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1. Fundament of Financial Mathematics -- Option Pricing liang_jin@mail.tongji.edu.cn 梁进

2. Chapter 1 Risk Management & Financial Derivative

3. 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 attitudes toward risks • Risk aversion • Risk seeking

4. 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., then the corresponding risk management instruments are: stock futures, bond futures, etc..

5. 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, simultaneously. = Underlying asset put or call Derivative call or put

6. 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 position - the buyer in a contract • short position - the seller in a contract • delivery price - the specified price • maturity - specified future time

7. Future Short position Long position K 0 K 0

8. Futures • same as a forward contract • have evolved from standardization of forward contracts • differences – • futures are generally 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

9. 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 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

10. Two Options • A call option - 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 date • exercise - the action to perform the buying or selling of the asset according to the option contract

11. 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.

12. Total Gain of an Option put option Call option 0 K 0 K p [Total gain]=[Gain of the option at expiration]-[Premium]

13. Option Pricing • risky asset’s 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 out

14. Types of Traders • Hedger - to invest on both sides to avoid loss • Speculator - to take action characterized by willing to risk with one's 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 arbitrageur tradessimultaneously at different markets to gain riskless instant profits

15. Hedger Example • In 90 days, A pays B£1000,000 • To avoid risk, A has 2 plans • Purchase 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. A pays a premium of $64,000 (4%)

16. 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 purchase a call option to buy 10,000 shares at the strike price $68.0 per share on August 22

17. 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 Return =(660-666)/666*100%=-0.9% • Strategy B loss all investment Return = - 100%

18. Chapter 2 Arbitrage-Free Principle

19. Financial Market • Two Kinds of Assets • Risk free asset • Bond • Risky asset • Stocks • Options • …. • Portfolio – an investment strategy to hold different assets

20. Investment • At time 0, invest S • When t=T, • Payoff = • Return = • For a risky asset, the return is uncertain, i.e., S is a random variable

21. 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

22. Arbitrage Opportunity • Self-financing - during [0, T] no add or withdraw fund • Arbitrage Opportunity - A self-financing investment, and Probability Prob

23. Arbitrage Free Theorem • Theorem2.1 • the market is arbitrage-free in time [0, T], • are any 2 portfolios satisfying &

24. Proof of Theorem • Suppose false, i.e., • Denote • B is a risk-free bond satisfying • Construct a portfolio at

25. Proof of Theorem cont. • r – risk free interest rate, at t=T • Then • From the supposition

26. Proof of Theorem cont. • It follows • Contradiction!

27. Corollary 2.1 • Market is arbitrage free • if portfolios satisfying • then for any

28. Proof of Corollary • Consider • Then • By Theorem, for • Namely

29. Proof of Corollary 2.1 • In the same way • Then • Corollary has been proved.

30. Option Pricing • European Option Pricing • Call-Put Parity for European Option • American Option Pricing • Early Exercise for American Option • Dependence of Option Pricing on the Strike Price

31. Assumptions • The market is arbitrage-free • All transactions are free of charge • The risk-free interest rate r is a constant • The underlying asset pays no dividends

32. Notations • ------ the risky asset price, • ------ European call option price, • ------ European put option price, • ------ American call option price, • ------ American put option price, • K ------ the option's strike price, • T ------ the option's expiration date, • r ------ the risk-free interest rate.

33. Theorem 2.2 • For European option pricing, the following valuations are true:

34. Proof of Theorem 2.2 • lower bound of (upper leaves to ex.) • consider two portfolios at t=0:

35. Proof of Theorem 2.2 cont. • At t=T, • and • By Theorem 2.1 • i.e.

36. Proof of Theorem 2.2 cont. cont. • Now consider a European call option c • Since • and • By Theorem 2.1 when t<T • i.e. • Together with last inequality, 2.2 proved.

37. Theorem 2.3 • For European Option pricing, there holds call-put parity

38. Proof of Theorem 2.3 • 2 portfolios when t=0 • when t=T

39. Proof of Theorem 2.3 cont. • So that • By Corollary 2.1 • i.e. call-put parity holds

40. Theorem 2.4 • For American option pricing, • if the market is arbitrage-free, • then

41. Proof of Theorem 2.4 • Take American call option as example. • Suppose not true, i.e., s.t • At time t, take cash to buy the American call option and exercise 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.

42. American Option v.s. European 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

43. Theorem 2.5 • If a stock S does not pay dividend, then • i.e., the ``early exercise" term is of no use for American call option on a non-dividend-paying stock.

44. Proof of Theorem 2.5 • By above inequalities, there holds • This indicates it is unwise to early exercise this option

45. Theorem 2.6 • If C,P are non-dividend-paying American call and put options respectively, • then,

46. Proof of Theorem 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.

47. Proof of Theorem 2.6 (left side) • Construct two portfolios at time t • If in [t, T], the American put option $P$ is not early exercised, then

48. Proof of Theorem 2.6 (left side) cont.1 Namely, when t=T

49. Proof of Theorem 2.6 (left side) cont.2 • If the American put option P is early exercised at time , then • By Theorem 2.2,2.5

50. Proof of Theorem 2.6 (left side) cont.3 • According to the arbitrage-free principle and Theorem 2.1, there must be • That is, • The Theorem has been proved.