Asset Pricing. Chapter XI. The Martingale Measure: Part I. June 20, 2006

Transcription

1 Chapter XI. The Martingale Measure: Part I June 20, 2006

2 1 (CAPM) 2 (Risk Neutral) 3 (Arrow-Debreu) ECF 1 (1 + r 1 f + π) ; ECF 2 (1 + r 2 f + ; ECF 3 ECF τ Π τ π)2 (1 + r 3 f ; or + π)3 (1 + rτ f. )τ X θτ Θτ ÊCF τ (1 + rτ f ; )τ q(θ τ )CF (θ τ ), p j,t = E CF j,t+1 cov( CF j,t+1, r M )[ E r M r f σ M 2 ], 1 + r f

3 Existence of Risk Neutral Probabilities 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

4 Existence of Risk Neutral Probabilities Existence of a set of numbers πj RN, Σπj RN = 1 s.t q e i (0) = q e i (0) = π RN 1 1 (1 + r f ) E π RN (qe i (θ, 1)) = ( q e i (θ 1, 1) 1 + r f ) +...+π RN J No solution if: qs e (0) = qk e (0) with 1 (1 + r f ) J j=1 ( q e i (θ J, 1) 1 + r f π RN j q e i (θ j, 1) (1) ), i = 1, 2,..., N, q e k (θ j, 1) q e s (θ j, 1) for all j, and q e k (θ ĵ, 1) > q e s (θĵ, 1) (3) = arbitrage opportunity (2)

5 Consider a portfolio, P, composed of n b P risk-free bonds and ni P units of risky security i, i = 1, 2,..., N. N V P (0) = np b qb (0) + np i qe i (0), (4) i=1 N V P (θ j, 1) = np b qb (1) + np i qe i (θ j, 1). (5) i=1

6 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 (θĵ, 1) > 0, for at least one ĵ {1, 2,..., J}.

7 { } J A probability measure πj RN defined on the set of states (θ j, j=1 j = 1, 2,..., J), is said to be a risk-neutral probability measure if (i) j > 0, for all j = 1, 2,..., J, and (7) { q (ii) qi e e } (0) = E i (θ, 1) π RN, 1 + r f π RN for all fundamental risky securities i = 1, 2,..., N in S.

8 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): q e (0): 4 q e (θ j, 1): 3 7 complete markets no arbitrage opportunities "objective" state probabilities?

9 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): q1 e (0): 2 qe 1 (θ j, 1): q2 e (0): 3 qe 2 (θ j, 1): The solution to this set of equations, «««2 = π RN π RN π RN «««3 = π RN π RN π RN = π RN 1 + π RN 2 + π RN 3. π RN 1 =.3, π RN 2 =.6, π RN 3 =.1,

10 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): q1 e(0): 2 qe 1 (θ j, 1): = π RN 1 ( ) ( ) ( ) π2 RN + π3 RN = π1 RN + π2 RN + π3 RN System indeterminate; many solutions

11 2.2 π RN 1 = 2π RN 2 + 3π RN 3 1 π RN 1 = π RN 2 + π RN 3, π RN 1 > 0 π RN 2 =.8 2π RN 1 > 0 π RN 3 =.2 + π RN 1 > 0 0 < π RN 1 <.4, (π RN 1, πrn 2, πrn 3 ) {(λ,8 2λ,.2 + λ) : 0 < λ <.4} Risk Neutral probabilities are not uniquely defined!

12 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): q1 e(0): 2 qe 1 (θ j, 1): q2 e(0): 2.5 qe 2 (θ j, 1): an arbitrage opportunity No solution (or solution with π RN i = 0 for some i)

13 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.

14 Suppose the set of securities S is free of arbitrage opportunities. Then for any portfolio ˆP in S VˆP(0) 1 = (1 + r f ) E π RN ṼˆP(θ, 1), (8) for any risk-neutral probability measure π RN on S.

15 Let ˆP be an arbitrary portfolio in S, and let it be composed of n bˆp bonds and n iˆp shares of fundamental risky asset i. In the absence of arbitrage, ˆP must be priced equal to the value of its constituent securities, in other words, VˆP(0) = n bˆpq b (0) + N n iˆp qi e (0) = n bˆp E π RN i=1 ( ) q b (1) 1+r f + N n iˆp E π RN i=1 ( ) q e i (θ,1) 1+r f, for any risk neutral probability measure π RN, n bˆp q b P (1)+ N n iˆp q e i (θ,1) (ṼˆP(θ, ) i=1 = E π RN 1+r f = 1 (1+r f ) E π 1) RN.

16 What if risk neutral measure is not unique? remains valid: each of the multiple of risk neutral measures assign the same value to the fundamental securities an thus to the portfolio itself!

17 Proposition 11.3: Consider an arbitrary period t = 1 payoff x(θ, 1) and let M represent the set of all risk-neutral probability measures on the set S. Assume S contains no arbitrage opportunities. If 1 (1 + r f ) E π RN x(θ, 1) = 1 (1 + r f ) Eˆπ RN x(θ, 1) for any π RN, ˆπ RN M, then there exists a portfolio in S with the same t = 1 payoff as x(θ, 1).

18 Proposition 11.4: Consider a set of securities S without arbitrage opportunities. Then S is complete if and only if there exists exactly one risk-neutral probability measure. Proof Suppose S is complete and there were two risk-neutral probability measures, {πj RN : j = 1, 2,..., J} and { π j RN : j = 1, 2,..., J}. Then there must be at least one state ĵ for which πĵ RN π ĵ RN. Since the market is complete, one must be able to construct a portfolio P in S such that V P (0) > 0, and { V P (θ j, 1) = 0 j ĵ V P (θ j, 1) = 1 j = ĵ.

19 This is simply the statement of the existence of an Arrow-Debreu security associated with θĵ. But then {πj RN :j = 1, 2,..., J} and { π j RN :j = 1, 2,..., J} cannot both be risk-neutral measures as, by, V P (0) = 1 (1 + r f ) E π RN ṼP(θ, 1) = π RN ĵ π RN ĵ (1 + r f ) (1 + r f ) = 1 (1 + r f ) E π RN ṼP(θ, 1) = V P (0), a contradiction. Thus, there cannot be more than one risk-neutral probability measure in a complete market economy.

20 : Back to example q j (0) = πrn j (1+r f ) π RN 1 =.3, π RN 2 =.6, π RN 3 =.1, q 1 (0) =.3/1.1 =.27; q 2 (0) =.6/1.1 =.55; q 3 (0) =.1/1.1 =.09. Conversely: JX p rf = q j (0), j=1 and thus (1 + r f ) = 1 p rf = 1 JP q j (0) j=1 We define the risk-neutral probabilities {π RN (θ)} according to π RN j = q j (0) JP q j (0) j=1 (9)

21 Table 11.6 The Exchange Economy of Section 8.3 Endowments and Preferences Endowments Preferences t = 0 t = 1 Agent U 1 (c 0, c 1 ) = 1 2 c ( 1 3 ln(c1 1 ) ln(c1 2 )) Agent U 2 (c 0, c 1 ) = 1 2 c ( 1 3 ln(c2 1 ) ln(c2 2 )) π1 RN =.24, and πrn 2 = Suppose a stock were traded where q e (θ 1, 1) = 1, and q e (θ 2, 1) = 3. By risk-neutral valuation (or equivalently, using Arrow-Debreu prices), its period t = 0 price must be [ ].24 q e.30 (0) =.54 (1) (3) = 1.14; the price of the risk-free security is q b (0) =.54.

22 Table 11.7 Initial Holdings of Equity and Debt Achieving Equivalence with Arrow-Debreu Equilibrium Endowments t = 0 Consumption ˆn i e ˆn i b Agent 1: 10 1 / 2 1 / 2 Agent 2: max(10 + 1q 1 (0) + 2q 2 (0) c 1 1 q 1(0) c 1 2 q 2(0)) +.9( 1 3 c c1 2 ) s.t. c 1 1 q 1(0) + c 1 2 q 2(0) 10 + q 1 (0) + 2q 2 (0) The first order conditions are c1 1 : q 1(0) = c2 1 : q 2(0) = from which it follows that π1 RN = = 1 3 while πrn 2 = = 2 3