1. Chapter 10 Arrow-Debreu pricing II: The Arbitrage Perspective

2. 10.21 Market Completeness and Complex Security (i) Completeness : financial markets are said to be complete if, for each state of nature , there exists a market for contingent claim or Arrow-Debreu security , i.e., for a claim promising delivery of one unit of the consumption good (or, more generally, the numeraire) if state  is realized and nothing otherwise. (ii) Complex security : a complex security is one that pays off in more than one state of nature. (5, 2, 0, 6) = 5(1, 0, 0, 0) + 2(0, 1, 0, 0) + 0(0, 0, 1,0) + 6(0, 0, 0, 1) P S = 5q 1 + 2q 2 + 6q 4 Complex security S:

3. Proposition 10.1. If markets are complete, any complex security or any cash flow stream can be replicated as a portfolio of Arrow-Debreu securities. Proposition 10.2. If M = N, and all the M complex securities are linearly independent, then (i) it is possible to infer the prices of the A-D state- contingent claims from the complex securities' prices and (ii) markets are effectively complete. Linearly independent = no complex security can be replicated as a portfolio of some of the other complex securities.

4. (3, 2, 0)(1, 1, 1)(2, 0, 2) (1, 0, 0) = w 1 (3, 2, 0) + w 2 (1, 1, 1) +w 3 (2, 0, 2) Thus,1 = 3w 1 + w 2 + 2 w 3 0 = 2 w 1 + w 2 0 = w 2 + 2 w 3

5. (8.4)

6. 10.3 Constructing State Contingent Claims Prices in a Risk-Free World: Deriving the Term Structure

7. (i) 7 7 / 8 % bond priced at 109 25 / 32, or $1'098.8125 / $ 1'000 face value (ii) 5 5 / 8 % bond priced at 100 9 / 32, or $1'002.8125 / $ 1'000 face value The coupons of these bonds are respectively,.07875 * $ 1'000 = $ 78.75 / year.05625 * $ 1'000 = $ 56.25 / year

8. Price of $1.00 in 5 years = $ 0.765

9. Recovering the term structure: Suppose we observe risk-free bonds with 1,2, 3 and 4 year maturities, all selling at par, with coupons of 6%, 6.5%, 8.2% and 9.5% respectively Term structure is constructed as per: r 1 = 6% r 2 solves yielding r 2 = 6.5113% Similarly r 3 = 8.2644% and r 4 = 9.935%

11. P(4yr.8%)=$950.21 Replicating 80 80 80 1080

12. Evaluating a risk-free project as a portfolio of A-D securities = discounting at the term structure. Evaluating a CF: 60 25 150 300

13. Appendix 10.1. Forward Prices and Forward Rates, etc.

15. 10.4 The Value Additivity Theorem, for some constant coefficients A and B. (10.2) p c = Ap a + Bp b. (10.3)

16. Diversifiable risk is not priced Suppose a and b are negatively correlated c is less risky, yet p c must be « in line » with p a and p b Suppose a and b are perfectly negatively correlated. Can be combined to form d, risk free p d must be such that holding d earns the riskless rate How can the risk of a and b be remunerated?

17. Ch.8 Options and Market Completeness 8.1 Introduction 8.2 Using Options to Complete the Market: An Abstract Setting 8.3 Synthesizing State-Contingent Claim: A First Approximation 8.4 Recovering Arrow-Debreu Prices from Option Prices: A Generalization 8.5 Arrow-Debreu Pricing in a Multiperiod Setting 8.6 Conclusions

18. 10.5 Using Options to Complete the Market: An Abstract Setting Proposition 10.3. A necessary as well as sufficient condition for the creation of a complete set of A-D securities is that there exists a single portfolio with the property that options can be written on it and such that its payoff pattern distinguishes among all states of nature. Proposition 10.4: If it is possible to create, using options, a complete set of traded securities, simple put and call options written on the underlying assets are sufficient to accomplish this goal.

19. 10.6 Synthesizing State-Contingent Claim: A First Approximation It is assumed that S T discriminates across all states of nature so that Proposition 8.1 applies; without loss of generality, we may assume that S T takes the following set of values: where S  is the price of this complex security if state  is realized at date T. Assume also that call options are written on this asset with all possible exercise prices, and that these options are traded. Let us also assume that for every state .

20. Consider, for any state, the following portfolio P: Buy one call with E = Sell two calls with E = Buy one call with E =. At any point in time, the value of this portfolio, V P, is

21. Figure 10-1 Payoff Diagram for All Options in the Portfolio P

22. 10.7 Recovering Arrow-Debreu Prices from Option Prices: A Generalization (i) Suppose that S T, the price of the underlying portfolio (we may think of it as a proxy for M), assumes a "continuum" of possible values. (ii) Let us construct the following portfolio : for some small positive number  >0, buy one call with sell one call with buy one call with.

23. Figure 10-2 Payoff Diagram: Portfolio of Options

24. A Generalization (iii) Let us thus consider buying 1 /  units of the portfolio. The total payment, when, is, for any choice of . We want to let, so as to eliminate the payments in the ranges and,.The value of 1 /  units of this portfolio is :

26. The value today of this cash flow is : (10.4) (iv)

27. Figure 10-3 Payoff Diagram for the Limiting Portfolio