Asset Pricing Chapter XI. The Martingale Measure: Part I June 20, - PowerPoint PPT Presentation

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Asset Pricing Chapter XI. The Martingale Measure: Part I June 20, 2006 Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing 1 (CAPM) E ˜ E ˜ E ˜ E ˜ CF 1 CF 2 CF 3 CF τ − Π τ ; 2 + π ) 2 ; 3 + π ) 3 ; or . ( 1 + r f ( 1 + r f ( 1 + r f ( 1 + r f τ ) τ 1 + π ) 2 (Risk Neutral) E ˜ ˆ CF τ τ ) τ ; ( 1 + r f 3 (Arrow-Debreu) X q ( θ τ ) CF ( θ τ ) , θτ ∈ Θ τ r M )[ E ˜ rM − rf “ ” ˜ − cov ( ˜ E CF j , t + 1 CF j , t + 1 , ˜ ] σ 2 M p j , t = , 1 + r f Asset Pricing

11.1 Introduction 11.2 The setting and the intuition Existence of Risk Neutral Probabilities 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing The setting and the intuition 2 dates J possible states of nature at date 1 State j = θ j with probability π j Risk free security q b ( 0 ) = 1, q b ( 1 ) ≡ q b ( θ j , 1 ) = ( 1 + r f ) i=1,..., N fundamental securities with prices q e ( 0 ) , q e i ( θ j , 1 ) Securities market may or may not be complete S is the set of all fundamental securities, including bond and linear combination thereof Asset Pricing

11.1 Introduction 11.2 The setting and the intuition Existence of Risk Neutral Probabilities 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Existence of a set of numbers π RN , Σ π RN = 1 s.t j j J 1 1 � q e ( 1 + r f ) E π RN ( q e π RN q e i ( 0 ) = i ( θ, 1 )) = i ( θ j , 1 ) (1) j ( 1 + r f ) j = 1 � q e � q e i ( θ 1 , 1 ) i ( θ J , 1 ) � � q e i ( 0 ) = π RN + ...... + π RN , i = 1 , 2 , ..., N , 1 J 1 + r f 1 + r f (2) No solution if: q e s ( 0 ) = q e k ( 0 ) with q e k ( θ j , 1 ) ≥ q e s ( θ j , 1 ) for all j , and q e  , 1 ) > q e k ( θ ˆ s ( θ ˆ  , 1 ) (3) = arbitrage opportunity Asset Pricing

Definition 11.1 11.1 Introduction Definition 11.2 11.2 The setting and the intuition Proposition 11.1 11.3 Notation, Definitions and Basic Results Proposition 11.2 Arrow-Debreu Pricing Proof of Proposition 11.2 Uniqueness Consider a portfolio, P , composed of n b P risk-free bonds and n i P units of risky security i , i = 1 , 2 , ..., N . N V P ( 0 ) = n b P q b ( 0 ) + � n i P q e i ( 0 ) , (4) i = 1 N V P ( θ j , 1 ) = n b P q b ( 1 ) + � n i P q e i ( θ j , 1 ) . (5) i = 1 Asset Pricing

Definition 11.1 11.1 Introduction Definition 11.2 11.2 The setting and the intuition Proposition 11.1 11.3 Notation, Definitions and Basic Results Proposition 11.2 Arrow-Debreu Pricing Proof of Proposition 11.2 Uniqueness Definition 11.1 A portfolio P in S constitutes an arbitrage opportunity provided the following conditions are satisfied: ( i ) V P ( 0 ) = 0 , (6) ( ii ) V P ( θ j , 1 ) ≥ 0 , for all j ∈ { 1 , 2 , . . ., J } , ( iii ) V P ( θ ˆ  , 1 ) > 0 , for at least one ˆ  ∈ { 1 , 2 , . . ., J } . Asset Pricing

Definition 11.1 11.1 Introduction Definition 11.2 11.2 The setting and the intuition Proposition 11.1 11.3 Notation, Definitions and Basic Results Proposition 11.2 Arrow-Debreu Pricing Proof of Proposition 11.2 Uniqueness Definition 11.2 � J � π RN A probability measure j = 1 defined on the set of states ( θ j , j j = 1 , 2 , ..., J ) , is said to be a risk-neutral probability measure if π RN ( i ) > 0 , for all j = 1 , 2 , ..., J , and (7) j � ˜ q e i ( θ, 1 ) � ( ii ) q e i ( 0 ) = E π RN , 1 + r f for all fundamental risky securities i = 1 , 2 , ..., N in S . Asset Pricing

Definition 11.1 11.1 Introduction Definition 11.2 11.2 The setting and the intuition Proposition 11.1 11.3 Notation, Definitions and Basic Results Proposition 11.2 Arrow-Debreu Pricing Proof of Proposition 11.2 Uniqueness Table 11.1: Fundamental Securities for Example 11.1 Period t = 0 Prices Period t = 1 Payoffs θ 1 θ 2 q b ( 0 ) : 1 q b ( 1 ) : 1.1 1.1 q e ( 0 ) : 4 q e ( θ j , 1 ) : 3 7 complete markets no arbitrage opportunities "objective" state probabilities? Asset Pricing

Definition 11.1 11.1 Introduction Definition 11.2 11.2 The setting and the intuition Proposition 11.1 11.3 Notation, Definitions and Basic Results Proposition 11.2 Arrow-Debreu Pricing Proof of Proposition 11.2 Uniqueness Table 11.2: Fundamental Securities for Example 11.2 Period t = 0 Prices Period t = 1 Payoffs θ 1 θ 2 θ 3 q b ( 0 ) : 1 q b ( 1 ) : 1.1 1.1 1.1 q e q e 1 ( 0 ) : 2 1 ( θ j , 1 ) : 3 2 1 q e q e 2 ( 0 ) : 3 2 ( θ j , 1 ) : 1 4 6 „ 3 „ 2 „ 1 « « « 2 = π RN + π RN + π RN 1 2 3 1 . 1 1 . 1 1 . 1 „ 1 „ 4 „ 6 « « « 3 = π RN + π RN + π RN 1 2 3 1 . 1 1 . 1 1 . 1 1 = π RN + π RN + π RN . 1 2 3 The solution to this set of equations, π RN π RN π RN = . 3 , = . 6 , = . 1 , 1 2 3 Asset Pricing

Definition 11.1 11.1 Introduction Definition 11.2 11.2 The setting and the intuition Proposition 11.1 11.3 Notation, Definitions and Basic Results Proposition 11.2 Arrow-Debreu Pricing Proof of Proposition 11.2 Uniqueness Table 11.3: Fundamental Securities for Example 11.3 Period t = 0 Prices Period t = 1 Payoffs θ 1 θ 2 θ 3 q b ( 0 ) : 1 q b ( 1 ) : 1.1 1.1 1.1 q e q e 1 ( 0 ) : 2 1 ( θ j , 1 ) : 1 2 3 � 1 � 2 � 3 � � � π RN + π RN + π RN = 2 1 2 3 1 . 1 1 . 1 1 . 1 π RN + π RN + π RN 1 = 1 2 3 System indeterminate; many solutions Asset Pricing

Definition 11.1 11.1 Introduction Definition 11.2 11.2 The setting and the intuition Proposition 11.1 11.3 Notation, Definitions and Basic Results Proposition 11.2 Arrow-Debreu Pricing Proof of Proposition 11.2 Uniqueness 2 . 2 − π RN = 2 π RN + 3 π RN 1 2 3 1 − π RN = π RN + π RN , 1 2 3 π RN > 0 1 π RN = . 8 − 2 π RN > 0 2 1 π RN = . 2 + π RN > 0 3 1 0 < π RN < . 4 , 1 ( π RN , π RN , π RN ) ∈ { ( λ, 8 − 2 λ, . 2 + λ ) : 0 < λ < . 4 } 1 2 3 Risk Neutral probabilities are not uniquely defined! Asset Pricing

Definition 11.1 11.1 Introduction Definition 11.2 11.2 The setting and the intuition Proposition 11.1 11.3 Notation, Definitions and Basic Results Proposition 11.2 Arrow-Debreu Pricing Proof of Proposition 11.2 Uniqueness Table 11.4: Fundamental Securities for Example 11.4 Period t = 0 Prices Period t = 1 Payoffs θ 1 θ 2 θ 3 q b ( 0 ) : 1 q b ( 1 ) : 1.1 1.1 1.1 q e q e 1 ( 0 ) : 2 1 ( θ j , 1 ) : 2 3 1 q e q e 2 ( 0 ) : 2 . 5 2 ( θ j , 1 ) : 4 5 3 an arbitrage opportunity No solution (or solution with π RN = 0 for some i) i Asset Pricing

Definition 11.1 11.1 Introduction Definition 11.2 11.2 The setting and the intuition Proposition 11.1 11.3 Notation, Definitions and Basic Results Proposition 11.2 Arrow-Debreu Pricing Proof of Proposition 11.2 Uniqueness Proposition 11.1 Consider the two-period setting described earlier in this chapter. Then there exists a risk-neutral probability measure on S , if and only if there are no arbitrage opportunities among the fundamental securities. May not be unique! Until now: Fundamental securities in S Now: Portfolio of fundamental securities. Asset Pricing

Definition 11.1 11.1 Introduction Definition 11.2 11.2 The setting and the intuition Proposition 11.1 11.3 Notation, Definitions and Basic Results Proposition 11.2 Arrow-Debreu Pricing Proof of Proposition 11.2 Uniqueness Proposition 11.2 Suppose the set of securities S is free of arbitrage opportunities. Then for any portfolio ˆ P in S 1 ( 1 + r f ) E π RN ˜ V ˆ P ( 0 ) = V ˆ P ( θ, 1 ) , (8) for any risk-neutral probability measure π RN on S . Asset Pricing

Definition 11.1 11.1 Introduction Definition 11.2 11.2 The setting and the intuition Proposition 11.1 11.3 Notation, Definitions and Basic Results Proposition 11.2 Arrow-Debreu Pricing Proof of Proposition 11.2 Uniqueness Proof of Proposition 11.2 Let ˆ P be an arbitrary portfolio in S , and let it be composed of n b ˆ P bonds and n i P shares of fundamental risky asset i . In the ˆ absence of arbitrage, ˆ P must be priced equal to the value of its constituent securities, in other words, � ˜ N N q e � q b ( 1 ) � � i ( θ, 1 ) P ( 0 ) = n b P q b ( 0 ) + n i P q e i ( 0 ) = n b n i � � V ˆ P E π RN + P E π RN , ˆ ˆ ˆ 1 + r f ˆ 1 + r f i = 1 i = 1 for any risk neutral probability measure π RN ,  N  n b P q b ( 1 )+ n i q e P ˜ P i ( θ, 1 ) ˆ ˆ   � � ˜ 1 = E π RN i = 1  = ( 1 + r f ) E π RN V ˆ P ( θ, 1 ) . 1 + r f  Asset Pricing

Definition 11.1 11.1 Introduction Definition 11.2 11.2 The setting and the intuition Proposition 11.1 11.3 Notation, Definitions and Basic Results Proposition 11.2 Arrow-Debreu Pricing Proof of Proposition 11.2 Uniqueness What if risk neutral measure is not unique? Proposition 11.2 remains valid: each of the multiple of risk neutral measures assign the same value to the fundamental securities an thus to the portfolio itself! Asset Pricing