buy 200 call option contracts from the market, and make a total profit of: 20,000* ,000*3.49 = $9500 The bank can continue buying additional call options until the rest of the market participants detect the abnormal trading, and update their information about firm type. At this point the call option price will rise to its fair value of $7.62, and the informational rents for the bank will disappear. Example 2. The difference between firm types is based on the volatility of returns This example will illustrate how the investment bank can earn profits if it learns valuable information about the volatility of stock returns of a firm. Suppose that there are two types of firms. Firms of type A that have a standard deviation of returns σ A = 0.2, while for firms of type B, σ B = 0.6. If there are an equal number of firms of type A and type B, then E(σ) = 0.4. Assume that the current stock price of both firm types is $50, and the riskfree rate is 5%. 13
15 The investment bank offers every firm to buy put options written on the firm s stock with a maturity of one year and a strike price of $45. The price of these put options, given the publicly available information and using the Black and Scholes (1973) formula is $4.30. The firms know their type and they can compute the true value of this option. The true value of the option of firms of type A is $1.12, while for firms of type B it is $7.70. If the bank offers a price of $4.30 to all firms, only firm of type A will agree to sell. The bank loses $3.18 per option. Let the bank buy 100 contracts from one firm, and incur a total loss of $31,800. Now, the bank knows that this firm is of type A, and has the low volatility of 0.2. The rest of the market thinks that this firm on average has a volatility of 0.4. The bank then sells month put contracts with the same strike price of $45 at the market price of $1.614, while the true price of the puts is $ The bank makes a profit on the sale of these shortterm puts of 50,000*( ) = $66,900. The net gain from the transaction is then 66,900 31,800 = $35,
16 IV. Empirical predictions of the model The theoretical model presented in Section III produces several predictions, which are outlined below. The identity of traders on capital markets in the USA is not publicly available information. Therefore, it is impossible to document the actual trades of the investment banks after they have bough put options from the firms. We have to focus on indirect predictions about informed trading around the date of the put option sale. Prediction 1. Firms that have sold put options have more liquid option trading markets. Black (1975) and Easley, O Hara, and Srinivas (1998) suggest that informed traders prefer to profit on their information by trading in options as opposed to stocks, because derivatives allow for higher leverage. On the other hand, transaction costs like the bidask spread are on average several times larger when trading in options compared to trading in stocks. Informed traders then will prefer to trade in firms with more liquid derivative markets. In such firms an informed party can make more money before the other participants in the market detect her activities. Therefore, in order to assure that the informational rent earned by the bank are high enough, the bank will propose to buy options only from firms that have a highly liquid derivative market, with a large daily trading volume and a significant amount of uninformed trading. This prediction of the model can be tested by comparing the liquidity of the derivative markets for a sample firms that have recently sold put options and a sample of control firms that are with a similar size, are from the same industry, and have recently initiated market stock buybacks. 15
17 Prediction 2. There is an abnormal volume of trading in the option market around the date of the put option sale. If the investment bank acquires private information about firm type, it will proceed and buy a large number of call options or sell a large number of put options. A test of the model will then involve documenting the change in trading volume in various types of options after the date of the put option sale. Cao, Chen, and Griffin (2000) implement a similar approach to detect informed trading before takeover announcements. In ongoing work we intend to apply their methodology to provide empirical evidence on Prediction 2 of our model. Prediction 3. The Option market of firms that have sold put options leads the stock market in terms of information. Another indirect way to detect informed trading is to look at the relationship of option and stock prices. If informed investors prefer to trade on the option market, the option prices will reveal information before the stock prices. As a result option prices will lead stock prices. There are various timeseries test that can be utilized to document this relationship. A recent paper by Pirinski (2000) uses an extension of a VAR model to test whether option prices changes can predict stock prices changes before earnings announcements. A similar approach can be taken here to establish that there is an increase in informed trading in firms that have recently sold put options. 16
18 Prediction 4. The time from the sale of put options to its disclosure to the public will be inversely related to the size of the expected losses of the investment bank from the trade and to the liquidity of the firm stock and publicly traded options. This prediction relies on the anecdotal evidence that firms cooperate with the investment bank in the trade and on the fact that SEC does not require immediate disclosure of derivative transactions by firms. The bank will be willing to incur more losses when buying put options from the managers if it can assure that there will be enough time to recover those losses through informed trading. This implies that if firm managers want to maximize the proceeds from the sale, they will have to agree to postpone the public disclosure of the trade. Also, similar to Prediction 1, the bank can make larger profits faster in firms with more liquid securities. In such firms, it is not necessary for the managers to postpone the disclosure of the put option transaction. V. Conclusion This study provides an explanation why an investment bank may agree to purchase from a company longterm put options written on the stock of the company. The issuer is more informed about the payoff of the put options. Therefore, the investment bank has ex ante negative expected profits from this trade. Our model demonstrates that the investment bank is behaving in a perfectly rational manner, because the information gained from this loss making activity can be used to make larger profits in trading with other uninformed parties. The idea that an uninformed player may engage in negative profit interactions in order to gain valuable information from its informed counterparties has been largely 17
19 ignored by the extensive literature modeling asymmetric information. Nevertheless, this notion potentially has many applications beyond the analyzed by this study case where an investment bank offers firms to buy put options written on their stock. For example, marketing companies incur costs to offer free subscriptions for various specialized magazines to a large pool of consumers. From the selections chosen by the consumers the marketing companies learn valuable information about consumer types. This information is used afterwards for promotional purposes. Also, banks with large credit card portfolios sell high interest rate credit cards to extremely risky clients, although they expect to lose on these sales. Banks engage in these transactions in order to test the completeness of their risk evaluation models. After improving their models, the banks are able to price risk better and avoid heavy losses due to model error. One direction in which the model can be extended is as follows. If the distribution of firm quality (or risk) is continuous instead of binomial, it may be optimal for I to offer each firm a premium for the put options that is smaller than the fair premium computed using public information. If the premium is lower than the fair premium, only the firms with the best prospects will accept to sell put options. Thus, when I decreases the premium, it reduces the set of firms that will accept to trade and simultaneously increases the average quality of the accepting firms. The optimal premium the bank has to balance these two effects and maximize the expected profits of the bank 7. The solution for the optimal premium that the investment bank should propose given a certain distribution of firm values is left for future work. 7 This maximization problem is similar to the problem of choosing the optimal bids in an auction a lower bid decreases the probability of winning but increases the profits if it wins. 18
20 References Akerlov, George, 1970, The market for lemons: Quality uncertainty and the market mechanism, Quarterly Journal of Economics, 89, Black, Fisher, 1975, Fact and fantasy in use of options, Financial Analyst Journal, 31, and Black, Fisher, and Myron Scholes, 1973, The pricing of options and corporate liabilities, Journal of Political Economy, 81, Bodie, Zvi, Alex Kane, and Alan Marcus, 1999, Investments, 4 th Edition, Irvin/McGraw Hill, Cao, Charles, Zhiwu Chen, and John Griffin, 2000, The informational content of option volume prior to takeovers, Working paper Easley, David, Maureen O Hara, and P.S. Srinivas, 1998, Option volume and stock prices: Evidence on where informed traders trade, Journal of Finance, 53, Gibson, Scott, and Raj Singh, 2000, Using put warrants to reduce corporate financing costs, Working paper, University of Minnesota Gyoshev, Stanley B. and George P. Tsetsekos, 2000, Enhancing share repurchase programs: Use of firm s own stock as an underlying asset in issuing put derivatives, Drexel University Working Paper Series Lo, Andrew, and Jiang Wang, 1995, Implementing option pricing models when asset returns are predictable, Journal of Finance, 50, Pirinski, Christo, 2000, Do equity options lead underlying stocks?, Working paper 19
21 Rotschild, M., J.E. Stiglitz, 1976, Equilibrium in competitive insurance markets: An essay in the economics of imperfect information, Quarterly Journal of Economics, 80, Wilson, C., 1977, A model of insurance markets with incomplete information, Journal of Economic Theory, 16,
22 0. Nature picks type A or B A B 1. IB offers to buy Puts 2. Firms decide to agree or not Agree Disagree Agree Disagree 3. Bank decides whether to trade on Information Trade Not Trade Trade Not Trade Trade Not Trade Trade Not Trade 4. Payoffs (P I 1, P A 1) (P I 2, P A 1) (P I 3, 0) (0, 0) (P I 5, P B 1) (P I 6, P B 1) (P I 7, 0) (0, 0) Figure 1. Extensive Form of the Game Played by the Investment Bank and the Firms Repurchasing Stocks The description of the information, players and payoffs is in Section II. From inequality (4) and equation (5) follows that P A 1>0, and P B 1<0. This ensures that firms of type B will always disagree, while firms of type A will always agree to sell put options to the investment bank. Because this separating equilibrium is unique, the bank acquires private information about firm type, and then the payoffs for the bank that matter P I 1, P I 7 are both greater than 0. 21
FINANCIAL ECONOMICS OPTION PRICING
OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the
More informationFundamentals of Futures and Options (a summary)
Fundamentals of Futures and Options (a summary) Roger G. Clarke, Harindra de Silva, CFA, and Steven Thorley, CFA Published 2013 by the Research Foundation of CFA Institute Summary prepared by Roger G.
More informationTwoState Options. John Norstad. jnorstad@northwestern.edu http://www.norstad.org. January 12, 1999 Updated: November 3, 2011.
TwoState Options John Norstad jnorstad@northwestern.edu http://www.norstad.org January 12, 1999 Updated: November 3, 2011 Abstract How options are priced when the underlying asset has only two possible
More informationChapter 20 Understanding Options
Chapter 20 Understanding Options Multiple Choice Questions 1. Firms regularly use the following to reduce risk: (I) Currency options (II) Interestrate options (III) Commodity options D) I, II, and III
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008. Options
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes describe the payoffs to European and American put and call options the socalled plain vanilla options. We consider the payoffs to these
More informationMarket Microstructure: An Interactive Exercise
Market Microstructure: An Interactive Exercise Jeff Donaldson, University of Tampa Donald Flagg, University of Tampa ABSTRACT Although a lecture on microstructure serves to initiate the inspiration of
More informationDerivative Users Traders of derivatives can be categorized as hedgers, speculators, or arbitrageurs.
OPTIONS THEORY Introduction The Financial Manager must be knowledgeable about derivatives in order to manage the price risk inherent in financial transactions. Price risk refers to the possibility of loss
More informationChapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO ENDOFCHAPTER QUESTIONS
Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO ENDOFCHAPTER QUESTIONS 81 a. An option is a contract which gives its holder the right to buy or sell an asset at some predetermined
More informationHedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging
Hedging An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in
More information9 Questions Every ETF Investor Should Ask Before Investing
9 Questions Every ETF Investor Should Ask Before Investing 1. What is an ETF? 2. What kinds of ETFs are available? 3. How do ETFs differ from other investment products like mutual funds, closedend funds,
More informationBlackScholesMerton approach merits and shortcomings
BlackScholesMerton approach merits and shortcomings Emilia Matei 1005056 EC372 Term Paper. Topic 3 1. Introduction The BlackScholes and Merton method of modelling derivatives prices was first introduced
More informationAnswers to Concepts in Review
Answers to Concepts in Review 1. Puts and calls are negotiable options issued in bearer form that allow the holder to sell (put) or buy (call) a stipulated amount of a specific security/financial asset,
More informationFinance 400 A. Penati  G. Pennacchi Market MicroStructure: Notes on the Kyle Model
Finance 400 A. Penati  G. Pennacchi Market MicroStructure: Notes on the Kyle Model These notes consider the singleperiod model in Kyle (1985) Continuous Auctions and Insider Trading, Econometrica 15,
More informationFactors Affecting Option Prices
Factors Affecting Option Prices 1. The current stock price S 0. 2. The option strike price K. 3. The time to expiration T. 4. The volatility of the stock price σ. 5. The riskfree interest rate r. 6. The
More informationBuying Call or Long Call. Unlimited Profit Potential
Options Basis 1 An Investor can use options to achieve a number of different things depending on the strategy the investor employs. Novice option traders will be allowed to buy calls and puts, to anticipate
More informationOptions Pricing. This is sometimes referred to as the intrinsic value of the option.
Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the PutCall Parity Relationship. I. Preliminary Material Recall the payoff
More informationLecture Notes: Basic Concepts in Option Pricing  The Black and Scholes Model
Brunel University Msc., EC5504, Financial Engineering Prof Menelaos Karanasos Lecture Notes: Basic Concepts in Option Pricing  The Black and Scholes Model Recall that the price of an option is equal to
More informationNine Questions Every ETF Investor Should Ask Before Investing
Nine Questions Every ETF Investor Should Ask Before Investing UnderstandETFs.org Copyright 2012 by the Investment Company Institute. All rights reserved. ICI permits use of this publication in any way,
More informationFutures Price d,f $ 0.65 = (1.05) (1.04)
24 e. Currency Futures In a currency futures contract, you enter into a contract to buy a foreign currency at a price fixed today. To see how spot and futures currency prices are related, note that holding
More informationGeneral Forex Glossary
General Forex Glossary A ADR American Depository Receipt Arbitrage The simultaneous buying and selling of a security at two different prices in two different markets, with the aim of creating profits without
More informationBlack Scholes Merton Approach To Modelling Financial Derivatives Prices Tomas Sinkariovas 0802869. Words: 3441
Black Scholes Merton Approach To Modelling Financial Derivatives Prices Tomas Sinkariovas 0802869 Words: 3441 1 1. Introduction In this paper I present Black, Scholes (1973) and Merton (1973) (BSM) general
More informationAnswers to Concepts in Review
Answers to Concepts in Review 1. (a) In the money market, shortterm securities such as CDs, Tbills, and banker s acceptances are traded. Longterm securities such as stocks and bonds are traded in the
More informationReview of Basic Options Concepts and Terminology
Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some
More informationReview for Exam 1. Instructions: Please read carefully
Review for Exam 1 Instructions: Please read carefully The exam will have 21 multiple choice questions and 5 work problems. Questions in the multiple choice section will be either concept or calculation
More informationContents. 2 What are Options? 3 Ways to use Options. 7 Getting started. 8 Frequently asked questions. 13 Contact us. 14 Important Information
Options For individuals, companies, trusts and SMSFs The Options and Lending Facility Contents 2 What are Options? 3 Ways to use Options 7 Getting started 8 Frequently asked questions 13 Contact us 14
More informationFAQ. (Continued on page 2) An Investment Advisory Firm
FAQ An Investment Advisory Firm What is QASH Flow Advantage? It is a timetested model that includes three strategic components: A portfolio of carefully selected ExchangeTraded Funds (ETFs) for diversification
More informationFIN 432 Investment Analysis and Management Review Notes for Midterm Exam
FIN 432 Investment Analysis and Management Review Notes for Midterm Exam Chapter 1 1. Investment vs. investments 2. Real assets vs. financial assets 3. Investment process Investment policy, asset allocation,
More information1 Introduction to Option Pricing
ESTM 60202: Financial Mathematics Alex Himonas 03 Lecture Notes 1 October 7, 2009 1 Introduction to Option Pricing We begin by defining the needed finance terms. Stock is a certificate of ownership of
More informationChapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.
Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of riskadjusted discount rate. Part D Introduction to derivatives. Forwards
More informationIntroduction to Equity Derivatives on Nasdaq Dubai NOT TO BE DISTRIUTED TO THIRD PARTIES WITHOUT NASDAQ DUBAI S WRITTEN CONSENT
Introduction to Equity Derivatives on Nasdaq Dubai NOT TO BE DISTRIUTED TO THIRD PARTIES WITHOUT NASDAQ DUBAI S WRITTEN CONSENT CONTENTS An Exchange with Credentials (Page 3) Introduction to Derivatives»
More informationInformed Trading around corporate event announcements: Stock versus Options
Informed Trading around corporate event announcements: Stock versus Options Tanguy de Launois 1 and Hervé Van Oppens 2 9th May 2003 1 Université Catholique de Louvain, 1 place des Doyens, 1348 LouvainlaNeuve
More informationFinancial Markets. Itay Goldstein. Wharton School, University of Pennsylvania
Financial Markets Itay Goldstein Wharton School, University of Pennsylvania 1 Trading and Price Formation This line of the literature analyzes the formation of prices in financial markets in a setting
More informationCommitment to Overinvest and Price Informativeness
Commitment to Overinvest and Price Informativeness James Dow Itay Goldstein Alexander Guembel London Business University of University of Oxford School Pennsylvania FMG financial stability conference,
More informationChapter 7. Sealedbid Auctions
Chapter 7 Sealedbid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)
More informationINVESTMENT DICTIONARY
INVESTMENT DICTIONARY Annual Report An annual report is a document that offers information about the company s activities and operations and contains financial details, cash flow statement, profit and
More informationStapled Finance. Paul Povel and Raj Singh. Bauer College of Business, Univ. of Houston Carlson School of Management, Univ.
Paul Povel and Raj Singh Bauer College of Business, Univ. of Houston Carlson School of Management, Univ. of Minnesota What Is? What is? From the Business Press Explanations Acquisition financing, prearranged
More informationt = 1 2 3 1. Calculate the implied interest rates and graph the term structure of interest rates. t = 1 2 3 X t = 100 100 100 t = 1 2 3
MØA 155 PROBLEM SET: Summarizing Exercise 1. Present Value [3] You are given the following prices P t today for receiving risk free payments t periods from now. t = 1 2 3 P t = 0.95 0.9 0.85 1. Calculate
More informationCompare and Contrast of Option Decay Functions. Nick Rettig and Carl Zulauf *,**
Compare and Contrast of Option Decay Functions Nick Rettig and Carl Zulauf *,** * Undergraduate Student (rettig.55@osu.edu) and Professor (zulauf.1@osu.edu) Department of Agricultural, Environmental, and
More informationStrategic Equity Investments
Strategic Equity Investments Wojciech Grabowski, Assistant Professor, Department of Economics, University of Warsaw I take a closer look at recent investment results of two major corporations in the technology
More informationMargin Requirements & Margin Calls
Margin Requirements & Margin Calls Dr. Patrick Toche References : Zvi Bodie, Alex Kane, Alan J. Marcus. Essentials of Investment. McGraw Hill Irwin. Chapter 3 of the BodieKaneMarcus textbook will be
More information11 Option. Payoffs and Option Strategies. Answers to Questions and Problems
11 Option Payoffs and Option Strategies Answers to Questions and Problems 1. Consider a call option with an exercise price of $80 and a cost of $5. Graph the profits and losses at expiration for various
More informationRisks involved with futures trading
Appendix 1: Risks involved with futures trading Before executing any futures transaction, the client should obtain information on the risks involved. Note in particular the risks summarized in the following
More informationECON4510 Finance Theory Lecture 7
ECON4510 Finance Theory Lecture 7 Diderik Lund Department of Economics University of Oslo 11 March 2015 Diderik Lund, Dept. of Economics, UiO ECON4510 Lecture 7 11 March 2015 1 / 24 Market efficiency Market
More informationMarkus K. Brunnermeier
Institutional tut Finance Financial Crises, Risk Management and Liquidity Markus K. Brunnermeier Preceptor: Dong BeomChoi Princeton University 1 Market Making Limit Orders Limit order price contingent
More informationFinancial Market Microstructure Theory
The Microstructure of Financial Markets, de Jong and Rindi (2009) Financial Market Microstructure Theory Based on de Jong and Rindi, Chapters 2 5 Frank de Jong Tilburg University 1 Determinants of the
More informationMargin Requirements & Margin Calls
Margin Requirements & Margin Calls Dr. Patrick Toche References : Zvi Bodie, Alex Kane, Alan J. Marcus. Essentials of Investment. McGraw Hill Irwin. Chapter 3 of the BodieKaneMarcus textbook will be
More informationInvesting on hope? Small Cap and Growth Investing!
Investing on hope? Small Cap and Growth Investing! Aswath Damodaran Aswath Damodaran! 1! Who is a growth investor?! The Conventional definition: An investor who buys high price earnings ratio stocks or
More informationApplied Economics For Managers Recitation 5 Tuesday July 6th 2004
Applied Economics For Managers Recitation 5 Tuesday July 6th 2004 Outline 1 Uncertainty and asset prices 2 Informational efficiency  rational expectations, random walks 3 Asymmetric information  lemons,
More informationOne Period Binomial Model
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing
More informationLEAPS LONGTERM EQUITY ANTICIPATION SECURITIES
LEAPS LONGTERM EQUITY ANTICIPATION SECURITIES The Options Industry Council (OIC) is a nonprofit association created to educate the investing public and brokers about the benefits and risks of exchangetraded
More informationChapter 2 An Introduction to Forwards and Options
Chapter 2 An Introduction to Forwards and Options Question 2.1. The payoff diagram of the stock is just a graph of the stock price as a function of the stock price: In order to obtain the profit diagram
More information6. Foreign Currency Options
6. Foreign Currency Options So far, we have studied contracts whose payoffs are contingent on the spot rate (foreign currency forward and foreign currency futures). he payoffs from these instruments are
More informationRené Garcia Professor of finance
Liquidity Risk: What is it? How to Measure it? René Garcia Professor of finance EDHEC Business School, CIRANO Cirano, Montreal, January 7, 2009 The financial and economic environment We are living through
More informationCHAPTER 21: OPTION VALUATION
CHAPTER 21: OPTION VALUATION 1. Put values also must increase as the volatility of the underlying stock increases. We see this from the parity relation as follows: P = C + PV(X) S 0 + PV(Dividends). Given
More informationImperfect information Up to now, consider only firms and consumers who are perfectly informed about market conditions: 1. prices, range of products
Imperfect information Up to now, consider only firms and consumers who are perfectly informed about market conditions: 1. prices, range of products available 2. characteristics or relative qualities of
More informationChap 3 CAPM, Arbitrage, and Linear Factor Models
Chap 3 CAPM, Arbitrage, and Linear Factor Models 1 Asset Pricing Model a logical extension of portfolio selection theory is to consider the equilibrium asset pricing consequences of investors individually
More informationEquilibrium in Competitive Insurance Markets: An Essay on the Economic of Imperfect Information
Equilibrium in Competitive Insurance Markets: An Essay on the Economic of Imperfect Information By: Michael Rothschild and Joseph Stiglitz Presented by Benjamin S. Barber IV, Xiaoshu Bei, Zhi Chen, Shaiobi
More informationShould Banks Trade Equity Derivatives to Manage Credit Risk? Kevin Davis 9/4/1991
Should Banks Trade Equity Derivatives to Manage Credit Risk? Kevin Davis 9/4/1991 Banks incur a variety of risks and utilise different techniques to manage the exposures so created. Some of those techniques
More informationWhen firms need to raise capital, they may issue securities to the public by investment bankers.
CHAPTER 3. HOW SECURITIES ARE TRADED When firms need to raise capital, they may issue securities to the public by investment bankers. Primary market is a market for new securities. Secondary market is
More informationBUSM 411: Derivatives and Fixed Income
BUSM 411: Derivatives and Fixed Income 2. Forwards, Options, and Hedging This lecture covers the basic derivatives contracts: forwards (and futures), and call and put options. These basic contracts are
More informationWhy Do Firms Announce OpenMarket Repurchase Programs?
Why Do Firms Announce OpenMarket Repurchase Programs? Jacob Oded, (2005) Boston College PhD Seminar in Corporate Finance, Spring 2006 Outline 1 The Problem Previous Work 2 3 Outline The Problem Previous
More informationCHAPTER 20. Financial Options. Chapter Synopsis
CHAPTER 20 Financial Options Chapter Synopsis 20.1 Option Basics A financial option gives its owner the right, but not the obligation, to buy or sell a financial asset at a fixed price on or until a specified
More informationIndex Options Beginners Tutorial
Index Options Beginners Tutorial 1 BUY A PUT TO TAKE ADVANTAGE OF A RISE A diversified portfolio of EUR 100,000 can be hedged by buying put options on the Eurostoxx50 Index. To avoid paying too high a
More informationChapter 15 OPTIONS ON MONEY MARKET FUTURES
Page 218 The information in this chapter was last updated in 1993. Since the money market evolves very rapidly, recent developments may have superseded some of the content of this chapter. Chapter 15 OPTIONS
More informationSummary of Interview Questions. 1. Does it matter if a company uses forwards, futures or other derivatives when hedging FX risk?
Summary of Interview Questions 1. Does it matter if a company uses forwards, futures or other derivatives when hedging FX risk? 2. Give me an example of how a company can use derivative instruments to
More informationOverlapping ETF: Pair trading between two gold stocks
MPRA Munich Personal RePEc Archive Overlapping ETF: Pair trading between two gold stocks Peter N Bell and Brian Lui and Alex Brekke University of Victoria 1. April 2012 Online at http://mpra.ub.unimuenchen.de/39534/
More informationI. Introduction. II. Financial Markets (Direct Finance) A. How the Financial Market Works. B. The Debt Market (Bond Market)
University of California, Merced EC 121Money and Banking Chapter 2 Lecture otes Professor Jason Lee I. Introduction In economics, investment is defined as an increase in the capital stock. This is important
More informationOptions: Valuation and (No) Arbitrage
Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial
More informationDiscussions of Monte Carlo Simulation in Option Pricing TIANYI SHI, Y LAURENT LIU PROF. RENATO FERES MATH 350 RESEARCH PAPER
Discussions of Monte Carlo Simulation in Option Pricing TIANYI SHI, Y LAURENT LIU PROF. RENATO FERES MATH 350 RESEARCH PAPER INTRODUCTION Having been exposed to a variety of applications of Monte Carlo
More informationLecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 7: Bounds on Options Prices Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Option Price Quotes Reading the
More informationLecture 4: Properties of stock options
Lecture 4: Properties of stock options Reading: J.C.Hull, Chapter 9 An European call option is an agreement between two parties giving the holder the right to buy a certain asset (e.g. one stock unit)
More informationMarket Efficiency and Behavioral Finance. Chapter 12
Market Efficiency and Behavioral Finance Chapter 12 Market Efficiency if stock prices reflect firm performance, should we be able to predict them? if prices were to be predictable, that would create the
More informationSolutions for EndofChapter Questions and Problems: Chapter Five
Solutions for EndofChapter Questions and Problems: Chapter Five 2. What are money market mutual funds? In what assets do these funds typically invest? What factors have caused the strong growth in this
More informationwww.optionseducation.org OIC Options on ETFs
www.optionseducation.org Options on ETFs 1 The Options Industry Council For the sake of simplicity, the examples that follow do not take into consideration commissions and other transaction fees, tax considerations,
More informationOption pricing. Vinod Kothari
Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate
More informationSession IX: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics
Session IX: Stock Options: Properties, Mechanics and Valuation Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Stock
More informationWho Should Consider Using Covered Calls?
Who Should Consider Using Covered Calls? An investor who is neutral to moderately bullish on some of the equities in his portfolio. An investor who is willing to limit upside potential in exchange for
More informationFor example, someone paid $3.67 per share (or $367 plus fees total) for the right to buy 100 shares of IBM for $180 on or before November 18, 2011
Chapter 7  Put and Call Options written for Economics 104 Financial Economics by Prof Gary R. Evans First edition 1995, this edition September 24, 2011 Gary R. Evans This is an effort to explain puts
More informationPorter, White & Company
Porter, White & Company Optimizing the Fixed Income Component of a Portfolio White Paper, September 2009, Number IM 17.2 In the White Paper, Comparison of Fixed Income Fund Performance, we show that a
More informationDESCRIPTION OF FINANCIAL INSTRUMENTS AND INVESTMENT RISKS
DESCRIPTION OF FINANCIAL INSTRUMENTS AND INVESTMENT RISKS A. General The services offered by Prochoice Stockbrokers cover a wide range of Financial Instruments. Every type of financial instrument carries
More informationStock Prices and Business Investment
Stock Prices and Business Investment BY YARON LEITNER I s there a link between the stock market and business investment? Empirical evidence indicates that there is. A firm tends to invest more when its
More informationGAMMA.0279 THETA 8.9173 VEGA 9.9144 RHO 3.5985
14 Option Sensitivities and Option Hedging Answers to Questions and Problems 1. Consider Call A, with: X $70; r 0.06; T t 90 days; 0.4; and S $60. Compute the price, DELTA, GAMMA, THETA, VEGA, and RHO
More informationFinance 2 for IBA (30J201) F. Feriozzi Resit exam June 18 th, 2012. Part One: MultipleChoice Questions (45 points)
Finance 2 for IBA (30J201) F. Feriozzi Resit exam June 18 th, 2012 Part One: MultipleChoice Questions (45 points) Question 1 Assume that capital markets are perfect. Which of the following statements
More informationOption pricing in detail
Course #: Title Module 2 Option pricing in detail Topic 1: Influences on option prices  recap... 3 Which stock to buy?... 3 Intrinsic value and time value... 3 Influences on option premiums... 4 Option
More informationCHAPTER 1: INTRODUCTION, BACKGROUND, AND MOTIVATION. Over the last decades, risk analysis and corporate risk management activities have
Chapter 1 INTRODUCTION, BACKGROUND, AND MOTIVATION 1.1 INTRODUCTION Over the last decades, risk analysis and corporate risk management activities have become very important elements for both financial
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationStock Market Dashboard BackTest October 29, 1998 March 29, 2010 Revised 2010 Leslie N. Masonson
Stock Market Dashboard BackTest October 29, 1998 March 29, 2010 Revised 2010 Leslie N. Masonson My objective in writing Buy DON T Hold was to provide investors with a better alternative than the buyandhold
More informationGeneral Risk Disclosure
General Risk Disclosure Colmex Pro Ltd (hereinafter called the Company ) is an Investment Firm regulated by the Cyprus Securities and Exchange Commission (license number 123/10). This notice is provided
More informationFIN 500R Exam Answers. By nature of the exam, almost none of the answers are unique. In a few places, I give examples of alternative correct answers.
FIN 500R Exam Answers Phil Dybvig October 14, 2015 By nature of the exam, almost none of the answers are unique. In a few places, I give examples of alternative correct answers. Bubbles, Doubling Strategies,
More informationComplex Products. NonComplex Products. General risks of trading
We offer a wide range of investments, each with their own risks and rewards. The following information provides you with a general description of the nature and risks of the investments that you can trade
More informationHow to Win the Stock Market Game
How to Win the Stock Market Game 1 Developing ShortTerm Stock Trading Strategies by Vladimir Daragan PART 1 Table of Contents 1. Introduction 2. Comparison of trading strategies 3. Return per trade 4.
More information2. How is a fund manager motivated to behave with this type of renumeration package?
MØA 155 PROBLEM SET: Options Exercise 1. Arbitrage [2] In the discussions of some of the models in this course, we relied on the following type of argument: If two investment strategies have the same payoff
More informationCompany Fundamentals. THE CMC Markets Trading Smart Series
Company Fundamentals THE CMC Markets Trading Smart Series How to evaluate company growth potential At any given point in time, share prices tend to represent the sum of expectations about its value from
More informationCurrency Options. www.mx.ca
Currency Options www.mx.ca Table of Contents Introduction...3 How currencies are quoted in the spot market...4 How currency options work...6 Underlying currency...6 Trading unit...6 Option premiums...6
More information15.401 Finance Theory
Finance Theory MIT Sloan MBA Program Andrew W. Lo Harris & Harris Group Professor, MIT Sloan School Lectures 10 11 11: : Options Critical Concepts Motivation Payoff Diagrams Payoff Tables Option Strategies
More informationHedging With a Stock Option
IOSR Journal of Business and Management (IOSRJBM) eissn: 2278487X, pissn: 23197668. Volume 17, Issue 9.Ver. I (Sep. 2015), PP 0611 www.iosrjournals.org Hedging With a Stock Option Afzal Ahmad Assistant
More informationStrategies in Options Trading By: Sarah Karfunkel
Strategies in Options Trading By: Sarah Karfunkel Covered Call Writing: I nvestors use two strategies involving stock options to offset risk: (1) covered call writing and (2) protective puts. The strategy
More informationChapter 7: Option pricing foundations Exercises  solutions
Chapter 7: Option pricing foundations Exercises  solutions 1. (a) We use the putcall parity: Share + Put = Call + PV(X) or Share + Put  Call = 97.70 + 4.16 23.20 = 78.66 and P V (X) = 80 e 0.0315 =
More information