Asset Pricing Chapter X. Arrow-Debreu pricing II: The Arbitrage - PowerPoint PPT Presentation

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization Asset Pricing Chapter X. Arrow-Debreu pricing II: The Arbitrage Perspective June 22, 2006 Asset Pricing

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization 10.1 Market Completeness and Complex Security Completeness : financial markets are said to be complete if, for each state of nature θ , there exists a θ , i.e., for a claim promising delivery of one until of the consumption good (or, more generally, the numeraire) if state θ is realized and nothing otherwise. 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 = 5 q 1 + 2 q 2 + 6 q 4 . Asset Pricing

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization Proposition 10.1 If markets are complete, any complex security or any cash flow stream can be replicated as a portfolio of Arrow-S 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 form 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. Asset Pricing

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization ( 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 = 3 w 1 + w 2 + 2 w 3 0 = 2 w 1 + w 2 0 = w 2 + 2 w 3 w 1 1 w 2 1 w 3       3 1 2 1 0 0 1  = w 1 2 w 2 2 w 3 2 1 0 0 1 0    2   w 1 3 w 2 3 w 3 0 1 2 0 0 1 3 Asset Pricing

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization t = 0 1 2 3 ... T ˜ ˜ ˜ ˜ − I 0 CF 1 CF 2 CF 3 ... CF T T N � � NPV = − I 0 + q t ,θ CF t ,θ . (1) t = 1 θ = 1 Asset Pricing

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure Table 10.2: Risk-Free Discount Bonds As Arrow-Debreu Securities Current Bond Price Future Cash Flows t = 0 1 2 3 4 ... T − q 1 $ 1 , 000 − q 2 $ 1 , 000 ... − q T $ 1 , 000 where the cash flow of a “ j -period discount bond” is just t = 0 1 ... j j + 1 ... T − q j 0 0 $ 1 , 000 0 0 0 Asset Pricing

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization (i) 7 7 8 % bond priced at 109 25 32 , or $ 1097 . 8125 / $ 1 , 000 of face value (ii) 5 5 8 % bond priced at 100 9 32 , or $ 1002 . 8125 / $ 1 , 000 of face value The coupons of these bonds are respectively, . 07875 ∗ $ 1 , 000 = $ 78 . 75 / year . 05625 ∗ $ 1 , 000 = $ 56 . 25/year Asset Pricing

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization Table 10.3: Present And Future Cash Flows For Two Coupon Bonds Bond Type Cash Flow at Time t t = 0 1 2 3 4 5 7 7 / 8 bond: − 1 , 097 . 8125 78 . 75 78 . 75 78 . 75 78 . 75 1 , 078 . 75 5 5 / 8 bond: − 1 , 002 . 8125 56 . 25 56 . 25 56 . 25 56 . 25 1 , 056 . 25 Table 10.4 : Eliminating Intermediate Payments Bond Cash Flow at Time t t = 0 1 2 3 4 5 − 1x 7 7 / 8 bond: + 1 , 097 . 8125 − 78 . 75 − 78 . 75 − 78 . 75 − 78 . 75 − 1 , 078 . 75 + 1 . 4x 5 5 / 8 bond: − 1 , 403 . 9375 78 . 75 78 . 75 78 . 75 78 . 75 1 , 478 . 75 Difference: − 306 . 125 0 0 0 0 400 . 00 Price of £1 in 5 years = £0.765 Asset Pricing

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization Table 10.5: Date Claim Prices vs. Discount Bond Prices Price of a N year claim Analogous Discount Bond Price ($1,000 Denomina- tion) N = 1 q 1 = $1/1.06 = $.94339 $ 943.39 q 2 = $1/(1.065113) 2 = $.88147 N = 2 $ 881.47 q 3 = $1/(1.072644) 3 = $.81027 N = 3 $ 810.27 q 4 = $1/1.09935) 4 =$ .68463 N = 4 $ 684.63 Table 10.6: Discount Bonds as Arrow-Debreu Claims Bond Price (t = 0) CF Pattern t = 1 2 3 4 1-yr discount -$943.39 $1,000 2-yr discount -$881.47 $1,000 3-yr discount -$810.27 $1,000 4-yr discount -$684.63 $1,000 Asset Pricing

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization Replicating 80 80 80 1080 Table 10.7: Replicating the Discount Bond Cash Flow Bond Price ( t = 0 ) CF Pattern t = 1 2 3 4 08 1-yr discount ( . 08 )( − 943 . 39 ) = − $ 75 . 47 $80 (80 state 1 A-D claims) 08 2-yr discount ( . 08 )( − 881 . 47 ) = − $ 70 . 52 $80 (80 state 2 A-D claims) 08 3-yr discount ( . 08 )( − 810 . 27 ) = − $ 64 . 82 $80 1.08 4-yr discount ( 1 . 08 )( − 684 . 63 ) = − $ 739 . 40 $1,080 Asset Pricing

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization Evaluating a CF: 60 25 150 300 „ $ . 94339 at t=0 „ $ . 88147 at t=0 « « p = ($ 60 at t=1 ) + ($ 25 at t=2 ) + ... $ 1 at t=1 $ 1 at t=2 1 . 00 1 . 00 = ($ 60 ) + ($ 25 ) ( 1 + r 2 ) 2 + ... 1 + r 1 1 . 00 1 . 00 = ($ 60 ) + ($ 25 ) ( 1 . 065113 ) 2 + ... 1 . 06 Evaluating a risk-free project as a portfolio of A-D securities=discounting at the term structure. Asset Pricing

10.1 Market Completeness and Complex Security 10.2 Constructing State Contingent Claims Prices in a risk-Free World: Deriving the term Structure 10.3 The Value Additivity Theorem 10.4 Using Options to Complete the Market: An Abstract Setting 10.5 Synthesizing State-Contingent Claim: A First Approximation 10.6 Recovering Arrow-Debreu Prices form Option Prices: A Generalization Appendix 10.1 Forward Prices and Forward Rates ( 1 + r 2 ) 2 ( 1 + r 1 )( 1 + 1 f 1 ) = ( 1 + r 1 )( 1 + 1 f 2 ) 2 ( 1 + r 3 ) 3 = ( 1 + r 2 ) 2 ( 1 + 2 f 1 ) ( 1 + r 3 ) 3 , etc . = Asset Pricing