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1、InvestmentsLecture 12Futures and Options: Objectives1.Understand uses of futures and options;2.Understand characteristics of futures and options investments, including:nRisknPricingnHedging3.Understand and apply mathematical and statistical techniques required to nValue futures and options;nHedge us
2、ing futures and options.Derivative SecuritiesnFinancial instruments that derive their values from other traded claims are called derivatives.nTypically, the value of these instruments is very closely related to the value of the underlying asset.nAs a result derivatives are useful for:nSpeculating on
3、 the underlying asset, and;nHedging the underlying asset.nFurthermore, arbitrage opportunities may be possible if the underlying asset and the derivative asset are not priced consistently.Speculators and HedgersnSpeculators are individuals hope to make a profit by closing out their positions at a pr
4、ice that is better than the initial price. They do not produce or use the asset in their daily course of business.nHedgers are individuals who use derivatives to offset an otherwise risky position in the underlying asset. They either produce or use the asset in their daily course of business.Example
5、: Wheat ForwardsnIn a wheat forward contract two counter-parties agree to exchange some quantity of wheat at some date in the future at a price negotiated today.nA wheat farmer has exposure to the future spot price of wheat.nThe spot price is the market price of wheat for immediate delivery.nThe cro
6、p planted in the spring and harvested in the fall will be sold at fall spot prices. Since these spot prices are uncertain, the profits on the farmers crop are risky.nA risk-averse farmer can hedge this risk by selling wheat now using a forward contract.Wheat ForwardsnWho might take the opposite side
7、 of this trade?nA bread producer may wish to hedge production costs.nA weather forecaster may speculate that the future spot price will be well above the forward price and therefore use this contract as part of a trading strategy (buy using the forward contract and sell in the future spot market).Ar
8、bitrageursnTwo basic types of arbitrage trades:nInvest nothing and make positive future profits;nReceive profits today without any future obligations.nArbitrageurs use derivative contracts to extract arbitrage profits.nTheir actions, along with normal supply and demand forces, ensure consistent rela
9、tionships among the underlying asset prices and the derivative security prices.Wheat ForwardsnWho might be in a position to derive arbitrage profits from wheat forward contracts?nIf you have a technology for storing wheat and the forward price is high relative to todays spot price, you may want to:n
10、Borrow money to buy wheat now,nSell it with the forward contract,nStore it until the fall,nDeliver the wheat and use the proceeds to pay back your lenders.nNotice that the cost of storage and lending rates will place a bound on how high the forward price can be (a sort of no-arbitrage bound).Futures
11、Futures Contracts - DefinitionnA futures contract is an agreement between two parties to buy or sell an asset at a certain time in the future for a certain price.nCharacteristics of futures contracts:nTraded on an exchange;nContracts are standardized;nClearing houses eliminate default risk;nMargin i
12、s required.Futures - Common ExamplesnCommodity futures:nWheat;nCrude oil;nGold;nLive cattle.nFinancial futures:nS&P 500 index futures;nT-bill futures;Futures Contract SpecificationnComponents of contract specification:nAsset;nContract size;nDelivery arrangements;nCash or physical delivery;nPlace;nTi
13、me.nPrice quotes;nPrice movement limits;nPosition limits.Futures Contracts - MarginnWhen you enter into a futures contract, the broker typically require that you deposit funds into a margin account.nYou may or may not earn interest on this account.nAt contract initiation you deposit the initial marg
14、in (also called performance margin).nThis account is “marked-to-market” periodically.nPeriodic profits, as represented by changes in the futures price, are credited to or debited from your account.Margin (contd)nYou can withdraw any funds in excess of the performance margin.nIf your margin account b
15、alance falls below the maintenance margin you will receive a margin call, in which case you must deposit additional funds (the variation margin) to bring your balance back to the initial margin level.nIf you do not honour a margin call, your position is closed out.Margin ExampleTime from contract in
16、itiationMarginAccountBalance InitialMaintenanceMargin ExamplenFutures on indexnTwo contracts, contract size = $50nInitial margin - $1500 / contractnMaintenance margin - $1000Important PointsnAs an investor you have more than your initial margin at risk prior to maturity.nPrevious example, could loos
17、e up to 1000 in index terms or (1000 x $50 x 2) = $100,000nYou can withdraw excess margin.Futures PayoutnAlthough, in practice, the lifetime payout from a futures accrues over time and accumulates in the margin account, it is useful to think of the payout from the contract as being received at the m
18、aturity date.nIt is important to understand the relationship between this hypothetical payout and the spot price.Futures PayoffsnLet F be the futures price and ST be the spot price at maturity.nThe payout from a long position in a futures contract:ST FnThe payout from a short position in a futures c
19、ontract:F STGraphicallyRisk of a FuturesnSuppose thatnyou are long an S&P 500 E-Mini index futures with one month to maturity. The current index value is 1242.98 and the futures price is 1250.75nThe initial margin is $4,313 and earns the risk-free rate of interest.nYou make no margin payments until
20、maturity when you settle the position.nWhat is the Beta of the investment?Beta of FuturesnThe return on the investment:nThe Beta isE-Mini BetanIn this case, the beta of the contract, with margin, is:nYour investment is almost 15 times more risky than an investment in the market!OptionsOption Contrac
21、tsnIn an option contract the writer grants the buyer the option, but not the obligation, to buy from or to sell to the writer a specific asset at a specific price (called the strike or exercise price) within a specified period of time.Common Options1.Call OptionnBuyer has the right to buy the asset
22、at a given price (the exercise price) at a given date.nWriter has commitment to sell the underlying asset to the holder at the exercise price if exercised.2.Put OptionnBuyer has the right to sell the asset at a given price (the exercise price) at a given date.nWriter has commitment to buy the underl
23、ying asset for the holder at the exercise price if exercised.Terminology1.American OptionOption that can be exercised at any time prior to expiration date.2.European OptionOption that can be exercised only at expiration date.3.In-the-moneystock price exercise price4.Out-of-the-moneystock price exerc
24、ise price5.At the moneystock price exercise priceNotationnS (or St): Current stock price at time t.nK: exercise or strike price.nT: time to expiration (maturity)nC (c): value of American (European) call.nP (p): value of American (European) put.Values of options at expirationnA Payout Graph shows the
25、 cashflows resulting from a position in an option as a function of of the underlying assets price.neg. Call Options:nIf ST K then payoff to call owner = 0nIf ST K then payoff to call owner = ST - KnWhere nST = value of stock at expiration, andnK = Strike PriceOptionValue ifExercisedKShare priceBUY C
26、ALLKShare priceSELL CALLExample: Call OptionnCall option on MOT: K =$90nValue of option at different stock prices:Stock Price$80$90$100$110$120Option Payoff $0 $0 $10 $20 $30OptionValue ifExercised Value of the put option at expiration:Payoff to put owner = K - ST if ST KPayoff to put owner = 0 if S
27、T KShare priceKKBUY PUTSELL PUTShare priceOptionValue ifExercisedOptionValue ifExercisedExample: Put OptionPayoffsnFor the owner of a call option:nFor the owner of a put option:Exotic OptionsnAsian OptionsnBarrier OptionsnLookback OptionsnCurrency Translated OptionsnBinary OptionsDerivative Strategi
28、esDerivative StrategiesnWe will examine how derivatives may be used to:nEliminate risk (hedge);nModify risk (partially hedge);nReplicate other payoffs and create synthetic payoffs.Hedging: Determining ExposurenConsider an individual who wishes to hedge. In order to determine exactly how to hedge, we
29、 must determine the sensitivity of their payoffs to the underlying hedging instrument. (i.e. calculate the hedge ratio.)nPayoff diagrams are useful for determining these ratios.Hedging Example 1nSuppose you are an index portfolio manager with $4,000,000 invested in the S&P 500 stocks. You have chose
30、n to eliminate any risk in the portfolio by using the E-mini S&P 500 contract (remember that each contract provides a payout equal to $50 times the S&P 500 level). Determine the number of contracts you should buy or sell in order to achieve the hedge.Step 1: Describe the PayoffsPortfolio return:Futu
31、res Contract Payoff: Step 1: Describe PayoffsnGraphically:Step 2: Determine the Hedge RationIf the sensitivity is to be zero, we need the payoff to be zero, regardless of the realized return on the market. In equation terms:Hedging Example 2nSuppose you are an index portfolio manager with $4,000,000
32、 invested in a stock portfolio. You have chosen to eliminate the market risk in the portfolio by using the E-mini S&P 500 contract. Using a market model regression you have determined that the portfolio beta is 1.175. How many contracts you should buy or sell in order to achieve the hedge.PayoffsPar
33、tial Hedging ExamplenYou need not offset all the risk in the portfolio. In fact, by using index futures you can tune your portfolio to have any exposure you desire.nExample: Given the data in the previous example, describe how you can use the futures contract to change the portfolio beta to 0.5.Grap
34、hicallyCreating Synthetic PayoffsnDerivatives can be used to replicate payoffs.nExample 1: Describe how you would use a futures contract that mimics the payoff to investing $50,000 in the S&P 500.Payoff GraphExamplenDescribe how you can use a futures contract and investments in t-bills to create a w
35、ell diversified portfolio with a beta of 0.5.Options StrategiesnBasic option payoffs can be added together to create compound payoffs that have almost arbitrary characteristics. Well examine:nPut-Call parity;nCovered strategies;nSpreads.Put-Call ParitynSuppose you simultaneously buy a call and write
36、 a put, both having a strike price K and maturity T.nSummary of payoffs from this position at T:nSo the total payoffs to these positions will be ST K.ST KPayoff from call purchased0ST KPayoff from put written-(K ST)0TotalST KST K51An AlternativenBorrow today and repay K at maturitynBuy 1 share of st
37、ocknYour payoffs from these positions at time T:nPayoff = Asset - Liability = ST KnThe two positions give you identical payoffs.nBy no arbitrage, the costs of establishing these positions must be identical52The Put-Call Parity Relationship1.Cost to establish option positions:a.Purchase call option f
38、or C0b.Sell put option for P0c.Total cost of establishing position: C0-P02.Cost to establish levered equity position:a.Cost of stock: S0b.Borrowed funds:c.Total cost of establishing levered stock position:3.Costs of establishing identical payoff positions must be identical:53If the Put-Call Parity d
39、oes not hold, then arbitrage opportunities will arise.Example: Suppose that the security price is $31, the exercise price is $30, the risk-free rate is 10% per annum, the price of a 3-month European call option is $3, and the price of a 3-month European put option is $2.25. Do we have mispricing? If
40、 yes, could we make arbitrage profits?Value of Portfolio A:Value of Portfolio B:Portfolio B is overpriced relative to portfolio A. What is the arbitrage strategy?Buy securities in Portfolio A and short the securities in Portfolio B.Covered StrategiesnProtective PutnBuying a put on a long position in
41、 the underlying asset.nActs like insurance by guaranteeing a floor on the total value of the portfolio.Protective PutStock PricePayoffCovered CallnWrite call option on a long position in the underlying asset.nGenerates income in the form of the option premium.Covered CallSpreadsnSpreads involve the
42、use of two or more options to create complex payoffs.nBull call spread:nBuy a call with a low strike price and sell a call with a higher strike price.nCan also be achieved with put options.nBear call spread:nMirror image payoff to Bull SpreadBull SpreadStraddlenBuy a call and put with the same strik
43、e price and expiration date.nAllows bets on volatility, since payoff is dependent on movement of the underlying asset.StraddleButterfly SpreadnThree calls:nLong low strike and high strike calls.nShort two mid-strike calls.nAllows bets on specific future prices.问题n所“套保”的对象是不是企业真实需求的产品，比如东航、中国国际航空股份有限
44、公司等，买卖的是航油“套保”工具；中信泰富因有在澳洲的投资项目，因而对澳元汇率进行了“套保”。n“套保”的方向是否和企业需求一致。比如，2004年中国航油（新加坡）股份有限公司衍生品交易巨亏5.5亿美元，其操作的方向是卖出航油看涨期权，而该公司实际上是航油的需求方，显然与公司经营目标相悖，是典型的投机行为。n“套保”的规模是否与企业的现货需求匹配，“套保”规模最多较现货需求放大5-10。中信泰富之所以被认为是投机，原因是交易合同涉及金额达97亿澳元，远远超过该公司澳洲实业投资所需的30亿澳元。Zero-cost collarn买入一个看涨期权锁定油价的上行风险，同时卖出一个看跌期权，两者价格相抵。 n东航：其一般做法是买入一个看涨期权，但该期权有封顶，即超过一定价格即中止合约；同时卖出2倍甚至大于2倍的看跌期权，即价格跌到一定程度，企业就要向交易对手支付超过2倍的价差亏损。这事实上和中信泰富在澳元衍生品交易的模式极为相似，也就是近年来在亚洲风行的Accumulator（累计期权）。
