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1、Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,1,Chapter 14-15 Forward,Future and Option,ObjectiveUnderstanding the definitions of forward,future and optionUnderstanding the basic idea ofsynthetic security,Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,2,Contents,
2、1 Distinction Between Forward&Futures Contracts2 Futures Price3 Financial future,4 Definition of an option5 How an option works6 Two-state option-pricing,Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,3,1 Distinction Between Forward&Futures Contracts,Forward:parties agree to exchan
3、ge some item in the future at a delivery price specified nowthe forward price is defined as the delivery price which makes the current market value of the contract zerono money is paid in the present by either party to the otherthe face value of the contract is the quantity of the item specified in
4、the contract multiplied by the forward pricethe party who agrees to buy the specified takes the long position,and the party who agrees to sell the item takes the short position,Who pays what to whom,If the spot price on the contract maturity date is higher than the forward price,the party who is lon
5、g makes money.But if the spot price on the contract maturity date is lower than the forward price,the party who is short makes money.,Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,5,Future:terms,Listing:Open,High,Low,Settle,Change,Lifetime high,Lifetime low,Open interestMark-to-ma
6、rket at the end of each trading day based on that days settlement price.Margin requirement:the exchange requires that there be enough collateral posted in each account to cover any losses.Margin call:if the collateral in your account falls below a prespecified level,you will receive a margin call fr
7、om the broker asking you to add money.,An illustration,Based on table 13.1.You place an order to take a long position in a July wheat futures contract on June 22,2006.the broker requires you to deposit money in your account,say$1,500,as margin.On June 23,the future price closes 2.25cents per bushel
8、lower,thus you have lost 1.25*5,000=$112.50 that day.The broker takes that amount out of your account(mark to market).The money is transferred to the exchange,which transfers it to one of the parties who was on the short side of the contract.,Daily realization of gains and losses,Such a process mini
9、mizes the possibility of contract default.And no matter how great their face value,the market value of future contracts is always 0 at the beginning of each day.,Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,8,Summarized characteristics for future,standard contractsimmune from the
10、 credit worthiness of buyer and seller becauseexchange stands between traderscontracts marked to market dailymargin requirements,Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,9,2 Futures Price,Arbitrageurs place an upper bound on futures prices by locking in a sure profit on futur
11、es prices if the spread between the futures price and spot price becomes greater than the cost of carry:F-S Cthe cost of carry varies as a function of time and warehousing organization,Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,10,3 Financial Futures,We now focus on financial f
12、uturesstandardized contracts for future delivery of stocks,bonds,indices,and foreign currency they have no intrinsic value,but represent claims on future cash flowsthey have very low storage costssettlement is usually in cash,Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,11,Financ
13、ial Futures,With no storage cost,the relationship between the forward and the spot isAny deviation from this will result in an arbitrage opportunity,Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,12,Financial Futures:Example,Consider shares in Bablonics,Inc,trading at$50 each,($5,0
14、00 for a round lot);assume 6-month T-bills yield 6%(compounded semiannually),Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,13,Bablonics,Inc(Continued),1 Purchase one round lot of stock at spotThis results in a negative cash flow today of$5,000(out),and will generate a cash flow of
15、 100*Spot6m(in)in six months,Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,14,Bablonics,Inc(Continued),2 Cover todays negative cash flow by selling short$5,000 worth of 6-month T-bills with a face value of 5000(1+0.06/2)0.5=$5,150The cash flow today is$5,000(in),and the cash flow
16、in six months will be$5,150(out),Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,15,Bablonics,Inc(Continued),3 Cover the risk exposure by selling 100 shares forward at the equilibrium price of 5000*(1+0.06/2)0.5=$5,150There is no cash flow today,but the value of this forward contrac
17、t in six months will be$(Spot6m-5,150),Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,16,Bablonics,Inc(Continued),-$5,000(long stock)+$5,000(short bond)+$0(short forward)=$0,Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,17,Bablonics,Inc(Continued),Cash Flow in 6-
18、Months+$Spot6m(settle long stock)-$5,150(settle short bond)+($5,150-$Spot6m)(settle forward)=$0,Copyright 2009 Pearson Education,Inc.Publishing as Prentice Hall,18,Bablonics,Inc(Conclusion),If your net risk-free investment was zero,and you receive nothingthat is what you should expectand you expect
19、to:received positive value with no risk,then the rule of one price has been violatedlose value with no risk,then reverse the direction of all transactions,and again you profit with no risk,4 Definition of an Option,Recall that an American European call(put)option is the right,but not the obligation
20、to buy(sell)an asset at a specified price any time before its expiration date on its expiration date,5 How Options Work,The Language of OptionsContingent Claim:Any asset whose future pay-off depends upon the outcome of an uncertain eventCall:an option to buyPut:an option to sellStrike or Exercise Pr
21、ice:the fixed price specified in an option contractExpiration or Maturity Date:The date after which an option cant be exercisedAmerican Option:an option that can be exercised at any time up to and including maturity date,European Option:an option that can only be exercised on the maturity dateTangib
22、le Value:The hypothetical value of an option if it were exercised immediatelyAt-the-Money:an option with a strike price equal to the value of the underlying assetOut-of-the-Money:an option thats not at-the-money,but has no tangible valueIn-the-Money:an option with a tangible valueTime Value:the diff
23、erence between an options market value and its tangible valueExchange-Traded Option:A standardized option that an exchange stands behind in the case of a defaultOver the Counter Option:An option on a security that is not an exchange-traded option,Investing with Options,The payoff diagram(terminal co
24、nditions,boundary conditions)for a call and a put option,each with a strike(exercise price)of$100,is derived next,Option Payoff Diagrams,The value of an option at expiration follows immediately from its definitionIn the case of a call option with strike of$100,if the stock price is$90($110),then exe
25、rcising the option results purchasing the share for$100,which is$10 more expensive($10 less expensive)than buying it,so you wouldnt(would)exercise your right,Call Option Payoff Diagram,Put Option Payoff Diagram,An example,Suppose you have$100,000 to invest.The riskless interest rate is 5%per year an
26、d the stock pays no dividends.Spot stock price is$100 and call premium is$10Strategies:1.invest all$100,000 in the stock2.invest all$100,000 in calls3.invest$10,000 in calls and the rest in the risk-free asset,Payoff Diagrams for Alternative Bullish Stock Strategies,Strategy 2:leverage 10 times(slop
27、e 10 times the slope of strategy 1)Strategy 3:minimum portfolio rate of return=-5.5%,6 Two-State(Binary)Option-Pricing,We are now going to derive a relatively simple model for evaluating optionsThe assumptions will at first appear totally unrealistic,but using some underhand mathematics,the model ma
28、y be made to price options to any desired level of accuracyThe advantage of the method is that it does not require learning stochastic calculus,and yet it illustrates all the key steps necessary to derive any option evaluation model,Binary Model Assumptions,Assume:the exercise price is equal to the
29、forward price of the underlying stockoption prices then depend only on the volatility and time to maturity,and do not depend on interest ratesthe put and call have the same price,Binary Model Assumptions,More specifically we assume:share price=strike price=$100time to maturity=1 yeardividend rate=in
30、terest rate=0stock prices either rise or fall by 20%in the year,and so are either$80 or$120 at yearend,Binary Model:Call,Strategy:replicate the call using a portfolio of the underlying stock the riskless bondby the law of one price,the price of the actual call must equal the price of the synthetic c
31、all,Binary Model:Call,Implementation:the synthetic call,C,is created bybuying a fraction x of shares,of the stock,S,and simultaneously selling short risk free bonds with a market value ythe fraction x is called the hedge ratio,Binary Model:Call,Specification:We have an equation,and given the value o
32、f the terminal share price,we know the terminal option value for two cases:By inspection,the solution is x=1/2,y=40,Binary Model:Call,Solution:We now substitute the value of the parameters x=1/2,y=40 into the equationto obtain:,Binary Model:Put,Strategy:replicate the put using a portfolio of the und
33、erlying stock and riskless bondby the law of one price,the price of the actual put must equal the price of the synthetic put replicated aboveMinor changes to the call argument are made in the next few slides for the put,Binary Model:Put,Implementation:the synthetic put,P,is created bysell short a fr
34、action x of shares,of the stock,S,and simultaneously buy risk free bonds with a market value ythe fraction x is called the hedge ratio,Binary Model:Put,Specification:We have an equation,and given the value of the terminal share price,we know the terminal option value for two cases:By inspection,the
35、solution is x=1/2,y=60,Binary Model:Put,Solution:We now substitute the value of the parameters x=1/2,y=60 into the equationto obtain:,Dynamic Replication and the Binomial Model,We now take the next step towards greater realism by dividing the year into 2 sub-periods of half a year each.This gives 3
36、possible outcomesOur first task is to find a self-financing investment strategy that does not require injection or withdrawal of new funds during the life of the optionWe first create a decision tree:,Decision Tree for Dynamic Replication of Call Option,Reading the Decision Tree,The tree is constructed backwards because we know only the future contingent call pricesFor Example,when constructing the weights for time 6-months,the option prices for 12-months are usedFor consistency with the next model,the discrete stock prices are usually fixed ratios,i.e.121,110,100,90.91,82.64,
