2. 1 CHAPTER FIVE: Options and Dynamic No-Arbitrage

3. 2 A Brief Introduction of Options An 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 price. Call and put Exercise price Expiration date American options (C and P) vs. European options ( c and p)

4. 3 The payoff profiles of call and put CallPut LongShort ++ __ XX STST STST 0 0 LongShort + + __ XX STST STST 0 0 In-the-money, out-of-the-money, at-the-money, intrinsic value and time value — A Brief Introduction of Options (Cont.)

5. 4 The Basic No-Arbitrage 1), 2) 3), 4) If, then, 5)

6. 5 The Basic No-Arbitrage (Cont.) The underlying is a non-dividend-paying stock Suppose, then Arbitrage Immediate Cash Flow Position Cash Flow on the expired date Short a stock Long an European call Long riskless security Net cash flows Arbitrage Opportunity !

7. 6 The Basic No-Arbitrage (Cont.) Proposition If the period to expiration is very long, the value of an European call is almost equal to its underlying. Proposition An American call on a non-dividend-paying stock should never be exercised prior to the expiration date.

8. 7 The relationship between American options and European options ? and Conclusion:

9. 8 The Parity of Call and Put The underlying is a non-dividend-paying stock S can be replicated by c, p and riskless security Suppose Position Cash flow at Cash flow at time T time t Buy a share Short a call Long a put Short treasury Net cash flow Arbitrage!

10. 9 Relationship between exercise and forward price Non-dividend-paying stock’s American call and put ?

11. 10 Non-dividend-paying stock’s American call and put (Cont.) Position Cash flow at Cash flow at time when put exercised time t Short a share Long an Amer. call Short an Amer. put Long treasury Net cash flow

12. 11 Non-dividend-paying stock’s American call and put (Cont.) Underlying is dividend-paying stock Present value of dividends at time t Present value of a long stock forward position Present value of a short stock forward position

13. 12 Underlying is dividend-paying stock For European call and put For American call and put Holds for non- dividend-paying stock underlying Dividend paid Proved! How to prove it? Please see the next page!

14. 13 Proof of Position Cash flow at Cash flow at time when put exercised time t Short a share Effect of dividends Long an Euro. call Short an Amer. put Long treasury Net cash flow

15. 14 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, what’s the optimal strategy to early exercise an American call? Answer: Please read the last paragraph of page 74 of the textbook.

16. 15 Dynamic No-Arbitrage t=0 t=1 t=2 Bond A Bond B

17. 16 Replication step by step Using Bond A and riskless security with market value to replicate Bond B’s value in the above step

18. 17 Replication step by step (Cont.) Replicating the blow binomial tree by using Bond A and riskless security with market value Replicating the left binomial tree by using Bond A and riskless security with market value

19. 18 Self-financing Notes: 1.Dynamic replication is forward while the procedure of pricing is backward 2.Short sale is available for self-financing

20. 19 Option Pricing—Binomial Trees — One-Step Binomial Model Non-dividend-paying stock’s European call Using 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.

21. 20 Is probability relevant to option pricing? Probability distribution Answer: 1.Directly: No! 2.Indirectly: Yes! Probability distribution is not relevant to No arbitrage pricing

22. 21 — One-Step Binomial Model (Cont.) Notation No Arbitrage Replicating : Short sale of riskless security

23. 22 Risk-Neutrality — Risk-Aversion A Mini Case — Tossing a Coin Head Tail Fair Game Risk premiumRisk discount InvestmentGambling Investors: risk-averseGamblers: risk-prefer From real economy be charged by casino risk-neutral

24. 23 — Risk-Neutral Pricing risk-neutral probability mean or expectation on risk-neutral probability discounted by risk-free rate Analysis becomes very simple! and In an imaginary world A risk- neutral world

25. 24 — 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 people’s risk-preference, then it can be resolved in an imaginary risk-neutral world and the solution would be still valid in the real world. Proposition : No-Arbitrage equilibrium in financial markets is fully irrelevant to people’s risk-preference. Therefore, risk- neutral pricing is valid equilibrium pricing. Risk- neutral pricing and no-arbitrage pricing must be equivalent to each other.

26. 25 — Risk-Neutral Pricing (Multi-Step Binomial Model ) t=0 t=1 t=2 The Underlying Stock The Call

27. 26 — Risk-Neutral Pricing (Cont. )Cont. Generalizing:

28. 27 — A Mini Case The Underlying StockThe Call t=0 t=1 t=2 Risk-Neutral Pricing:

29. 28 — A Mini Case (Cont.)Cont. Dynamic No-Arbitrage Pricing:

30. 29 — Implication of Risk-Neutral Pricing Mean or mathematical expectation with probability in the real world Discount rates with risk premium Risk-free rate used as discount rates without risk premium Question: Does risk-neutral probability exist and is it unique? Mean or mathematical expectation with risk-neutral probability in the imaginary world

31. 30 Fundamental Theorems of Financial Economics The 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 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.

32. 31 — Problem and Inverse Problem many investors make portfolio changes each portfolio’s change is limited 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 equilibrium Inverse Problem: Knowing the market prices of securities, determine the market’s risk-neutral probabilities. Problem: Knowing the market’s 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 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.

33. 32 — Equivalent Martingale Definition: The risk-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).

34. 33 Summary of Chapter Five 1.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 Economics