1风险管理与金融衍生品.pptx

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1、liang_Chapter 1Risk Management&Financial Derivative RisknRisk-uncertainty of the outcomenbring unexpected gainsncause unforeseen lossesnRisks in Financial Marketnasset(stocks,),interest rate,foreign exchange,credit,commodity,nTwo attitudes toward risks n Risk aversionn Risk seekingFinancial Derivati

2、ves Many forms of financial derivatives instruments exist in the financial markets.Among them,the 3 most fundamental financial derivatives instruments:nForward contractsnFuturenOptionsIf the underlying assets are stocks,bonds etc.,then the corresponding risk management instruments are:stock futures,

3、bond futures,etc.Risk Managementnrisk management -underlying assets nMethod hedging-using financial derivatives i.e.holds two positions of equal amounts but opposite directions,one in the underlying markets,and the other in the derivatives markets,simultaneously.Underlying asset put or callDerivativ

4、e call or put=Forward Contractsnan agreement to buy or sell at a specified future time a certain amount of an underlying asset at a specified price.nan agreement to replace a risk by a certaintyntraded OTCnlong position-the buyer in a contractnshort position-the seller in a contractndelivery price-t

5、he specified pricenmaturity-specified future time FutureKK00Long positionShort positionFuturesnsame as a forward contractnhave evolved from standardization of forward contractsndifferences nfutures are generally traded on an exchangena future contract contains standardized articlesnthe delivery pric

6、e on a future contract is generally determined on an exchange,and depends on the market demandsOptionsnan agreement that the holder can buy from(or sell to)the seller(the buyer)of the option at a specified future time a certain amount of an underlying asset at a specified price.But the holder is und

7、er no obligation to exercise the contract.na right,no obligation nthe holder has to pay premium for this right nis a contingent claimnHas a much higher level of leverage Two Options nA call option-a contract to buy at a specified future time a certain amount of an underlying asset at a specified pri

8、cen A put option-a contract to sell at a specified future time a certain amount of an underlying asset at a specified price.nexercise price-the specified price nexpiration date-the specified date nexercise-the action to perform the buying or selling of the asset according to the option contract Opti

9、on Typesn European options-can be exercised only on the expiration date.n American options-can be exercised on or prior to the expiration date.nOther options Asia option etc.Total Gain of an OptionKK00Call optionput optionpTotal gain=Gain of the option at expiration-PremiumOption Pricingnrisky asset

10、s price is a random variablenthe price of any option derived from risky asset is also random nthe price also depends on time tnthere exists a function such thatnknown nHow to find outTypes of TradersnHedger-to invest on both sides to avoid lossnSpeculator-to take action characterized by willing to r

11、isk with ones money by frequently buying and selling derivatives(futures,options)for the prospect of gaining from the frequent price changes.nArbitrage-based on observations of the same kind of risky assets,taking advantage of the price differences between markets,the arbitrageur trades simultaneous

12、ly at different markets to gain riskless instant profitsHedger ExamplenIn 90 days,A pays B 1000,000 nTo avoid risk,A has 2 plansnPurchase a forward contract to buy 1000,000 with$1,650,000 90 days later nPurchase a call option to buy 1000,000 with$1,600,000 90 days later.A pays a premium of$64,000(4%

13、)current rate($/)90-day laterRate($/)no hedging$forward cont.hedging$call optionhedging$1.60up 1.70down 1.551,700,0001,550,0001,650,0001,650,0001,664,0001,614,000Speculator ExamplenStock A is$66.6 on April 30,may grownA speculator has 2 plansnbuys 10,000 shares with$666,000 on April 30npays a premiu

14、m of$39,000 USD to purchase a call option to buy 10,000 shares at the strike price$68.0 per share on August 22Speculator Example cont.nSituation I:The stock$73.0 on 8/22.nStrategy A Return=(730-666)/666*100%=9.6%nStrategy B Return =(730-680-39)/39*100%=28.2%nSituation II:The stock$66.0 on 8/22.nStra

15、tegy A Return=(660-666)/666*100%=-0.9%nStrategy B loss all investment Return=-100%Chapter 2Arbitrage-Free PrincipleFinancial Marketn Two Kinds of AssetsnRisk free asset nBondnRisky assetnStocksnOptionsn.n Portfolio an investment strategy to hold different assetsInvestmentnAt time 0,invest SnWhen t=T

16、,nPayoff=nReturn=n For a risky asset,the return is uncertain,i.e.,S is a random variableA Portfolio na risk-free asset Bnn risky assetsn a portfolio is called a investment strategynon time t,wealth:portion of the cor.AssetArbitrage OpportunitynSelf-financing-during 0,T no add or withdraw fund nArbit

17、rage Opportunity-A self-financing investment,and Probability ProbArbitrage Free Theorem nTheorem 2.1n the market is arbitrage-free in time 0,T,n are any 2 portfolios satisfying&Proof of TheoremnSuppose false,i.e.,n DenotenB is a risk-free bond satisfyingnConstruct a portfolio at Proof of Theorem con

18、t.nr risk free interest rate,at t=TnThennFrom the suppositionProof of Theorem cont.nIt followsnContradiction!Corollary 2.1nMarket is arbitrage free nif portfolios satisfying nthen for any Proof of CorollarynConsidernThennBy Theorem,for nNamely Proof of Corollary 2.1nIn the same waynThen nCorollary h

19、as been proved.Option PricingnEuropean Option PricingnCall-Put Parity for European Option nAmerican Option PricingnEarly Exercise for American Option nDependence of Option Pricing on the Strike PriceAssumptions1.The market is arbitrage-free2.All transactions are free of charge3.The risk-free interes

20、t rate r is a constant4.The underlying asset pays no dividendsNotationsn -the risky asset price,n -European call option price,n -European put option price,n -American call option price,n -American put option price,n K -the options strike price,n T -the options expiration date,n r -the risk-free inte

21、rest rate.Theorem 2.2nFor European option pricing,the following valuations are true:Proof of Theorem 2.2nlower bound of (upper leaves to ex.)nconsider two portfolios at t=0:n n Proof of Theorem 2.2 cont.nAt t=T,nand nBy Theorem 2.1n i.e.Proof of Theorem 2.2 cont.cont.nNow consider a European call op

22、tion cnSincenandnBy Theorem 2.1 when t=that of the European option.nThereforeTheorem 2.5nIf a stock S does not pay dividend,thenni.e.,the early exercise term is of no use for American call option on a non-dividend-paying stock.Proof of Theorem 2.5nBy above inequalities,there holdsnThis indicates it

23、is unwise to early exercise this optionTheorem 2.6nIf C,P are non-dividend-paying American call and put options respectively,nthen,Proof of Theorem 2.6(right side)nIt follows from call-put party,and Theorem 2.5,nthus the right side of the inequality in Theorem 2.6 is proved.Proof of Theorem 2.6(left

24、 side)nConstruct two portfolios at time tnIf in t,T,the American put option$P$is not early exercised,thenProof of Theorem 2.6(left side)cont.1Namely,when t=TProof of Theorem 2.6(left side)cont.2nIf the American put option P is early exercised at time ,thennBy Theorem 2.2,2.5Proof of Theorem 2.6(left

25、 side)cont.3nAccording to the arbitrage-free principle and Theorem 2.1,there must benThat is,nThe Theorem has been proved.Theorem 2.7nLet be the price of a European call option with the strike price K.nFor with the same expiration date,Financial Meaning of Theorem 2.7nFor 2 European call options wit

26、h the same expiration date,nthe option with strike price ,leaves its holder profit room and is therefore priced ,nthe difference between the two options shall not exceed the difference between the strike prices.Proof of Theorem 2.7(left side)nLeave the right side part to readernConstruct two portfol

27、ios at t:nwhen t=T:Proof of Theorem 2.7(left side)cont.1nCase 1 Proof of Theorem 2.7(left side)cont.2nCase 2 So,Proof of Theorem 2.7(left side)cont.3nCase 3nThus,when t=TnBy Theorem 2.1&Arbitrage Free Principle,for 0tT,Theorem 2.8nFor two European put options with the same expiration date,if thenThe

28、orem 2.9nEuropean call(put)option price is a convex function of K,i.e.,Proof of Theorem 2.9nOnly prove the first one,the second one left to the readernConstruct two portfolios at t=0nOn the expiration date t=T,nDiscuss in 4 casesProof of Theorem 2.9 cont.1nCase 1nCase 2 Therefore Proof of Theorem 2.

29、9 cont.2nCase 3nCase 4Proof of Theorem 2.9 cont.3nIn all Cases,when t=T nButnBy Arbitrage Free P.and Theorem 2.1 Theorem 2.10nEuropean call(put)option price is a linear homogeneous function of the underlying asset price and the strike price K.i.e.for Financial Meaning of Theorem 2.10nConsider buying

30、 European options,with each option to purchase one share of a stock on the expiration date at strike price K;Also consider buying 1 European option to purchase shares of the same stock at strike price K on the expiration date;The money spent on the options in these two cases must be equal.nProof leaves to exercise