A derivative is an instrument whose value - PowerPoint Presentation

1. A derivative is an instrument whose value is derived from the value of another asset.Why Derivatives Are Important?Derivatives play a key role in transferring risks in the economy - HedgingThe underlying assets include stocks, currencies, interest rates, commodities, debt, electricity, the weather, etcMany financial transactions have embedded derivativesThe real options approach to assessing capital investment decisions has become widely accepted1

2. How Derivatives Are TradedOn exchanges such as the Chicago Board Options ExchangeIn the over-the-counter (OTC) market where traders working for banks, fund managers and corporate treasurers contact each other directly2

3. Size of OTC vs Exchange-Traded Markets3Source: Bank for International Settlements. Chart shows total principal amounts for OTC market and value of underlying assets for exchange market

4. Forward PriceThe forward price for a contract is the delivery price that would be applicable to the contract if it were negotiated today (i.e., it is the delivery price that would make the contract worth exactly zero)The forward price may be different for contracts of different maturities (as shown by the table)4

5. 5Euro Spot Rate as of 9/30/19 = 1.09Questions:US Company sold products in Italy. Will be paid 1M Euros in JanuaryUS Company wants to open a 2M euro factory in Italy in MarchFerrari sold $10M worth of cars for November delivery

6. Euro Forwards (5/22/12) Spot= 1.2766Delivery Date

7. Euro Forwards Curve7

8. TerminologyThe party that has agreed to buy has what is termed a long position. They are the holder.The party that has agreed to sell has what is termed a short position. They are the writer.8

9. ExampleOn January 1st the treasurer of a corporation enters into a long forward contract to buy £1 million in six months at an exchange rate of 1.4422 USD/GBPThis obligates the corporation to pay $1,442,200 for £1 million on July 1st What are the possible outcomes?9

10. Profit from a Long Forward Position (K= delivery price, which is the forward price at time contract is entered into)10ProfitPrice of Underlying at Maturity, STK

11. 11ProfitPrice of Underlying at Maturity, STKProfit from a Short Forward Position (K= delivery price, which is the forward price at time contract is entered into)

12. Futures ContractsAgreement to buy or sell an asset for a certain price at a certain timeSimilar to forward contractOne difference a forward contract is traded OTC a futures contract is traded on an exchange12

13. Exchanges Trading FuturesCME Group (formerly Chicago Mercantile Exchange and Chicago Board of Trade)NYSE EuronextBM&F (Sao Paulo, Brazil)TIFFE (Tokyo)and many more (see list at end of book)13

14. Examples of Futures ContractsAgreement to:Buy 100 oz. of gold @ US$1400/oz. in December Sell £62,500 @ 1.4500 US$/£ in MarchSell 1,000 bbl. of oil @ US$90/bbl. in April14* All futures have different contract sizes

15. 1. Stock: An Arbitrage Opportunity?Suppose that:The price of Exelon Stock is $60Exelon does not pay a dividendThe 1-year forward price of Exelon Stock is $63The 1-year US interest rate is 5%Is there an arbitrage opportunity? 15

16. 1. Stock: An Arbitrage Opportunity?What would you do if:The price of Exelon Stock is $60Exelon does not pay a dividendThe 1-year forward price of Exelon Stock is $67The 1-year US interest rate is 5%Is there an arbitrage opportunity?What if the forward were $58 and you already own the stock?16

17. 2. Gold: Another Arbitrage Opportunity?Suppose that:The spot price of gold is US$1,400The 1-year forward price of gold is US$1,500The 1-year US interest rate is 5% Is there an arbitrage opportunity?17

18. The Forward Price of Gold If the spot price of gold is S and the forward price for a contract deliverable in T years is F, then F = S (1+r )T where r is the 1-year (domestic currency) risk-free rate of interest. In our examples, S = 1400, T = 1, and r =0.05 so thatF = 1400(1+0.05) = 147018

19. 1. Oil: An Arbitrage Opportunity?Suppose that:The spot price of oil is US$95The 1-year futures price of oil is US$105The 1-year US$ interest rate is 5% The storage costs of oil are 2% per year Is there an arbitrage opportunity?19

20. OptionsThe owner of a call option has the right to buy a certain asset by a certain date for a certain price (the strike price or exercise price)Therefore, the seller of a call is obligated to sell.The owner of a put option has the right to sell a certain asset by a certain date for a certain price (the strike price or exercise price)Therefore, the seller of a put is obligated to buy.20

21. American vs European OptionsA European option can be exercised only at maturity An American option can be exercised at any time during its lifeA Bermudian option restricts early exercise to certain datesStock options: One contract is for 100 shares21

22. DuPont Stock Option Prices (DD was trading at $52.86)22

23. S&P 500 ETF (SPY) Option Prices (SPY was trading at $140.08)23

24. Option Payoffs at Expiration24Call Options A European call option on a stock with X=$40 a share at time = T. •If ST< $40 option is not exercised. Why? •If ST> $40 option is exercised.Put Options A European put option on a stock with X=$40 a share at time = T. •If ST> $40 option is not exercised. Why? •If ST< $40 option is exercised.

25. Options vs Futures/ForwardsA futures/forward contract gives the holder the obligation to buy or sell at a certain priceAn option gives the holder the right to buy or sell at a certain price25

26. Hedging ExampleA US company will pay £10 million for imports from Britain in 3 months and decides to hedge using a long position in a forward contract26

27. Arbitrage ExampleA stock is quoted at £100 on the London Stock Exchange and $140 on the NYSE.The current exchange rate is 1.4300What is the arbitrage opportunity?27

28. Mechanics of Futures Markets

29. Futures ContractsAvailable on a wide range of assetsExchange tradedSpecifications need to be defined:What can be delivered,Where it can be delivered, & When it can be deliveredSettled daily

30. Futures Contracts Exampleswww.cmegroup.com

31. Convergence of Futures to Spot (Figure 2.1, page 26) TimeTime(a)(b)FuturesPriceFuturesPriceSpot PriceSpot Price

32. Margin for futuresA margin is cash or marketable securities deposited by an investor with his or her brokerThe balance in the margin account is adjusted to reflect daily settlement“Mark-to-market”Margins minimize the possibility of a loss through a default on a contract

33. Example of a Futures Trade (page 27-29)An investor takes a long position in 2 December gold futures contracts on June 5contract size is 100 oz.futures price is US$1250initial margin requirement is $6,000 per contract (approx. 5% notional)maintenance margin is $4,500 per contract ($9,000 in total)

34. A Possible Outcome (Table 2.1, page 28)DayTrade Price ($)Settle Price ($)Daily Gain ($)Cumul. Gain ($)Margin Balance ($)Margin Call ($)11,250.0012,00011,241.00−1,800− 1,80010,20021,238.30 −540 −2,340 9,660…..…..…..…..……61,236.20 −780 −2,760 9,24071,229.90−1,260 −4,020 7,9804,02081,230.80 180 −3,84012,180…..…..…..…..……161,226.90 780 −4,62015,180

35. 6-month Coffee Futures = $2Contract Specification = 37,500 poundsInitial Margin = $7500Maintenance Margin = $5000 LongPRICEShort

36. Margin Cash Flows When Futures Price IncreasesLong TraderBrokerClearing HouseMember Clearing HouseClearing HouseMember BrokerShort Trader

37. Margin Cash Flows When Futures Price DecreasesLong TraderBrokerClearing HouseMember Clearing HouseClearing HouseMember BrokerShort Trader

38. Some TerminologyOpen interest: the total number of contracts outstanding equal to number of long positions or number of short positions. Does not double count.Settlement price: the price just before the final bell each day used for the daily settlement processVolume of trading: the number of trades in 1-dayNot the same as open interest which tracks open positions over timeNote: trades in derivatives must be labeled as “opening” or “closing” transactions. Without this specification, we would be unable to track open interest.

39. Crude Oil Trading on May 26, 2010OpenHighLowSettleChangeVolumeOpen IntJul 201070.0671.7069.2171.512.766,315388,902Aug 201071.2572.7770.4272.542.448,746115,305Dec 201074.0075.3473.1775.232.195,055196,033Dec 201177.0178.5976.5178.532.004,175100,674Dec 201278.5080.2178.5080.181.861,258 70,126

40. Key Points About FuturesThey are settled dailyMost futures trades do not last through the delivery date. Instead, they are closed out.Closing out a futures position involves entering into an offsetting trade

41. Collateralization in OTC MarketsIt is becoming increasingly common for transactions to be collateralized in OTC marketsConsider transactions between companies A and BThese might be governed by an ISDA Master agreement with a credit support annex (CSA)The CSA might require A to post collateral with B and/or vice versa

42. Clearing Houses and OTC MarketsTraditionally transactions have been cleared bilaterally in OTC marketsFollowing the 2007-2009 crisis, there has been a requirement for most standardized OTC derivatives transactions to be cleared centrally though clearing houses.The framework of global clearing houses is currently being negotiated

43. Bilateral Clearing vs Central Clearing House

44. DeliveryIf a futures contract is not closed out before maturity, it is usually settled by delivering the assets underlying the contract. When there are alternatives about what is delivered, where it is delivered, and when it is delivered, the party with the short position chooses. A few contracts (for example, those on stock indices and currencies) are settled in cash

45. QuestionsWhen a new trade is completed what are the possible effects on the open interest?Can the volume of trading in a day be greater than the open interest?

46. Market Order to Buy executes at best askMarket Order to Sell executes at best bidLimit Orders to BuyLimit Orders to SellMarket Orders & Limit Orders

47. Forward Contracts vs Futures Contracts (Table 2.3, page 41) Contract usually closed outPrivate contract between 2 partiesExchange tradedNon-standard contractStandard contractUsually 1 specified delivery dateRange of delivery datesSettled at end of contractSettled dailyDelivery or final cashsettlement usually occursprior to maturityFORWARDSFUTURESSome credit riskVirtually no credit risk

48. Hedging Strategies Using Futures

49. Long & Short HedgesWhat is your natural exposure?Do the opposite to hedge! A long futures hedge is appropriate when you know you will purchase an asset in the future and want to lock in the priceA short futures hedge is appropriate when you know you will sell an asset in the future and want to lock in the price

50. Arguments in Favor of HedgingCompanies should focus on the main business they are in and take steps to minimize risks arising from interest rates, exchange rates, and other market variables

51. Arguments against HedgingShareholders are usually well diversified and can make their own hedging decisionsIt may increase risk to hedge when competitors do notExplaining a situation where there is a loss on the hedge and a gain on the underlying can be difficult

52. Basis RiskBasis is defined as the spot price minus the futures priceBasis risk arises because of the uncertainty about the basis when the hedge is closed out“Mother of all risks”

53. BasisThe difference between the spot price and the futures price.4 Month Futures

54. ContangoTheory: Investors are willing to pay more for the futures contract than the expected spot price (which is not observable).In practice, when a market is “in contango”, traders are saying that the futures prices are converging downward overtime towards the spot price. The other interpretation which is not 100% correct is to simply see an upward sloping futures curve and make the statement that the market is in contango.PriceFutures Curve Today40555045Spot12345678940555045123456789Months from todayExpected Spot

55. BackwardationTheory: Investors are willing to sell the futures contract for less than the expected spot price (which is not observable).In practice, when a market is “in backwardation”, traders are saying that the futures prices are converging upward overtime towards the spot price. The other interpretation which is not 100% correct is to simply see an downward sloping futures curve and make the statement that the market is in backwardation.PriceFutures Curve Today40555045Spot12345678940555045123456789Months from todayExpected Spot

56. Long Hedge for Purchase of an Asset DefineF1 : Futures price at time hedge is set upF2 : Futures price at time asset is purchasedS2 : Asset price at time of purchaseb2 : Basis at time of purchaseCost of assetS2Gain on FuturesF2 −F1 Net amount paidS2 − (F2 −F1) =F1 + b2

57. Short Hedge for Sale of an Asset DefineF1 : Futures price at time hedge is set upF2 : Futures price at time asset is soldS2 : Asset price at time of saleb2 : Basis at time of salePrice of assetS2Gain on FuturesF1 −F2 Net amount receivedS2 + (F1 −F2) =F1 + b2

58. Choice of ContractChoose a delivery month that is as close as possible to, but later than, the end of the life of the hedgeWhen there is no futures contract on the asset being hedged, choose the contract whose futures price is most highly correlated with the asset price. This is known as cross hedging.

59. Optimal Hedge Ratio (page 57) Proportion of the exposure that should optimally be hedged is where sS is the standard deviation of DS, the change in the spot price during the hedging period, sF is the standard deviation of DF, the change in the futures price during the hedging period r is the coefficient of correlation between DS and DF.

60. Optimal Number of ContractsQA Size of position being hedged (units)QFSize of one futures contract (units)VAValue of position being hedged (=spot price times QA)VFValue of one futures contract (=futures price times QF)Optimal number of contracts if no tailing adjustmentOptimal number of contracts after tailing adjustment to account for daily settlement of futures

61. Example (Pages 59-60)Airline will purchase 2 million gallons of jet fuel in one month and hedges using heating oil futuresFrom historical data sF =0.0313, sS =0.0263, and r= 0.928

62. Example continued The size of one heating oil contract is 42,000 gallonsThe spot price is 1.94 and the futures price is 1.99 (both dollars per gallon) so that Optimal number of contracts assuming no daily settlement Optimal number of contracts after tailing

63. Hedging Using Index Futures(Page 61) To hedge the risk in an equity portfolio the number of contracts that should be shorted is where VA is the value of the portfolio, b is its beta, and VF is the value of one futures contract

64. Example S&P 500 futures price is 1,000 Value of Portfolio is $5 million Beta of portfolio is 1.5 What position in futures contracts on the S&P 500 is necessary to hedge the portfolio?Tip: set B=(BT – BP) . This well ensure you make the appropriate choice between buying and selling

65. Changing BetaWhat position is necessary to reduce the beta of the portfolio to 0.75?What position is necessary to increase the beta of the portfolio to 2.0?Need to understand how beta is estimated!!! Time period regressed? Daily, Weekly, Monthly? Appropriate benchmark?

66. You manage a $20M portfolio of large-cap equities (B=1)and want to move half the portfolio into mid-cap equities (B=1.4). S&P 500 (large cap) futures = $1800 (multiplier 250) = $450,000S&P 400 (mid cap) futures = $1400 (multiplier 100) = $140,000Changing Portfolio Allocations

67. Sell $10M Large Cap Futures AND Buy $10M Mid Cap FuturesSell S&P 500 (large cap) futures = (0-1) ($10M/$450,000) = -22.22Buy S&P 400 (mid cap) futures = (1.4-0)($10M/$140,000) = 100Note: We assume the futures contract has the same beta as the underlying index. In the real world, this may not hold. To adjust, divide (BT – BP) by BFChanging Portfolio Allocations

68. Why Hedge Equity ReturnsMay want to be out of the market for a while. Hedging avoids the costs of selling and repurchasing the portfolioSuppose stocks in your portfolio have an average beta of 1.0, but you feel they have been chosen well and will outperform the market in both good and bad times. Hedging ensures that the return you earn is the risk-free return plus the excess return of your portfolio due to your skills (not the market)Isolate the portfolio alpha Beta neutralStrategy: Portable Alpha or Alpha Transfer

69. Stack and Roll (page 65-66)We can roll futures contracts forward to hedge future exposures.You may need to hedge out 5 years but there are only 2 years worth of contracts.Initially we enter into futures contracts to hedge exposures up to a time horizonJust before maturity we close them out and replace them with a new contract that reflects the next date exposure“Stack and Roll” hedging is NOT precise and you should not expect a perfect hedge. Why? Basis risk!

70. Stack and Roll ExampleIn October 2015 your company realizes it will sell 100,000 barrels of oil in June 2017. (assume no tailing and a hedge ratio of 1)There is very little liquidity in the longer dated contracts:Therefore, in October 2015 you short 100 March 2016 contracts and plan to roll them into the January 2017 contracts (in February 2016), and then again roll the January 2017 contracts into the July 2017 contracts (in December 2016), and ultimately close out the July 2017 contracts (in June 2017).Consider the change in spot: $69 to $66 = $3.00 gain if you are shortConsider your gain from being short BUT using stack and roll:(68.20-67.40)+(67-66.50)+(66.30-65.90) = $1.70

71. Liquidity Issues (See Business Snapshot 3.2)In any hedging situation there is a danger that losses will be realized on the hedge while the gains on the underlying exposure are unrealizedThis can create liquidity problemsOne example is Metallgesellschaft which sold long term fixed-price forward contracts on heating oil and gasoline and established a long hedge using stack and roll with short term futuresThe price of oil fell causing margin calls for their long hedge. The company came under financial strain to meet the margin calls because they had not yet received the cash flows from the short fixed-price forward contracts.The timing mis-match is this example cost the company $1.33B

72. Determination of Forward and Futures Prices

73. Short Selling (Page 102-103)Short selling involves selling securities you do not own. You believe the price will decrease and you will Buy later.Sell High (first)… Buy Low (later)Sell firstBuy later

74. Notation for Valuing Futures and Forward ContractsS0:Spot price todayF0:Futures or forward price todayT:Time until delivery dater:Risk-free interest rate for maturity T

75. Assumptions1. All participants are subject to the same tax rate2. The short-term risk-free rate, r, is constant3. All can borrow and lend at r4. No transaction costs5. All securities are perfectly divisible.6. Market participants will take advantage of arbitrage opportunities as they occurNote: These assumptions do not need to be true for ALL participants. They only need to apply to a few large participants (broker dealers) for arbitrage pricing to be enforced

76. An Arbitrage Opportunity?Suppose that:The spot price of a non-dividend-paying stock is $40The 3-month forward price is $43The 3-month US$ interest rate is 5% per annumIs there an arbitrage opportunity?

77. Another Arbitrage Opportunity?Suppose that:The spot price of nondividend-paying stock is $40The 3-month forward price is US$39The 1-year US$ interest rate is 5% per annumIs there an arbitrage opportunity?

78. The Forward Price If the spot price of an investment asset is S0 and the futures price for a contract deliverable in T years is F0, then F0 = S0erT where r is the T-year risk-free rate of interest. In our examples, S0 =40, T=0.25, and r=0.05 so thatF0 = 40e0.05×0.25 = 40.50

79. Arbitrage opportunities if forward is mispriced

80. Forward vs Futures PricesWhen the maturity and asset price are the same, forward and futures prices are usually assumed to be equal. (Eurodollar futures are an exception)When interest rates are uncertain they are, in theory, slightly different:A strong positive correlation between interest rates and the asset price implies the futures price is slightly higher than the forward priceWhy? Because the futures contract gets marked to market each day. As S increases, so does r, and therefore the interest earned on the increase makes the future valued higher than the forwardA strong negative correlation implies the reverse Pg. 112

81. Stock Index (Page 112-114)Can be viewed as an investment asset paying a dividend yieldThe futures price and spot price relationship is therefore F0 = S0 e(r–q )T where q is the average dividend yield on the portfolio represented by the index during life of contract

82. Stock Index (continued)For the formula to be true it is important that the index represent an investment assetIn other words, changes in the index must correspond to changes in the value of a tradable portfolioThe Nikkei index viewed as a dollar number does not represent an investment asset . The underlying portfolio trades in Yen!(See Business Snapshot 5.3, page 113)

83. Index ArbitrageWhen F0 > S0e(r-q)T an arbitrageur buys the stocks underlying the index and sells futuresWhen F0 < S0e(r-q)T an arbitrageur buys futures and shorts or sells the stocks underlying the index

84. Index Arbitrage(continued)Index arbitrage involves simultaneous trades in futures and many different stocks A computer algorithm is used to generate the tradesOccasionally simultaneous trades are not possible and the theoretical no-arbitrage relationship between F0 and S0 does not hold Think about days when the market crashes and exchanges are overloaded.Trades get delayed due to server issues, circuit breakers prohibit trading, etc.(see Business Snapshot 5.4 on page 114)

85. S&P 500 Index Contract Table as of 6/6/12One contract = 250 x Future

86. Futures and Forwards on Currencies (Page 112-115)A foreign currency is analogous to a security providing a yieldThe yield is the foreign risk-free interest rateIt follows that if rf is the foreign risk-free interest rate

87. Explanation of the Relationship Between Spot and Forward (Figure 5.1)Interest Rate Parity

88. Currency Forward ExampleConsider the 2-year rate in Australia is 5% and the 2-year rate in the US is 7%.The spot exchange rate is 0.62 USD per AUDF = 0.62e(0.07-0.05)*2 = 0.6453What would you do if the forward was trading at 0.63?Borrow 1000 AUD at 5% for 2 years (you will owe 1,105.17 AUD at maturity), convert to 620 USD and invest at 7% (worth 713.17 USD at maturity)Enter a long forward to buy 1,105.17 AUD in 2 years (Use this to payoff AUD loan). This will cost 0.63*1,105.17=$696.26. Arbitrage profit = $16.91 (713.17 – 696.26)

89. British Pound Contract Table Quoted USD per 1 GBPas of 6/6/12

90. ContangoTheory: Investors are willing to pay more for the futures contract than the expected spot price (which is not observable).In practice, when a market is “in contango”, traders are saying that the futures prices are converging downward overtime towards the spot price. The other interpretation which is not 100% correct is to simply see an upward sloping futures curve and make the statement that the market is in contango.PriceFutures Curve Today40555045Spot12345678940555045123456789Months from todayExpected Spot

91. BackwardationTheory: Investors are willing to sell the futures contract for less than the expected spot price (which is not observable).In practice, when a market is “in backwardation”, traders are saying that the futures prices are converging upward overtime towards the spot price. The other interpretation which is not 100% correct is to simply see an downward sloping futures curve and make the statement that the market is in backwardation.PriceFutures Curve Today40555045Spot12345678940555045123456789Months from todayExpected Spot

92. Review of Option TypesA European option can be exercised only at the end of its lifeAn American option can be exercised at any timeBermudian options, Asian options, etc.

93. Option PositionsLong call – Right to BuyLong put – Right to SellShort call – Obligation to SellShort put – Obligation to BuyOption buyers pay a premium to option sellers

94. Long Call Profit from buying one European call option: option price = $5, strike price = $100, option life = 2 months3020100-5708090100110120130Profit ($)Terminalstock price ($)

95. Short Call Profit from writing one European call option: option price = $5, strike price = $100-30-20-1005708090100110120130Profit ($)Terminalstock price ($)

96. Long Put Profit from buying a European put option: option price = $7, strike price = $703020100-7706050408090100Profit ($)Terminalstock price ($)

97. Short Put Profit from writing a European put option: option price = $7, strike price = $70-30-20-1070706050408090100Profit ($)Terminalstock price ($)

98. Net payoffsThe net payoff includes the option premiumNote that net payoffs mix payment at T0 and T (ignores the time value of money)XXShort PutLong PutShort CallLong Call

99. Cheat sheet

100. Long 55 Call at $5 Payoff55-5payoffLong CallS<55S>55Long Call0S-55Initial Cash Flow-5-5Net Payoff-5S-60

101. Short 65 Call at $5 Payoff655payoffShort CallS<65S>65Short Call0-(S-65)Initial Cash Flow+5+5Net Payoff+570-S

102. Long 65 Put at $465?payoffLong Put4S<65S>65Long Put65-S0Initial Cash Flow-4-4Net Payoff61-S-4

103. Short 45 Put at $5 Payoff45?payoffShort Put5S<45S>45Short Put-(45-S)0Initial Cash Flow+5+5Net PayoffS-40+5

104. Long stock at $55 and Short 55 Call at $7 (Covered Call))557payoffWritten CallLong StockPortfolioPositionS<55S>55Long StockSSShort Call0-(S-55)Initial CF-48-48PortfolioS-487

105. Long Stock at $52 & Long 55Put at $8 (Protective Put)PositionS<55S>55Long StockSSLong Put55-S0Initial CF-60-60Portfolio-5S-60555payoffLong PutLong StockPortfolio

106. Long Stock at $52, Long 50Put at $1, and Short 55 Call at 1 (Cashless Collar)PositionS<5050<S<55S>55Long StockSSSLong Put50-S00Short Call00-(S-55)Initial CF-52-52-52Portfolio-2S-52355-2payoffLong PutLong StockPortfolio50Short Call

107. Long Stock at $52 & Long Two 55Puts at $5 eachPositionS<55S>55Long StockSS2 Long Puts2(55-S)0Initial CF-62-62Portfolio48-SS-6255-10payoffLong PutsLong StockPortfolio

108. Short a 50 Call at $7 and Long a 55 Call at $5(Bear Spread with Calls)555payoffShort 50 CallLong 55 CallPortfolio507S<5050<S<55S>55Short 50 Call0-(S-50)-(S-50)Long 55 Call00S-55Initial Cash flow222Portfolio252-S-3

109. Short a 35 Put at $4 and Long a 30 Put at $1(Bull Spread with Puts)35-2payoffShort 35 PutLong 30 PutPortfolio304S<3030<S<35S>35Long 30 Put30-S00Short 35 Put-(35-S)-(35-S)0Initial Cash flow333Portfolio-2S-323

110. Short a 70 Call at $4 and Short a 70 Put at $3 (Short Straddle)PositionS<70S>70Written 70 Call0-(S-70)Written 70 Put-(70-S)0Initial CF77PortfolioS-6377-S70payoffShort 70 PutShort 70 CallPortfolio4

111. Long a 70 Put at $3 and Long a 75 Call at $1 (Long Strangle)PositionS<7070<S<75S>75Long 70 Put70-S00Long 75 Call00S-75Initial CF-4-4-4Portfolio66-S-4S-7970payoffLong 75 CallLong 70 PutPortfolio-475

112. Long a 55 Call at $10, Short Two 60 Calls at $7, and Long a 65 Call at $5 (Long Butterfly Spread with Calls)PositionS<5555<S<6060<S<65S>65Long 55 Call0S-55S-55S-552 Short 60 Calls00-2(S-60)-2(S-60)Long 65 Call000S-65Initial CF-1-1-1-1Portfolio-1S-5664-S-160payoffLong 55 CallLong 65 CallPortfolio-1055652 Short 60 Calls14-5

113. Sell a 55 Call at $10, Buy Two 60 Calls at $7, and Short a 65 Call at $5 (Short Butterfly Spread)PositionS<5555<S<6060<S<65S>65Short 55 Call0-(S-55)-(S-55)-(S-55)2 Long 60 Calls002(S-60)2(S-60)Short 65 Call000-(S-65)Initial CF1111Portfolio156-SS-64160payoffShort 55 CallShort 65 CallPortfolio1055652 Long 60 Calls-145

114. Long two 70 Puts at $3 and Long a 70 Call at $4 (Strip)PositionS<70S>70Buy 2 70 Puts2(70-S)0Buy 70 Call0S-70Initial CF-10-10Portfolio130-2SS-8070payoffLong 70 Call2 Long 70 PutsPortfolio-4

115. Long two 70 Calls at $4 and Long a 70 Put at $3 (Strap)PositionS<70S>70Buy 2 70 Calls02(S-70)Buy 70 Put70-S0Initial CF-11-11Portfolio59-S2S-15170payoff2 Long 70 CallsLong 70 PutPortfolio-8

116. Box SpreadA combination of a bull call spread and a bear put spreadIf all options are European a box spread is worth the present value of the difference between the strike pricesIf they are American this is not necessarily so (see Business Snapshot 11.1)

117. Box Spread with European OptionsLong 50 call at $6, short 55 call at $3, short 50 put at $1.5, long 55 put at $3PositionS<5050<S<55S>55Long 50 Call0S-50S-50Short 55 Call00-(S-55)Short 50 Put-(50-S)00Long 55 Put55-S55-S0Initial CF-4.5-4.5-4.5Portfolio0.500.500.505055$0.50What is the Rf rate?

118.

119.

120. A formal argumentConsider two portfolios:Port A: One European call option and a zero coupon bond that matures to K at time TPort B: One share of stockAt T, the zero coupon bond will be worth K. If ST >K the call gets exercised and Port A is worth ST. If ST <K, the option is worthless and Port A is worth K. Hence, at T Port A is worth max (ST, K)Port B is always worth ST, . So Port A is ALWAYS worth at least as much as Port B…..c + Ke-rt ≥ S…..orc ≥ S0 – Ke-rt

121. Put-Call Parity: No Dividends Consider the following 2 portfolios:Portfolio A: European call on a stock + zero-coupon bond that pays K at time TPortfolio C: European put on the stock + the stock

122. Values of PortfoliosST > KST < KPortfolio ACall optionST − K0Zero-coupon bondKKTotalSTKPortfolio CPut Option0K− STShareSTSTTotalSTK

123. The Put-Call Parity ResultBoth are worth max(ST , K ) at the maturity of the optionsThey must therefore be worth the same today. This means that c + Ke -rT = p + S0

124. The Concepts Underlying Black-Scholes-MertonThe option price and the stock price depend on the same underlying source of uncertaintyWe can form a portfolio consisting of the stock and the option which eliminates this source of uncertaintyThe portfolio is instantaneously riskless and must instantaneously earn the risk-free rateThis leads to the Black-Scholes-Merton differential equationAll can borrow and lend at rNo transaction costsAll securities are perfectly divisible.No dividends paid during life of the option

125. The Black-Scholes-Merton FormulasWhere N(x) is the cumulative probability function for a standardized normal variable

126. The N(x) FunctionN(x) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than xSee tables at the end of the book

127.

128. Black-Scholes exampleSuppose S=20, X=18, r=.08, t=.5, and =.2. What is the value of a European call option?d1=[ln(20/18)+(.08+.22/2).5]/[.2*(.5).5]=1.1d2 = 1.1- [.2*(.5).5] = 0.96from the probability table:N(d1) = .8643N(d2) = .8315C = 20(.8643) - 18e-.08(.5)(.8315) = $2.91

129. Properties of the equationIf S becomes very large, N(d1) and N(d2) approach 1.0call value approaches S-Xe-rtand 1-N(d1) approaches zeroIf S becomes very small, N(d1) and N(d2) approach zerocall value approaches zero1-N(d1) approaches 1.0We can also show that options have sensitivities to S, X, r, t, and .

130. Black-Scholes Values55-5payoffLong CallValue at Expiration(Intrinsic Value)Value just before ExpirationValue weeks before ExpirationDifference between value and intrinsic value is the time value of the option

131. Option ValuationFive (5) variables affect the value of optionsUnderlying (stock) priceExercise price (X)Time to maturity (T)Volatility of the underlying returns ()Interest rates (r)Which are most important?

132. Implied VolatilityThe implied volatility of an option is the volatility for which the Black-Scholes price equals the market priceTo calculate implied volatility you use an iterative approachTraders and brokers often quote implied volatilities rather than dollar prices