Foundations of Financial Economics Two period financial ... · Two period financial markets Paulo...
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Foundations of Financial Economics Two period financial markets Paulo Brito 1 [email protected] University of Lisbon April 3, 2020 1 / 63
Transcript of Foundations of Financial Economics Two period financial ... · Two period financial markets Paulo...
Foundations of Financial EconomicsTwo period financial markets
Paulo Brito
[email protected] of Lisbon
April 3, 2020
1 / 63
Topics
▶ Financial assets▶ Financial market▶ State prices▶ Portfolios▶ Absence of arbitrage opportunities▶ Complete financial markets▶ Arbitrage pricing▶ Equity premium▶ Equity premium puzzle
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Information
▶ Assume again that there is complete information regarding
time t = 0, but incomplete information regarding time t = 1.▶ Until now we have (mostly) assumed that the probability
distribution is given by nature.▶ But next we start assuming that information regarding time
t = 1 is provided by the financial market.▶ A financial market is defined by a collection of contingent
claims.▶ The general idea: under some conditions, we can extract an
implicit probability distribution of the states of naturefrom the structure of the financial market.
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Financial assetsPrices and payoffs
A financial asset or contract or contingent claim: is definedby the price and payoff pair (Sj,Vj) (for asset j):▶ for an observed price, Sj, paid at time t = 0▶ a contingent payoff, Vj, will be received at time t = 1,
Vj = (Vj1, . . . ,Vjs, . . . ,VjN)⊤
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Financial assetsReturns and rates of return
▶ Return Rj and rate of return rj of asset j are related as
Rj = 1 + rj,
▶ In the 2-period case: it is the payoff/price ratio
Rj =VjSj
=
Vj,1Sj
. . .Vj,sSj
. . .Vj,N
Sj
=
1 + rj,1. . .
1 + rj,s. . .
1 + rj,N
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Financial assetsTiming, information and flow of funds
Two alternative ways of representing (net) income flows:▶ as a price-payoff sequence
0
−Sj
1
Vj,1. . .Vj,s. . .
Vj,N
▶ as an investment-return sequence
0
−1
1
Rj,1. . .Rj,s. . .
Rj,N
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Financial assetsStatistics
▶ From the information, at time t = 0, on the probabilities for thestates of nature at time t = 1 we can compute:
▶ Expected return for asset j at time t = 1, from theinformation at time t = 0
E[Rj] =N∑
s=1πsRj,s = π1Rj,1 + . . . πsRj,s + . . .+ πNRj,N
▶ Variance of the return for asset j at time t = 1, from theinformation at time t = 0
V[Rj] =
N∑s=1
πs(Rjs−E[Rj])2 = π1 (Rj,1 − E[Rj])
2+. . . πN (Rj,N − E[Rj])
2
▶ Observation: an useful relationshipV[R] = E[R2]− (E[R])
2
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Classification of assetsRisk classification
As regards risk:▶ risk-less or risk-free asset: V⊤ = (v, . . . , v)⊤
▶ risky asset: V⊤ = (v1, . . . , vN)⊤ with at least two different
elements, i.e. there are at least two elements, vi and vj such thatvi ̸= vj for i ̸= j
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Classification of assetsTypes of assets
Particular types of assets as regards the income flows:▶ one period bonds with unit facial value:
S =1
1 + i , V⊤ = (1, 1, . . . , 1)⊤
▶ deposits:
S = 1, V⊤ = (1 + i, 1 + i, . . . , 1 + i)⊤
▶ equity: the payoff are dividends
Se, V⊤e = (d1, . . . , dN)
⊤
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Classification of assetsTypes of assets
Derivatives:▶ forward contract on an underlying asset with offered price p:
Sf, V⊤f = (v1 − p, . . . , vN − p)⊤
▶ european call option with exercise price p:
Sc, V⊤c = (max{v1 − p, 0}, . . . ,max{vN − p, 0})⊤
▶ european put option with exercise price p:
Sp, V⊤p = (max{p − v1, 0}, . . . ,max{p − vN, 0})⊤
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Financial marketDefinition: A financial market is a collection of K traded assets. Itcan be characterized by the structure of prices and payoffs of all Kassets : i.e by the pair (S,V):▶ vector of observed prices, at time t = 0 (we use row vectors for
prices)S︸︷︷︸
K×1
= (S1, . . . ,SK),
▶ and a matrix of contingent (i.e., uncertain) payoffs, at time t = 1
V︸︷︷︸N×K
=
V11 . . . VK1...
...V1N . . . VKN
▶ The participants observe S and know V, but they do not knowwhich payoff they will get for every asset (i.e, which line of V willbe realized)
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Financial marketThe payoff matrix contains the following information:▶ each column represents the payoff forasset j = 1, . . . ,K
V =
V11 . . . Vj1 . . . VK1...
......
V1s . . . Vjs . . . VKs...
......
V1N . . . VjN . . . VKN
▶ each row represents the outcomes (in terms of payoffs) for state of nature s = 1, . . . ,N
V =
V11 . . . Vj1 . . . VK1...
......
V1s . . . Vjs . . . VKs...
......
V1N . . . VjN . . . VKN
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Financial market▶ Equivalently, we can charaterize a financial market by the matrix
of returns
R︸︷︷︸N×K
=
R11 . . . RK1...
...R1N . . . RKN
where Rj,s =
Vj,sSj
= 1 + rj,s
▶ ThereforeR︸︷︷︸
N×K
= V︸︷︷︸N×K
(diag(S))−1︸ ︷︷ ︸K×K
where
diag(S)︸ ︷︷ ︸K×K
=
S1 . . . 0 . . . 0...
......
......
0 . . . Sj . . . 0...
......
......
0 . . . 0 . . . SK
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State pricesImplicit state valuation: the modern approach to financialeconomics characterizes asset markets by the prices of the statesof nature implicit in the relationship between present prices Sand future payoffs V, i.e.
Q := {x : S = xV}
▶ if Q is unique it is a vector
Q︸︷︷︸1×N
= (q1, . . . , qN)
satisfyingS︸︷︷︸
1×K
= Q︸︷︷︸1×N
V︸︷︷︸N×K
⇔ S⊤︸︷︷︸K×1
= V⊤︸︷︷︸K×N
Q⊤︸︷︷︸N×1
▶ Definition: Q is a state price vector if is it positive , i.e. qs > 0
for all s ∈ {1, . . . ,N}
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Arbitrage opportunitiesDefinition: if there is at least one qs ≤ 0, s = 1, . . . ,N then wesay there are arbitrage opportunitiesDefinition: we say there are no arbitrage opportunities ifqs > 0, for all s = 1, . . . ,NIntuition:
▶ existence of arbitrage (opportunities) means there are free ornegatively valued states of nature
▶ absence of arbitrage means every state of nature is costly andtherefore, positively priced.
Proposition 1Given (S,V), there are no arbitrage opportunities if and only if Q isa vector of state prices.
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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 areincomplete
Intuition:▶ completeness: there is an unique valuation of the states of
nature.▶ incompleteness: there is not an unique valuation of the states of
nature.
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Completeness of a financial marketConditions for completeness
We can classify completeness of a market by looking at thenumber of assets, number of states of nature and theindependence of payoffs:
1. if K = N and det (V) ̸= 0 then markets are complete and allassets are independent;
2. if K > N and det (V) ̸= 0 then markets are complete andthere are N independent and K − N redundant assets;
3. if K < N or K ≥ N and det (V) = 0 then markets areincomplete.
Proposition 2Given (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 marketNo arbitrage opportunities and completeness
▶ Consider a financial market (S,V), such that K = N and
det (V) ̸= 0▶ We defined state prices from S = QV▶ As K = N and det (V) ̸= 0 then V−1 exists and is unique▶ Therefore, we obtain uniquely
Q︸︷︷︸1×N
= S︸︷︷︸1×K
V−1︸︷︷︸K×N
that is
(q1, . . . , qN) = (S1, . . . ,SK)
V1,1 . . . V1,K. . . . . . . . .
VN,1 . . . VN,K
−1
▶ If Q ≫ 0 we say there are no arbitrage opportunities▶ And because Q is unique we say markets are complete
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Characterization of a financial marketNo arbitrage opportunities and completeness: alternative 1
Alternatively, because
V⊤︸︷︷︸K×N
Q⊤︸︷︷︸N×1
= S⊤︸︷︷︸K×1
If det (V) ̸= 0 thenQ⊤ =
(V⊤)−1 S⊤
that is q1...
qN
=
V1,1 . . . VN,1. . . . . . . . .
V1,K . . . VN,K
−1S1
...SK
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Characterization of a financial marketNo arbitrage opportunities and completeness: alternative 2
As Rjs = Vjs/Sj equivalently, we can write
Q︸︷︷︸1×N
R︸︷︷︸N×K
= 1⊤︸︷︷︸1×K
(q1, . . . , qN)
R1,1 . . . R1,K. . . . . . . . .
RN,1 . . . RN,K
=(1 1 1
) then
Q = 1⊤R−1
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Characterization of a financial marketNo arbitrage opportunities and completeness: alternative 3
Or, alternatively,R⊤Q⊤ = 1
thenQ⊤︸︷︷︸N×1
= (R⊤)−1︸ ︷︷ ︸N×K
1︸︷︷︸K×1
matricially q1...
qN
=
R1,1 . . . R1,K. . . . . . . . .
RN,1 . . . RN,K
−11
...1
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Characterization of a financial marketMarket incompleteness
▶ Consider a financial market (S,V), such that K < N, and in the
partitionV⊤︸︷︷︸
K×N
= ( V⊤1︸︷︷︸
(K×K)
V⊤2︸︷︷︸
(N−K×K)
)
assume that det (V1) ̸= 0▶ We defined state prices from V⊤Q⊤ = S⊤
▶ But now we have (V⊤
1 V⊤2)(Q⊤
1Q⊤
2
)= S⊤
where Q1 is (1 × K) and Q2 is (1 × N − K).
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Characterization of a financial marketMarket incompleteness
▶ Then
V⊤1 Q⊤
1 + V⊤2 Q⊤
2 = S⊤
▶ Because det (V1) ̸= 0 we can make
Q⊤1 = (V⊤
1 )−1 (S⊤ − V⊤2 Q⊤
2)
▶ There are only K independent prices: i.e, Q is indeterminate and
the degree of indeterminacy is N − K,▶ As Q is not uniquely determined (i.e, we can fix arbitrarilty
N − K state prices) the market is incomplete
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Characterization of a financial marketMarket completeness with redundant assets
▶ Consider a financial market (S,V), such that K > N, partition
V︸︷︷︸N×K
= ( V1︸︷︷︸(N×N)
V2︸︷︷︸(K−N×N)
)
and assume that det (V1) ̸= 0▶ We defined state prices from V⊤Q⊤ = S⊤
▶ But now we have (V⊤
1V⊤
2
)(Q⊤) = (S⊤
1S⊤
2
)where S1 is (1 × N) and S2 is (1 × K − N).
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Characterization of a financial marketMarket completeness with redundant assets
▶ Then {
V⊤1 Q⊤ = S⊤
1V⊤
2 Q⊤ = S⊤2
▶ Because det (V1) ̸= 0 we can determine Q uniquelyQ⊤ = (V⊤
1 )−1S⊤1
▶ Which implies that the prices of the remaining K − N assets can
be obtained from the prices S1
S⊤2 = V⊤
2 Q⊤ = V⊤2((V⊤
1 )−1S⊤1)
▶ There are K − N redundant assets (i.e, they do not add new
information on Q)▶ As Q is uniquely determined the market is complete
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Characterization of a financial marketSuggestion on how to apply this theory
▶ First: obtain RT which is a (K × N) matrix▶ Second: check the number of assets, K and the number of states
of nature N▶ if N < K then markets are incomplete▶ if N ≥ K check the number of independent rowsn of R
▶ if dim (V) < N then markets are incomplete▶ if dim (V) ≥ N then markets are complete
▶ Third: obtaining the state prices Q▶ if K = N and detR ̸= 0 compute Q⊤ = (R⊤)−11▶ if K > N and R̃ a N × N partition of R has det R̃ ̸= 0 then
compute Q⊤ = (R̃⊤)−11▶ if K = N but detR = 0, then, solve the independent
equations in R⊤Q⊤ = 1▶ if K < N solve the independent equations in R⊤Q⊤ = 1.
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Example 1
▶ Let S = (1, 1) and V =
(1 11 2
)▶ then
R =
(1 11 2
), R⊤ =
(1 11 2
)
▶ Therefore
Q⊤ =
(1 11 2
)−1(11
)=
(2 −1−1 1
)(11
)=
(10
) then markets are complete but there are arbitrageopportunities.
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Example 2
▶ Let S = (1, 1) and V =
(2 11 2
)▶ then
R =
(2 11 2
)= R⊤
▶ Therefore
Q⊤ =
(2 11 2
)−1(11
)=
( 23 − 1
3− 1
323
)(11
)=
( 1313
)markets are complete and there are no arbitrage opportunities
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Example 3▶ Let S = (1, 1, 2) and V =
(2 1 31 2 3
)
▶ then R =
(2 1 3
21 2 3
2
) and R⊤ =
2 11 232
32
▶ we pick all the combinations of two assets
Q⊤ =
(2 11 2
)−1(11
)=
( 23 − 1
3− 1
323
)(11
)=
( 1313
)
Q⊤ =
(2 132
32
)−1(11
)=
(1 − 2
3−1 4
3
)(11
)=
( 1313
)and
Q⊤ =
(1 232
32
)−1(11
)=
(−1 4
31 − 2
3
)(11
)=
( 1313
)then markets are complete, there are no arbitrage opportunitiesand there is one redundant asset
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Example 4
▶ Let S = (1, 2) and V =
(2 41 2
)▶ then R =
(2 21 1
)and R⊤ =
(2 12 1
)two assets have the same
returns▶ then R⊤Q⊤ = 1⊤ becomes(
2 12 1
)(q1q2
)=
(11
)has an infinite number of solutions
Q⊤ =
((1 − k)/2
k
)for an arbitrary k, then markets are incomplete and if 0 < k < 1there are no arbitrage opportunities;
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Example 5
▶ Let S = (1) and V =
(21
)▶ then
R =
(21
)
▶ then we also have(q1, q2)
(21
)= 1
or2q1 + q2 = 1
has an infinite number of solutions Q = ((1 − k)/2, k), for anyk, then markets are incomplete and if 0 < k < 1 there are noarbitrage opportunities
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Implicit market probabilities: risk-free probabilities▶ Let Q = (q1, . . . , qN) be a state-price vector▶ If there are no arbitrage opportunities then qs > 0, i.e Q ≫ 0▶ A Radon-Nikodyn derivative is defined as
πQs = qs/q̄
where q̄ =∑N
s=1 qs.▶ Therefore, if there are no arbitrage opportunities then
0 < πQs < 1, and
N∑s=1
πQs = 1
that is πQs is a probability measure
▶ This is a probability measure implicit in the financial market(S,V)
▶ Therefore:▶ If markets are complete the probability measure is unique▶ If markets are incomplete the probability measure is not
unique 32 / 63
And asset pricing
▶ Using the definition for qs
Sj =
N∑s=1
qs Vsj =
N∑s=1
q̄ qsq̄ Vsj
then, for any asset
Sj = q̄EQ[Vj
]
▶ This means prices are proportional to the expected value of thepayoff, using an implicit market probability.
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Definition▶ Let Q = (q1, . . . , qN)
▶ Assume we have a (objective or subjective) probabilitydistribution for the states of nature
P = (π1, . . . , πN)
such that 0 < πs < 1 and∑N
s=1 πs = 1▶ The stochastic discount factor for state s is
ms =qsπs
, s = 1, . . . ,N
▶ The stochastic discount factor is the random variable
M︸︷︷︸1×N
= (m1, . . . ,ms, . . . ,mN)
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And asset pricing
▶ Using the definition for qs
Sj =N∑
s=1qs Vsj =
N∑s=1
πsqsπs
Vsj
then, for any asset
Sj = E[M Vj
]
▶ This means that the stochastic discount factor combines bothmarket and fundamental information.
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The state price approach to asset markets
Summing up:▶ The state price translates the information structure implicit in
financial transactions▶ The relevant information is related to:
▶ how costly is the insurance against the states of nature▶ the uniqueness of that insurance cost▶ suggests a method for pricing new assets: pricing by
redundancy
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Portfolios
▶ Definition: A portfolio is a vector specifying the positions, θ,in all the assets in the market
θ⊤ = (θ1, . . . , θK)⊤ ∈ RK
If θj > 0 there is a long position in asset jif θj < 0 there is a short position in asset j
▶
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Portfolios
▶ The stream of income generated by position θj is
0
−Sjθj
1
θjVj,1. . .
θjVj,s. . .
θjVj,N
▶ long position (θj > 0): pay Sjθj at time t = 0 and receivethe contingent payoff θjVj at time t = 1
▶ short position (θj < 0): receive Sjθj at time t = 0 and paythe contingent payoff θjVj at time t = 1
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Portfolios
A portfolio generates a (stochastic) stream of income {zθ0,Zθ1}
zθ0 = −C(θ,S) = −Sθ︸︷︷︸1×1
=
K∑j=1
Sjθj
Zθ1 = Vθ︸︷︷︸
N×1
=
K∑j=1
Vjθj
where
Zθ1 =
zθ1,1. . .zθ1,s. . .zθ1,N
=
∑K
j=1 Vj,1θj. . .∑K
j=1 Vj,sθj∑Kj=1 Vj,Nθj
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Portfolios
The flow of income generated by a portfolio θ is:
0
zθ0
1
zθ1,1. . .zθ1,s. . .zθ1,N
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Portfolios and arbitrage opportunities
Proposition 3Assume there are arbitrage opportunities. Then there exists atleast one portfolio ϑ such that
zϑ0 = 0 and Zϑ1 > 0
orzϑ0 > 0 and Zϑ
1 = 0.
(Obs. A positive vector X > 0 has non-negative elements, and has at least one equal to
zero. A strictly positive vector X ≫ 0 has only positive elements )
Intuitionwith a zero cost we can get a positive income in at least onestate of nature.If we have an initial income, we will pay a zero cost in thefuture, in every state of nature. 41 / 63
Example 1
▶ We consider Example 1 again: S = (1, 1) and V =
(1 11 2
)▶ We can build a portfolio ϑ = (ϑ1, ϑ2)
⊤ such thatzϑ0 = −Sϑ = −(ϑ1 + ϑ2) = 0, that is
ϑ1 = −x, ϑ2 = x
▶ The return at time t = 1 is
Zϑ1 = Vϑ =
(−x + x−x + 2x
)=
(0x
) then if we take a long position in the risky asset (x > 0) wehave a positive income at t = 1 with a zero investment.
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Portfolios and arbitrage opportunities
Proposition 4Assume there are no arbitrage opportunities. A portfolio withcost, zϑ0 = 0, generates a nonpositive income at time 1, Zϑ
1 .
▶ Non-positive Zϑ1 means that there is at least one state of nature,
s such that zθ1,s < 0.
Intuition: with a zero cost although we can get a positiveincome in several states of nature there is at least one state ofnature in which there is a negative income.
43 / 63
Example 2
▶ Consider the previous Example 2: S = (1, 1) and V =
(2 11 2
)▶ We can build a portfolio (ϑ1, ϑ2)
⊤ such that zϑ0 = −Sϑ = 0:
ϑ1 = −x, ϑ2 = x
▶ The return at time t = 1 is
Zϑ1 = Vϑ =
(−2x + x−x + 2x
)=
(−xx
) then whatever the position, long (x > 0) or short (x < 0), inthe risky asset we will not have positive income at t = 1 (in somestates of the nature we win, but we will lose in others).
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Replicating portfolios
Definition: Replicating porfolio: Assume we have acontingent income W = (w1, . . . ,wN)
⊤. To the portfolio, ϑ, buildusing the assets in the market, generating the same (contingent)income
ϑ ≡ {θ : Z1 = Vθ = W}
we call replicating portfolio. The cost of the replicatingportfolio is
C(ϑ,S) = Sϑ
45 / 63
Arbitrage asset pricing theory (APT)
▶ Deals with the determination of asset prices throughreplication.
▶ APT vs GEAP (general equilibrium asset pricing):▶ in APT we take (S,V) as given (S is exogenous)▶ in GEAP we take V as given and want to determine S from
the fundamentals (S is endogenous)▶ Both theories have three equivalent formulations:
▶ using state prices Q▶ using the stochastic discount factor M▶ using market or risk-neutral probabilities πQ
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Arbitrage asset pricingUsing state prices
Proposition 5Assume there is a financial market (S,V) such that there are noarbitrage opportunities. Then, given a new asset with payoff Vk itsprice can be determined by redundancy by using the expression
Sk = QVk
where Q = SV−1 ≫ 0 is determined from the market pair (S,V).
That is
Sk =
N∑s=1
qsVks
47 / 63
Arbitrage asset pricingUsing state prices
From what we have previously seen there are two important results
Proposition 6If markets are complete then the value, Sk, is unique. If marketsare incomplete then Sk is not unique.
Proposition 7Assume a financial market (S,V) is complete and there are K − Nredundant assets. Then the price of any asset is equal to thecost of their replicating portfolios build with N independentassets.
48 / 63
Arbitrage asset pricingUsing stochastic discount factors
▶ If Q is a state price vector and π is the vector or probabilities,we define the stochastic discount factor (SDF) for state s by
ms ≡qsπs
Proposition 8Assume there is a financial market (S,V) such that there are noarbitrage opportunities. Then, given a new asset with payoff Vk itsprice can be determined by redundancy by using the expression
Sk = E[MVk]
where M = (m1, . . . ,mN)⊤ is the stochastic discount factor.
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Arbitrage asset pricingUsing stochastic discount factors
▶ Proof: as Sk =∑N
s=1 qsVks and using the definition of SDF weget
Sk =N∑
s=1qsVks =
N∑s=1
πsmsVks
▶ Intuition: if there are no arbitrage opportunities the price of anasset is the expected value of the future payoffs discounted bythe stochastic discount factor (which is the market discount foruncertain payoffs)
50 / 63
Arbitrage asset pricingUsing risk-neutral probabilities
▶ Therefore,
Sk =
N∑s=1
qsVks = q̄N∑
s=1πQ
s Vks = q̄EQ[Vk]
or compactlySk = q̄EQ[Vk]
▶ Intuition: if there are no arbitrage opportunities there is an
equivalent probability measure such that the price of an asset isproportional of the expected value of the future payoffs
▶ Although q̄ is mysterious we can calculate it a intuitive way.
51 / 63
Arbitrage asset pricingExistence of risk-free asset
Proposition 9Assume there are no arbitrage opportunities and there is a risk-freebond with price 1/(1 + i) and face value 1. Then the price of any assetverifies the relationship
Sj =1
1 + iEQ[Vj]
Exercise: prove this.
52 / 63
Arbitrage asset pricingExistence of risk-free asset
Proposition 10Assume there are no arbitrage opportunities and there is arisk-free asset. Then there is a (market) probability measuresuch that the expected returns of every asset is equal to the return ofthe risk-free asset.
▶ Proof. From Proposition 8 we have
1 + i = EQ[Rj], for any, j = 1, . . . ,K
As this holds for any asset, therefore πQ has the property
1 + i = EQ[R1] = . . . = EQ[RK]
53 / 63
ApplicationArbitrage and completeness with two assets
▶ Assume that N = 2 and there is a risky and a risk-free asset
S =
(1
1 + i , p), V =
(1 d11 d2
)
▶ and that r1 < i < r2.
Then markets are complete and there are no arbitrageopportunities.
The return matrix is
R =
( 11
1+i
d1p
11
1+i
d2p
)=
(1 + i 1 + r11 + i 1 + r2
)
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ApplicationArbitrage asset pricing
▶ If there are risk-free assets and absence of arbitrage
opportunities, then any asset can be priced by redundancy as
Sk = q1Vk,1 + q2Vk,2
where
q1 =i − r2
(1 + i)(r1 − r2), q2 =
r1 − i(1 + i)(r1 − r2)
.
▶ Proof: assume there is a risk-free and a risky asset such that
QR = 1⊤ becomes(q1q2
)=
(1 + i 1 + i
1 + r1 1 + r2
)−1(11
) and if there are no arbitrage opportunities r1 < i < r2 implyingq1 > 0 and q2 > 0.
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ApplicationSimple Black and Scholes (1973) equation
▶ Consider an european call option with exercise price p on an
underlying asset with payoff
V⊤underlying = (d1, d2)
▶ Then the contingent payoff of the option is
V⊤option = (max{d1 − p, 0},max{d2 − p, 0})
▶ Question: assuming there are no arbitrage opportunities and
there is a risk-free asset what is the market price of the option ?▶ Answer: the price for a call option with exercise price p is
Soption =(r2 − i)max{d1 − p, 0}+ (r1 − i)max{d2 − p, 0}
(1 + i)(r1 − r2)
(prove this)56 / 63
Equity premium
▶ We call equity premium to the difference in the rates of returnbetween the risky and a risk-free asset
r − i =(
r1 − ir2 − i
)
▶ The Sharpe index is defined as
E[r − i]σ[r] =
∑2s=1 πs(rs − i)√∑2
s=1 πs((rs − i)− E[r − i])2
where σ[r − i] =√
V[r − i] is the standard deviation of theequity premium (prove that σ[r − i] = σ[r]).
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Risk neutral probabilities
Proposition 11If there are no arbitrage opportunities then there is a (market) riskneutral probability distribution such that the expected value of theequity risk premium is zero,
EQ[r − i] = 0
This is the reason why πQs are called risk neutral probabilities:
PQ = (πQ1 , . . . πQ
N) is a probability measure such that the expectedvalue of the equity premium is zero.
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Risk neutral probabilities
▶ Proof: LetR⊤ =
(1 + i 1 + i
1 + r1 1 + r2
)
▶ Then, because R⊤Q⊤ = 1{(1 + i)q1 + (1 + i)q2 = 1(1 + r1)q1 + (1 + r2)q2 = 1
▶ Then (1 + r1)q1 + (1 + r2)q2 = (1 + i)q1 + (1 + i)q2.▶ Therefore
q1(r1 − i) + q2(r2 − i) = 0
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Risk neutral probabilities
▶ Proof (cont)Define q̄ = q1 + q2
q1q̄ (r1 − i) + q1
q̄ (r2 − i) = 0
▶ Define again πQs = qs
q̄ . If there are no arbitrage opportunities,then qs > 0 and πQ
s > 0. Because∑2
s=1 πQs = 1 then πQ
s areprobabilities.
▶ At lastπQ
1 (r1 − i) + πQ2 (r2 − i) = 0
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The Sharpe index and the SDF
▶
Proposition 12Assume there are no arbitrage opportunities and there is a risk-freeasset. Then the Sharpe index verifies the relationship
E[r − i]σ[r] = −ρr−i,m
(E[M]
σ[M]
)−1
where ρr−i,m is the coefficient of correlation between the equitypremium and the stochastic discount factor.
▶
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The Sharpe index and the SDF
▶ Proof: Using our previous definition of the stochastic discount
factor ms = qs/πs then
E[M(r − i)] = 0
▶ But E[M(r − i)] = Cov[M(r − i)] + E[M]E[r − i]▶ Using the definition of correlation
ρr−i,m =Cov[M(r − i)]σ[M]σ[r − i] ∈ (−1, 1)
▶ Then ρr−i,m σ[M]σ[r − i] + E[M]E[r − i] = 0
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The equity premium and DGE models ▶ Intuition: the Sharpe index is equal symmetric to the product
between coefficient of variation of the stochastic discount factorand correlation coefficient between the equity premium and thestochastic discount factor, ρr−i,m
▶ The coefficient of variation of the stochastic discount factorcontains the aggregate market value of risk.
▶ E[M]/σ[M] can be derived from simple DSGE models (as we willsee next)
▶ Observe that the Sharpe index satisfies
E[R − Rf]
σ[R]
where Rf = 1 + i is the risk-free return and R can be taken as areturn on a market index. (seehttp://web.stanford.edu/~wfsharpe/art/sr/SR.htm
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