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Two Period Financial Markets Foundations of Financial Economics Two period financial markets Paulo Brito [email protected] University of Lisbon April 3, 2020 1 / 63 Topics I Financial assets I Financial market I State prices I Portfolios I Absence of arbitrage opportunities I Complete financial markets I Arbitrage pricing I Equity premium I Equity premium puzzle 2 / 63 Information I Assume again that there is complete information regarding time t = 0, but incomplete information regarding time t = 1. I Until now we have (mostly) assumed that the probability distribution is given by nature. I But next we start assuming that information regarding time t = 1 is provided by the financial market. I A financial market is defined by a collection of contingent claims. I The general idea: under some conditions, we can extract an implicit probability distribution of the states of nature from the structure of the financial market. 3 / 63 Financial assets Prices and payoffs A financial asset or contract or contingent claim: is defined by the price and payoff pair (Sj; Vj) (for asset j): I for an observed price, Sj, paid at time t = 0 I a contingent payoff, Vj, will be received at time t = 1, > Vj = (Vj1;:::; Vjs;:::; VjN) 4 / 63 Financial assets Returns and rates of return I Return Rj and rate of return rj of asset j are related as Rj = 1 + rj; I In the 2-period case: it is the payoff/price ratio 0 1 Vj;1 0 1 B Sj C 1 + rj;1 B ::: C B ::: C V B C B C R = j = B Vj;s C = B 1 + r C j Sj B j;s C Sj B C @ A @ ::: A ::: Vj;N 1 + rj;N Sj 5 / 63 Financial assets Timing, information and flow of funds Two alternative ways of representing (net) income flows: I as a price-payoff sequence 0 1 Vj;1 B C B ::: C B C B Vj;s C @ ::: A −Sj Vj;N 0 1 I as an investment-return sequence 0 1 Rj;1 B C B ::: C B C B Rj;s C @ ::: A −1 Rj;N 0 1 6 / 63 Financial assets Statistics I From the information, at time t = 0, on the probabilities for the states of nature at time t = 1 we can compute: I Expected return for asset j at time t = 1, from the information at time t = 0 XN E[Rj] = πsRj;s = π1Rj;1 + : : : πsRj;s + ::: + πNRj;N s=1 I Variance of the return for asset j at time t = 1, from the information at time t = 0 XN 2 2 2 V[Rj] = πs(Rjs−E[Rj]) = π1 (Rj;1 − E[Rj]) +: : : πN (Rj;N − E[Rj]) s=1 I Observation: an useful relationship V[R] = E[R2] − (E[R])2 7 / 63 Classification of assets Risk classification As regards risk: I risk-less or risk-free asset: V> = (v;:::; v)> I > > risky asset: V = (v1;:::; vN) with at least two different elements, i.e. there are at least two elements, vi and vj such that vi =6 vj for i =6 j 8 / 63 Classification of assets Types of assets Particular types of assets as regards the income flows: I one period bonds with unit facial value: 1 S = ; V> = (1; 1;:::; 1)> 1 + i I deposits: S = 1; V> = (1 + i; 1 + i;:::; 1 + i)> I equity: the payoff are dividends > > Se; Ve = (d1;:::; dN) 9 / 63 Classification of assets Types of assets Derivatives: I forward contract on an underlying asset with offered price p: > − − > Sf; Vf = (v1 p;:::; vN p) I european call option with exercise price p: > f − g f − g > Sc; Vc = (max v1 p; 0 ;:::; max vN p; 0 ) I european put option with exercise price p: > f − g f − g > Sp; Vp = (max p v1; 0 ;:::; max p vN; 0 ) 10 / 63 Financial market Definition:A financial market is a collection of K traded assets. It can be characterized by the structure of prices and payoffs of all K assets : i.e by the pair (S; V): I vector of observed prices, at time t = 0 (we use row vectors for prices) |{z}S = (S1;:::; SK); K×1 I and a matrix of contingent (i.e., uncertain) payoffs, at time t = 1 0 1 V11 ::: VK1 B . C |{z}V = @ . A × N K V1N ::: VKN I The participants observe S and know V, but they do not know which payoff they will get for every asset (i.e, which line of V will be realized) 11 / 63 Financial market The payoff matrix contains the following information: I each column represents the payoff forasset j = 1;:::; K 0 1 V11 ::: Vj1 ::: VK1 B C B . C B . C B C V = B V1s ::: Vjs ::: VKs C B . C @ . A V1N ::: VjN ::: VKN I each row represents the outcomes (in terms of payoffs) for state of nature s = 1;:::; N 0 1 V11 ::: Vj1 ::: VK1 B C B . C B . C B C V = BV1s ::: Vjs ::: VKsC B . C @ . A V1N ::: VjN ::: VKN 12 / 63 Financial market I Equivalently, we can charaterize a financial market by the matrix of returns 0 1 R11 ::: RK1 B . C |{z}R = @ . A × N K R1N ::: RKN Vj;s where Rj;s = = 1 + rj;s Sj I Therefore −1 |{z}R = |{z}V |(diag{z(S)) } × × N K N K K×K where 0 1 S1 ::: 0 ::: 0 B C B . C B . C B C |diag{z(S}) = B 0 ::: Sj ::: 0 C B . C K×K @ . A 0 ::: 0 ::: SK 13 / 63 State prices Implicit state valuation: the modern approach to financial economics characterizes asset markets by the prices of the states of nature implicit in the relationship between present prices S and future payoffs V, i.e. Q := fx : S = xVg I if Q is unique it is a vector |{z}Q = (q1;:::; qN) 1×N satisfying , > > > |{z}S = |{z}Q |{z}V |{z}S = |{z}V |{z}Q 1×K 1×N N×K K×1 K×N N×1 I Definition: Q is a state price vector if is it positive , i.e. qs > 0 for all s 2 f1;:::; Ng 14 / 63 Arbitrage opportunities Definition: if there is at least one qs ≤ 0, s = 1;:::; N then we say there are arbitrage opportunities Definition: we say there are no arbitrage opportunities if qs > 0, for all s = 1;:::; N Intuition: I existence of arbitrage (opportunities) means there are free or negatively valued states of nature I absence of arbitrage means every state of nature is costly and therefore, positively priced. Proposition 1 Given (S; V), there are no arbitrage opportunities if and only if Q is a vector of state prices. 15 / 63 Completeness of a financial market Definition: if Q is unique, then we say markets are complete Definition: if Q is not unique then we say markets are incomplete Intuition: I completeness: there is an unique valuation of the states of nature. I incompleteness: there is not an unique valuation of the states of nature. 16 / 63 Completeness of a financial market Conditions for completeness We can classify completeness of a market by looking at the number of assets, number of states of nature and the independence of payoffs: 1. if K = N and det (V) =6 0 then markets are complete and all assets are independent; 2. if K > N and det (V) =6 0 then markets are complete and there are N independent and K − N redundant assets; 3. if K < N or K ≥ N and det (V) = 0 then markets are incomplete. Proposition 2 Given (S; V), markets are complete if and only if dim(V) = N dim(V) = number of linearly independent columns (i.e., assets) 17 / 63 Characterization of a financial market No arbitrage opportunities and completeness I Consider a financial market (S; V), such that K = N and det (V) =6 0 I We defined state prices from S = QV I As K = N and det (V) =6 0 then V−1 exists and is unique I Therefore, we obtain uniquely −1 |{z}Q = |{z}S |V{z} 1×N 1×K K×N that is 0 1−1 V1;1 ::: V1;K @ A (q1;:::; qN) = (S1;:::; SK) ::::::::: VN;1 ::: VN;K I If Q ≫ 0 we say there are no arbitrage opportunities I And because Q is unique we say markets are complete 18 / 63 Characterization of a financial market No arbitrage opportunities and completeness: alternative 1 Alternatively, because > > > |{z}V |{z}Q = |{z}S K×N N×1 K×1 If det (V) =6 0 then ( )−1 Q> = V> S> that is 0 1 0 1 0 1 q −1 S 1 V1;1 ::: VN;1 1 B . C @ A B . C @ . A = ::::::::: @ . A V ::: V qN 1;K N;K SK 19 / 63 Characterization of a financial market No arbitrage opportunities and completeness: alternative 2 As Rjs = Vjs=Sj equivalently, we can write > |{z}Q |{z}R = |{z}1 1×N N×K 1×K 0 1 R1;1 ::: R1;K ( ) @ A (q1;:::; qN) ::::::::: = 1 1 1 RN;1 ::: RN;K then Q = 1>R−1 20 / 63 Characterization of a financial market No arbitrage opportunities and completeness: alternative 3 Or, alternatively, R>Q> = 1 then > > −1 |{z}Q =| (R{z) } |{z}1 × N×1 N×K K 1 matricially 0 1 0 1 0 1 q −1 1 1 R1;1 ::: R1;K B . C @ A B.C @ . A = ::::::::: @.A R ::: R qN N;1 N;K 1 21 / 63 Characterization of a financial market Market incompleteness I Consider a financial market (S; V), such that K < N, and in the partition > > > |{z}V = ( |{z}V1 |{z}V2 ) × K N (K×K) (N−K×K) assume that det (V1) =6 0 I We defined state prices from V>Q> = S> I But now we have ( ) ( ) > > > Q1 > V1 V2 > = S Q2 where Q1 is (1 × K) and Q2 is (1 × N − K).
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