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1、,梁进,Fundament of,Financial Mathematics,-Option Pricing,Chapter 1,Risk Management&Financial Derivative,Risk,Risk-uncertainty of the outcomebring unexpected gainscause unforeseen lossesRisks in Financial Marketasset(stocks,),interest rate,foreign exchange,credit,commodity,Two attitudes toward risks Ri
2、sk aversion Risk seeking,Financial Derivatives,Many forms of financial derivatives instruments exist in the financial markets.Among them,the 3 most fundamental financial derivatives instruments:Forward contractsFutureOptionsIf the underlying assets are stocks,bonds etc.,then the corresponding risk m
3、anagement instruments are:stock futures,bond futures,etc.,Risk Management,risk management-underlying assets Method 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,simultaneousl
4、y.,Underlying asset put or call,Derivative call or put,=,Forward Contracts,an agreement to buy or sell at a specified future time a certain amount of an underlying asset at a specified price.an agreement to replace a risk by a certaintytraded OTClong position-the buyer in a contractshort position-th
5、e seller in a contractdelivery price-the specified pricematurity-specified future time,Future,K,K,0,0,Long position,Short position,Futures,same as a forward contracthave evolved from standardization of forward contractsdifferences futures are generally traded on an exchangea future contract contains
6、 standardized articlesthe delivery price on a future contract is generally determined on an exchange,and depends on the market demands,Options,an 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
7、 a specified price.But the holder is under no obligation to exercise the contract.a right,no obligation the holder has to pay premium for this right is a contingent claimHas a much higher level of leverage,Two Options,A call option-a contract to buy at a specified future time a certain amount of an
8、underlying asset at a specified price A put option-a contract to sell at a specified future time a certain amount of an underlying asset at a specified price.exercise price-the specified price expiration date-the specified date exercise-the action to perform the buying or selling of the asset accord
9、ing to the option contract,Option Types,European options-can be exercised only on the expiration date.American options-can be exercised on or prior to the expiration date.Other options Asia option etc.,Total Gain of an Option,K,K,0,0,Call option,put option,p,Total gain=Gain of the option at expirati
10、on-Premium,Option Pricing,risky assets price is a random variablethe price of any option derived from risky asset is also random the price also depends on time tthere exists a function such thatknown How to find out,Types of Traders,Hedger-to invest on both sides to avoid lossSpeculator-to take acti
11、on characterized by willing to risk with ones money by frequently buying and selling derivatives(futures,options)for the prospect of gaining from the frequent price changes.Arbitrage-based on observations of the same kind of risky assets,taking advantage of the price differences between markets,the
12、arbitrageur trades simultaneously at different markets to gain riskless instant profits,Hedger Example,In 90 days,A pays B 1000,000 To avoid risk,A has 2 plansPurchase a forward contract to buy 1000,000 with$1,650,000 90 days later Purchase a call option to buy 1000,000 with$1,600,000 90 days later.
13、A pays a premium of$64,000(4%),Speculator Example,Stock A is$66.6 on April 30,may growA speculator has 2 plansbuys 10,000 shares with$666,000 on April 30pays a premium of$39,000 USD to purchase a call option to buy 10,000 shares at the strike price$68.0 per share on August 22,Speculator Example cont
14、.,Situation I:The stock$73.0 on 8/22.Strategy A Return=(730-666)/666*100%=9.6%Strategy B Return=(730-680-39)/39*100%=28.2%Situation II:The stock$66.0 on 8/22.Strategy A Return=(660-666)/666*100%=-0.9%Strategy B loss all investment Return=-100%,Chapter 2,Arbitrage-Free Principle,Financial Market,Two
15、Kinds of AssetsRisk free asset BondRisky assetStocksOptions.Portfolio an investment strategy to hold different assets,Investment,At time 0,invest SWhen t=T,Payoff=Return=For a risky asset,the return is uncertain,i.e.,S is a random variable,A Portfolio,a risk-free asset Bn risky assets a portfolio is
16、 called a investment strategyon time t,wealth:,portion of the cor.Asset,Arbitrage Opportunity,Self-financing-during 0,T no add or withdraw fund Arbitrage Opportunity-A self-financing investment,and Probability Prob,Arbitrage Free Theorem,Theorem 2.1 the market is arbitrage-free in time 0,T,are any 2
17、 portfolios satisfying&,Proof of Theorem,Suppose false,i.e.,DenoteB is a risk-free bond satisfyingConstruct a portfolio at,Proof of Theorem cont.,r risk free interest rate,at t=TThenFrom the supposition,Proof of Theorem cont.,It followsContradiction!,Corollary 2.1,Market is arbitrage free if portfol
18、ios satisfying then for any,Proof of Corollary,ConsiderThenBy Theorem,for Namely,Proof of Corollary 2.1,In the same wayThen Corollary has been proved.,Option Pricing,European Option PricingCall-Put Parity for European Option American Option PricingEarly Exercise for American Option Dependence of Opt
19、ion Pricing on the Strike Price,Assumptions,The market is arbitrage-free All transactions are free of chargeThe risk-free interest rate r is a constantThe underlying asset pays no dividends,Notations,-the risky asset price,-European call option price,-European put option price,-American call option
20、price,-American put option price,K-the options strike price,T-the options expiration date,r-the risk-free interest rate.,Theorem 2.2,For European option pricing,the following valuations are true:,Proof of Theorem 2.2,lower bound of(upper leaves to ex.)consider two portfolios at t=0:,Proof of Theorem
21、 2.2 cont.,At t=T,and By Theorem 2.1 i.e.,Proof of Theorem 2.2 cont.cont.,Now consider a European call option cSinceandBy Theorem 2.1 when tT i.e.Together with last inequality,2.2 proved.,Theorem 2.3,For European Option pricing,there holds call-put parity,Proof of Theorem 2.3,2 portfolios when t=0wh
22、en t=T,Proof of Theorem 2.3 cont.,So thatBy Corollary 2.1i.e.call-put parity holds,Theorem 2.4,For American option pricing,if the market is arbitrage-free,then,Proof of Theorem 2.4,Take American call option as example.Suppose not true,i.e.,s.tAt time t,take cash to buy the American call option and e
23、xercise it,i.e.,to buy the stock S with cash K,then sell the stock in the stock market to receive in cash.Thus the trader gains a riskless profit instantly.But this is impossible since the market is assumed to be arbitrage-free.Therefore,must be true.can be proved similarly.,American Option v.s.Euro
24、pean Option,For an American option and a European option with the same expiration date T and the same strike price K,since the American option can be early exercised,its gaining opportunity must be=that of the European option.Therefore,Theorem 2.5,If a stock S does not pay dividend,theni.e.,the earl
25、y exercise term is of no use for American call option on a non-dividend-paying stock.,Proof of Theorem 2.5,By above inequalities,there holdsThis indicates it is unwise to early exercise this option,Theorem 2.6,If C,P are non-dividend-paying American call and put options respectively,then,Proof of Th
26、eorem 2.6(right side),It follows from call-put party,and Theorem 2.5,thus the right side of the inequality in Theorem 2.6 is proved.,Proof of Theorem 2.6(left side),Construct two portfolios at time tIf in t,T,the American put option$P$is not early exercised,then,Proof of Theorem 2.6(left side)cont.1
27、,Namely,when t=T,Proof of Theorem 2.6(left side)cont.2,If the American put option P is early exercised at time,thenBy Theorem 2.2,2.5,Proof of Theorem 2.6(left side)cont.3,According to the arbitrage-free principle and Theorem 2.1,there must beThat is,The Theorem has been proved.,Theorem 2.7,Let be t
28、he price of a European call option with the strike price K.For with the same expiration date,Financial Meaning of Theorem 2.7,For 2 European call options with the same expiration date,the option with strike price,leaves its holder profit room and is therefore priced,the difference between the two op
29、tions shall not exceed the difference between the strike prices.,Proof of Theorem 2.7(left side),Leave the right side part to readerConstruct two portfolios at t:when t=T:,Proof of Theorem 2.7(left side)cont.1,Case 1,Proof of Theorem 2.7(left side)cont.2,Case 2 So,Proof of Theorem 2.7(left side)cont
30、.3,Case 3Thus,when t=TBy Theorem 2.1&Arbitrage Free Principle,for 0tT,Theorem 2.8,For two European put options with the same expiration date,if then,Theorem 2.9,European call(put)option price is a convex function of K,i.e.,Proof of Theorem 2.9,Only prove the first one,the second one left to the read
31、erConstruct two portfolios at t=0On the expiration date t=T,Discuss in 4 cases,Proof of Theorem 2.9 cont.1,Case 1Case 2 Therefore,Proof of Theorem 2.9 cont.2,Case 3Case 4,Proof of Theorem 2.9 cont.3,In all Cases,when t=T ButBy Arbitrage Free P.and Theorem 2.1,Theorem 2.10,European call(put)option pr
32、ice is a linear homogeneous function of the underlying asset price and the strike price K.i.e.for,Financial Meaning of Theorem 2.10,Consider buying 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.Proof leaves to exercise,
