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1、CHAPTER FIVE: Options and Dynamic No-Arbitrage1 A Brief Introduction of OptionsAn option is the right of choice exercised in future. The holder (buyer, or longer) of the option has a right but not an obligation to buy or sell a special amount of the asset with a special quality at a pre-determined p
2、rice. Call and put Exercise price Expiration date American options (C and P) vs. European options ( c and p)2The payoff profiles of call and putCallPutLongShort+_XXSTST00LongShort+_XXSTST00In-the-money, out-of-the-money, at-the-money, intrinsic value and time value A Brief Introduction of Options (C
3、ont.)3The Basic No-Arbitrage1) ,2) 3) ,4) If , then , 5) 4The Basic No-Arbitrage (Cont.)The underlying is a non-dividend-paying stockSuppose , then Arbitrage Immediate Cash Flow Position Cash Flow on the expired date Short a stockLong an European call Long riskless securityNet cash flowsArbitrage Op
4、portunity!5The Basic No-Arbitrage (Cont.)PropositionIf the period to expiration is very long, the value of an European call is almost equal to its underlying.PropositionAn American call on a non-dividend-paying stock should never be exercised prior to the expiration date.6 The relationship between A
5、merican options and European options?andConclusion:7The Parity of Call and Put The underlying is a non-dividend-paying stockS can be replicated by c, p and riskless securitySupposePosition Cash flow at Cash flow at time T time tBuy a shareShort a callLong a putShort treasuryNet cash flowArbitrage!8
6、Relationship between exercise and forward price Non-dividend-paying stocks American call and put?9 Non-dividend-paying stocks American call and put (Cont.)Position Cash flow at Cash flow at time when put exercised time tShort a shareLong an Amer. callShort an Amer. putLong treasuryNet cash flow10 No
7、n-dividend-paying stocks American call and put (Cont.) Underlying is dividend-paying stockPresent value of dividends at time tPresent value of a long stock forward positionPresent value of a short stock forward position11 Underlying is dividend-paying stockFor European call and putFor American call
8、and putHolds for non-dividend-paying stock underlyingDividend paidProved!How to prove it?Please see the next page!12 Proof ofPosition Cash flow at Cash flow at time when put exercised time tShort a shareEffect of dividendsLong an Euro. callShort an Amer. putLong treasuryNet cash flow13 Proposition!F
9、or an American call, when there are dividends with big amount, the call may be early exercised at a time immediately before the stock goes ex-dividend.Question:If there are n ex-dividend dates anticipated, whats the optimal strategy to early exercise an American call?Answer:Please read the last para
10、graph of page 74 of the textbook.14Dynamic No-Arbitrage t=0 t=1 t=2Bond ABond B 15 Replication step by stepUsing Bond A and riskless security with market value to replicate Bond Bs value in the above step16 Replication step by step (Cont.)Replicating the blow binomial tree by using Bond A and riskle
11、ss security with market valueReplicating the left binomial tree by using Bond A and riskless security with market value17 Self-financingNotes:1.Dynamic replication is forward while the procedure of pricing is backward2.Short sale is available for self-financing18Option PricingBinomial Trees One-Step
12、 Binomial Model Non-dividend-paying stocks European callUsing the underlying stock and riskless security with market value to replicate the European call?Sensitivity of the replicating portfolio to the change of the stock.19 Is probability relevant to option pricing?Probability distributionAnswer:1.
13、Directly: No!2.Indirectly: Yes!Probability distribution is not relevant to No arbitrage pricing 20 One-Step Binomial Model (Cont.) NotationNo ArbitrageReplicating :Short sale of riskless security21Risk-Neutrality Risk-Aversion A Mini Case Tossing a CoinHeadTailFair GameFair GameRisk premiumRisk disc
14、ountInvestment Gambling Investors: risk-averseGamblers: risk-preferFrom real economy be charged by casino risk-neutral22 Risk-Neutral Pricingrisk-neutral probabilitymean or expectation on risk-neutral probabilitydiscounted by risk-free rateAnalysis becomes very simple!andIn an imaginary worldA risk-
15、neutral world23 What Kind of Problems Can Be Resolved in an Imaginary Risk-Neutral World?Proposition : If a problem with its resolving procedure is fully irrelevant to peoples risk-preference, then it can be resolved in an imaginary risk-neutral world and the solution would be still valid in the rea
16、l world.Proposition : No-Arbitrage equilibrium in financial markets is fully irrelevant to peoples risk-preference. Therefore, risk-neutral pricing is valid equilibrium pricing. Risk-neutral pricing and no-arbitrage pricing must be equivalent to each other.24 Risk-Neutral Pricing (Multi-Step Binomia
17、l Model ) t=0 t=1 t=2The Underlying StockThe Call25 Risk-Neutral Pricing (Cont. )Generalizing: 26 A Mini Case The Underlying StockThe Call t=0 t=1 t=2 t=0 t=1 t=2 Risk-Neutral Pricing:27 A Mini Case (Cont.) Dynamic No-Arbitrage Pricing:28 Implication of Risk-Neutral PricingMean or mathematical expec
18、tation with probability in the real worldDiscount rates with risk premiumRisk-free rate used as discount rates without risk premiumQuestion:Does risk-neutral probability exist and is it unique?Mean or mathematical expectation with risk-neutral probability in the imaginary world29Fundamental Theorems
19、 of Financial EconomicsThe First Financial Economics Theorem:Risk-neutral probabilities exist if and only if there are no riskless arbitrage opportunities.The Second Financial Economics Theorem:The risk-neutral probabilities are unique if and only if the market is complete.The Third Financial Econom
20、ics Theorem:Under certain conditions, the ability to revise the portfolio of available securities over time can dynamically make up for the missing securities and effectively complete the market.30 Problem and Inverse Problem many investors make portfolio changes each portfolios change is limited th
21、e aggregation creates a large volume of buying and selling to restore equilibrium implying arbitrage opportunity exists each arbitrageur wants to take as large position as possible a few arbitrageurs bring the price pressures to restore equilibriumInverse Problem:Knowing the market prices of securit
22、ies, determine the markets risk-neutral probabilities.Problem:Knowing the markets risk-neutral probabilities, determine the market prices of securities.Unfortunately, are actual securities markets like this ? Are they incomplete ? So it would seem that we will not be able to solve the inverse proble
23、m; that is, although risk-neutral probabilities may exist, they are not unique. However, in 1954, economist Kenneth Arrow saved the day by stating the third fundamental theorem of financial economics, the critical idea behind modern securities pricing theory.31 Equivalent Martingale Definition:The r
24、isk-neutral valuation approach is sometimes referred to as using equivalent martingale measure, i.e., the risk-neutral probability is referred to an equivalent martingale measure (probability distribution).32Summary of Chapter Five1.No-Arbitrage The Key of Finance Theory, Especially For Derivatives Such as Options. 2.Dynamic No-Arbitrage Pricing Risk-Neutral Pricing.3.Does Risk-Neutral Probability Exist and Is It Unique?4.The Core of Finance Theory The Fundamental Theorems of Financial Economics33
