期权定价与动态套利均衡分析课件

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.Terms of Options 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 Introducti

3、on of Options (Cont.)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 f

4、lowsArbitrage Opportunity!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 relati

5、onship between American 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

6、flowArbitrage!8 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 treasuryNe

7、t cash flow10 Non-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 putFo

8、r American call 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 flow

9、13 Proposition!For 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 re

10、ad the last paragraph 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 B

11、ond A and riskless security with market valueReplicating the left binomial tree by using Bond A and riskless security with market value17 Self-financingNotes:Dynamic replication is forward while the procedure of pricing is backwardShort sale is available for self-financing18Option PricingBinomial Tr

12、ees One-Step 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 distribut

13、ionAnswer:Directly: No!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 premiumR

14、isk discountInvestment 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 worl

15、dA risk-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

16、 the real 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

17、 Binomial 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 mathematic

18、al expectation 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

19、Theorems 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 Financia

20、l Economics 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 li

21、mited the 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

22、 securities, 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 problem; that is, although risk-neutral probabilities may ex