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1、liang_,梁进,Fundament of,Financial Mathematics,- Option Pricing,Chapter 1,Risk Management & Financial Derivative,Risk,Risk - uncertainty of the outcome bring unexpected gains cause unforeseen losses Risks in Financial Market asset (stocks, ), interest rate, foreign exchange, credit, commodity, Two att
2、itudes toward risks Risk 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 contracts Future Options If the underlying assets are stocks, bonds etc.,
3、 then the corresponding risk management 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 i
4、n the derivatives markets, simultaneously.,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 certainty traded OTC long positi
5、on - the buyer in a contract short position - the seller in a contract delivery price - the specified price maturity - specified future time,Future,K,K,0,0,Long position,Short position,Futures,same as a forward contract have evolved from standardization of forward contracts differences futures are g
6、enerally traded on an exchange a future contract contains standardized articles the 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 s
7、pecified future time a certain amount of an underlying asset at 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 claim Has a much higher level of leverage,Two Options,A call option -
8、 a contract to buy at a specified future time a certain amount of an 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 d
9、ate exercise - the action to perform the buying or selling of the asset according 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
10、an Option,K,K,0,0,Call option,put option,p,Total gain= Gain of the option at expiration-Premium,Option Pricing,risky assets price is a random variable the price of any option derived from risky asset is also random the price also depends on time t there exists a function such that known How to find
11、out,Types of Traders,Hedger - to invest on both sides to avoid loss Speculator - to take action 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 observati
12、ons of the same kind of risky assets, taking advantage of the price differences between markets, the 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 plans Purchase a forward contract to buy 10
13、00,000 with $1,650,000 90 days later Purchase a call option to buy 1000,000 with $1,600,000 90 days later. A pays a premium of $64,000 (4%),Speculator Example,Stock A is $66.6 on April 30, may grow A speculator has 2 plans buys 10,000 shares with $666,000 on April 30 pays a premium of $39,000 USD to
14、 purchase a call option to buy 10,000 shares at the strike price $68.0 per share on August 22,Speculator Example cont.,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 Retu
15、rn =(660-666)/666*100%=-0.9% Strategy B loss all investment Return = - 100%,Chapter 2,Arbitrage-Free Principle,Financial Market,Two Kinds of Assets Risk free asset Bond Risky asset Stocks Options . Portfolio an investment strategy to hold different assets,Investment,At time 0, invest S When t=T, Pay
16、off = Return = For a risky asset, the return is uncertain, i.e., S is a random variable,A Portfolio,a risk-free asset B n risky assets a portfolio is called a investment strategy on 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
