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Asset Economy
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Asset economy
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A more realistic trading arrangement
Spot markets are markets for commodities that are available at t = 1.
Only goods available at particular state s are traded on them.
The spot markets for commodities are complemented by a set of Önancial markets.
Financial assets are contracts that deliver some state-contingent amount of money in the future.
For simplicity, consider a two-period model with S states and J Önancial assets.
Asset j is a vector
rj = 6r2j 7
with state contingent CF as components.
With J such assets the whole Önancial market is a matrix of assetsí
r : = 64 . . . . . . . . . 75 .
rS1 rSJ () Assets
2r1 r1J3
Arrow Securities
A risk-free asset delivers a Öxed amount of money in all states rf =[1,1,,1]0.
An elementary Önancial asset is Arrow security.
It delivers one unit of purchasing power conditional on a speciÖc event s, and zero otherwise
es = [00,1s,00]0 .
The payo§ matrix of the collection of all S Arrow securities is the
S S identity matrix
261 0 037
e=60 1 07=I. 4. . … .5
001 Assets
Arrow Securities
Are hardly ever traded, but some of them can be generated by combining other assets.
ìdelta securitiesî or ìdigital contractsî are very close to Arrow securities. Combining options into ìbutteráy spreadsîalso does the trick. (details later)
Because of their simplicity, Arrow securities are handy when pricing more general assets.
A general Önancial asset can be represented by a portfolio of Arrow securities.
A Önancial asset that pays one in state 1, three in state 2, and six in state 3 has the same state-contingent payo§ as a portfolio of one state-1, three state-2, and six state-3 Arrow securities.
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The law of one price
011 Letthepayo§vectorofassetj berj =@ 3 A.
011 001 001
ArrowSecuritiese1 =@ 0 A,e2 =@ 1 A,e3 =@ 0 A. 001
A portfolio of Arrow securities z = rj (1 unit of e1, 3 units of e2, 6 units of e3) is equivalent to asset rj itself.
The prices of such assets and portfolios must be the same. Otherwise we create arbitrage. If α = (α1,α2,α3) is the vector of Arrow security prices, the (arbitrage free) price of asset j
qj =αrj =α1r1j +α2r2j +α3r3j
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Decomposition
We can artiÖcially reproduce the cash áow of any asset rj with a portfolio of Arrow securities, holding rsj state s Arrow securities for each s.
Denote the prices of the Arrow securities
α = [α1,α2, ,αS].
We can use Arrow prices to compute the price of any asset by the law of one price.
Security rj can be decomposed into a portfolio, of rsj state s Arrow securities, for each s. By the law of one price,
One can extend this to a portfolio of assets z with payo§ r. The date
0 price of z
Q = α r z.
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The law of one price
The same no arbitrage pricing applies to portfolios of assets. 21313
Imagine assets with r = 4 1 2 1 5 and prices q = (q1,q2,q3). 103
Importantly, columns in r are the assets, rows are the states. Asset 1 is risk free, asset 2 delivers (3, 2, 0) in the three di§erent
states, etc. 0 z1 1
If one holds the portfolio z = @ z2 A , the payo§ is
0z1+3z2+z3 1 A
The price of such portfolio is q z = q1z1 + q2z2 + q3z3. () Assets
The law of one price
Assume no transaction costs and no bidñask spreads.
Any asset can be bought and sold at the same price (in any quantity).
Denote the period 0 price of asset j with qj .
Any two assets with the same payo§ vector must have the same price.
If a portfolio of assets z = (z1 , z2 , …, zJ )0 produces the same cash áow at every state as another portfolio y, that is
r z = r y, then the two portfolios cost the same
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Risk free asset
A risk free asset has payo§ vector
rf =[1,,1]0.
The price of risk-free asset is β. This is connected to the gross risk-free interest rate ρ
A risk-free asset is equivalent to holding a portfolio of one unit of
each Arrow security
Thus, the price of this bond must be the same as the sum of the prices of all Arrow securities
β = ρ 1 = ∑ αs .
Risk neutral probabilities
ρ is the risk-free interest rate and α is the vector of Arrow prices. The risk-neutral probabilities are the numbers
α ̃ s = ρ α s .
∑ α ̃ s = 1 .
we will also see later that α ̃s 0 for every s.
Let ~E be the expectation operator using the risk-neutral probabilities.
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Risk-neutral pricing and returns
We used to have
qj =αrj,now
qj = α ̃rj=β~Erj.
The price of security rj is the expected cash áow of the security, using the risk-neutral probabilities, discounted with the risk-free interest rate. The asset price is calculated as if the risk in r is not important. The rate of return is the payo§ over the price of the investment.
Rs = q for all s
q j = β ~E r j , 1 = β ~E R j , ~E R j = ρ .
The expected rate of return of any asset, evaluated with the risk-neutral probabilities, equals the risk-free rate of return.
Asset economy
An asset economy consists of I agents with utilities u, their state endowments ω, and a payo§ matrix r. The matrix r has S rows and J columns, if we have J Önancial assets.
Take r and asset prices q. A portfolio z costs q z today (produces a payo§of qz today),andyieldsapayo§ofrs z instates tomorrow.
All todayís and tomorrowís payo§s that can be achieved by any choice of portfolio z produces the market span M(q),
M(q)= qz anyz. r
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Market span Example
Captures the choice set, i.e. the set of allocations of purchasing power through time and the states that can be achieved by holding some portfolio.
r=1 1and(q1,q2) 12
be the Önancial markets. det r = 1, hence the markets are complete. (a1) and (a12) that solve
r a1 = 1 and r a12 = 0,
a21 0 a2 1
are the portfolios that mimic Arrow securities for states 1 and 2.
r 1 = 2 1 , hencea1= 2 anda12= 1. 11 a21 1 a2 1
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Market span Example
The prices of Arrow securities
α1 =2q1 q2 >0,α2 =q2 q1 >0.
to prevent arbitrage (the Arrowís Theorem on that will follow) Letq1 =32 andq2 =2sothatα1 =1,α2 =12.
Market Span for the agent with w0 of initial wealth
32z1+2z2 w0
y1 z1+z2
y2 z1+2z2
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Market span
[α+] is orthogonal to M(q), i.e. ([α+] x) = 0 for any x 2 M(q).
[α+] := [1,α1,α2,…,αS]. Consider some x 2 M(q), i.e.
x = q z for some z. r
[1,α1,…,αS]x =[1,α1,…,αS] qz =( q+αr)z =0
Recall decomposition.
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Arbitrage and Arrow prices
Example of arbitrage, S = 2, J = 2. Asset 1 pays 1 in each state, the other pays 32 in state 1 and 12 unit in state 2. The asset prices
q = (1, 2).
Buy two bonds and sell one share short, This costs nothing and pays o§12 instate1and32 instate2.
One can get unbounded proÖts like this.
More generally (q,r) allows arbitrage opportunities if there is z such
q z 0. r
The payo§ today or in any s is not negative, and it is strictly positive either today or in at least one s.
Allowing no arbitrage opportunities rules the above out.
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Arbitrage and Arrow prices Theorem
(Arrow) (q, r ) is arbitrage-free if and only if the Arrow prices αs > 0 for every s.
Start with the only if part. (q,r) is arbitrage-free only if Arrow prices αs >0foreverys.
Clearly (q,r) admits arbitrage if αs < 0 for some s.
Now the if part. If αs > 0 for every s then (q, r ) is arbitrage-free,
that is, there is no arbitrage portfolio.
Any [α+] with α r = q must be othogonal to M(q), i.e., to any cash áow vector.
Since αs > 0 for every s then no cash áow vector from M(q) can point into RS+1, otherwise they would not be orthogonal. Hence
there is no arbitrage portfolio.
Complete markets
Any Önancial asset can be represented by a portfolio of Arrow securities.
Is the converse also true? Is reverse decomposition possible?
Can we construct any Arrow security with a combination of general Önancial assets?
Can we compute the price of an Arrow security from the prices of those?
Yes, if markets are complete.
DeÖnition
The markets are complete if agents can insure each state separately, that is if they can trade assets in such a way as to a§ect the payo§ in one speciÖc state without a§ecting the payo§s in the other states.
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Complete markets
If markets are complete, there is a portfolioó for each state s a di§erent oneó that generates the state-contingent cash áows of the state-s Arrow security.
That is, for each s there exists a replicating portfolio zs such that rzs =es .
Collecting these portfolios for each state
r [z1,z2,…,zS] = e.
When does such product make sence? What are the dimensions of there matrices?
rr 1 = Iande=I [z1,z2,…,zS] = r 1,
if r 1 exists, i.e. r is invertible.
For this r has to have a full rank, that is we need at least S linearly
independent assets in r.
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Complete markets and uniqueness of Arrow prices
Markets are complete if and only if r is invertible. Recallforr=r1 r2 ,detr=r1r4 r2r36=0ifrisinvertible
r3 r4 andr 1= 1 r4 r2 .
detr r3 r1
The Arrow prices are then unique and follow
α = q r 1 .
All complete asset markets are equivalent to each other. They are all
equivalent to an economy containing every Arrow security.
The agents can move the wealth into any state s by an appropriate
choice of state s portfolios.
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Two assets, both risky. Payo§ matrix
If we had an Arrow security for each state, one could hold both and
have a portfolio that mimics a risk-free bond.
r is invertible. Why? How do we construct a risk free portfolio form the available assets?
The inverse of the payo§ matrix is
r=1 1. 14
z = r 1 = 43 53 1 5
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By construction z1 = 43 , 31 0 is the portfolio that generates A1 .
And z2 = 53 , 35 0 is the portfolio that generates A2 .
Holding [A1,A2] generates risk-free bond.
Hence holding z = z1 + z2 = 13 , 43 0 generates risk-free bond. Equivalently we seek z such that
rz=1.=)z=r 11= 1343 .
Buy (a multiple of) 4/3 shares of risky asset 2 and sell short (a
multiple of) 1/3 shares of risky asset 1.
Spanning through options
Completes any Önancial market
risky asset
is obviously incomplete markets.
Call option with strike price k has a payo§ of maxfrs k,0g
Consider call options on r with strike prices 3.5, 2.5, 1.5. The asset structure is complete.
r (3.5) r (2.5) r (1.5) r .5 1.5 2.5 4 0 .5 1.5 3 0 0 .5 2 0001
r = (4,3,2,1)0
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Spanning through options
If the primary asset r separates states i.e. has rs 6= rt for any s and t it is possible to generate complete markets by means of options.
Relabelthestatessothatr1 >r2 >…>rS >0.
For each s = 1,…,S 1 let cs 2 (rs+1,rs) and let asset s be the call
option with strike proce cs .
The ìmarketî payo§ matrix r is then
26r1 c1 r1 c2 r1 cS 1 r1 37 60r2 c2r2 cS 1 r27 6 . . … . . 7
640 0 rS 1 cS 1rS 175 0 0 0 rS
This is a matrix of full rank since
detR = (r1 c1)(r2 c2)…(rS 1 cS 1)rS > 0
Decision problem in Asset economy
An agent chooses a consumption bundle x0 for t = 0 and the planned consumption bundles for all s for t = 1, (x1 , …, xS ), as well as portfolio z.
The choice must satisfy the budget constraint at every t and every s. An agent faces an integrated consumption-portfolio problem. Formally
maxu(x) p0 (x0 ω0)+qz 0 ps (xs ωs ) rs z 0 for every s
Decision problem and resourse constraints
We expect = in each BC then
maxfu(x) jps (xs ωs)2M(q) allsg
holds for every agent.
What about markets as a whole?
We require that demand equals supply for each commodity in each state in equilibrium.
Every Önancial asset that is bought by an investor has to be issued by someone else. The net supply for every Önancial asset is zero.
We assume perfect foresight about spot prices conditional on the state.
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Radner Equilibrium
A Radner equilibrium is a four-tuple (p, q, x, z),
a matrix of spot prices p,
a vector of securities prices q,
a collection of consumption matrices x(i), for each i, a collection of portfolios of securities z(i), for each i,
(x(i),z(i)) solves iís optimization problem
maxfu(x) jps (xs ωs)2M(q) allsg
aggregate supply equates aggregate demand today and in each s
∑xms (i)=∑ωsm(i)foreachs=1,…,Sandm=1,…,M. i=1 i=1
each security is in zero net supply
∑ zj (i ) = 0 for each j = 1, …, J i=1
Equivalence of asset and contingent claim economies
In contingent claim economy the agent solves
(S) max u(x) ∑pˆs (xs ωs)0
In asset economy there are no contingent claims markets.
Instead there are spot markets, one for t = 0 and one for each s, and there are markets for S Arrow securities.
z is the portfolio of AS, zs can be any positive or negative real number.
Agent solves
maxu(x) p0 (x0 ω0)+αz 0 ,
ps(xs ωs)zs fors=1,…,S
where ps are spot market prices at s determined when s materializes.
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Equivalence of asset and contingent claim economies
α z is the value of the portfolio. It equals to the savings between t = 0 and t = 1.
This can be negative, which means a loan. The t = 1 the constraint must bind. Hence
(S) max u(x) p0(x0 ω0)+∑(αsps)(xs ωs)0 ,
αs is a scalar ps is vector here
Choose a normalization so that αs ps = pˆs for any s. This does not
change state s budget constraint.
The decision problem in the complete market asset economy and
contingent claims economy are the same.
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Equivalence of asset and contingent claim economies Theorem
Suppose the asset markets are complete.
i) If the consumption plans xs (i) and the prices p constitute an equilibrium in the contingent claims economy, then there exist asset prices q and portfolio holdings z (i) such that (x,z,p,q) constitute Radner equilibrium.
ii) Conversely, if the consumption plans xs (i), portfolio holdings z (i), asset prices q and spot prices p constitute Radner equilibrium, then there are Arrow prices α 0 such that consumption plans x, and the contingent commodity prices (αs ps ) constitute an equilibrium in the contingent claims economy.
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Decomposition
Split the decision problem
maxfu(x) jps (xs ωs)2M(q) allsg
into the consumption and the Önancial problems.
ws is the s-contingent value of the endowment, at spot market prices,
ws (i) := ps ωs (i) for s = 0,…,S. Introduce the indirect utility
v(ys)=maxfu(xs)jpsxs ys fors=0,…,S.g
v(ys) is the value of the maximized utility if at most ys can be spent in state s. Here we choose how to spend ys, that is solve for xs.The choice is about the composition of consumption.
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Decomposition
y := (y0 , y1 , …, yS ) is a distribution of incomes spent today and tomorrow in each s.
The choice of y is about saving and risk exposure, i.e. his Önancial decisions.
Only the Önancial decision problem.
maxfv(y) j(y w)2M(q)g
This problem is equivalent to the original decision problem in Asset economy.
maxfv(y) j(y w)2M(q)g= maxfmaxfu(xs) jps xs =ys allsgj(y w)2M(q)g= maxfu(x) jps (xs ωs)2M(q) allsg
This drastically simpliÖes the original economy.
Decomposition
Let (p,q,x,z) be an equilibrium of the original economy.
Consider the new economy (v,w), v describe the preferences, w – the
endowments.
This is a contingent claim economy with I agents but only one
commodityó wealthó today and in each s.
Let α be the Arrow price vector, and let [α+ ] = [1, α] .
Then (α+,y), with ys = ps xs, is an equilibrium of this one-good economy.
The Arrow price αs is then interpreted as the value at t = 0 of a dollar at t = 1 available in state s.
max v(y) (y0 w0)+∑αs(ys ws)=0 s=1
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Decomposition
(y0 w0)+∑αs (ys ws)=[α+](y w)=0
The constraint states that [α+] is orthogonal to y w. But this
states that y w 2 M(q). Hence i solves maxfv(y) j(y w)2M(q)g.
This is satisÖed in the Radner equilibrium (p, q, x, z).
In addition, the market for ìwealthî clears.
Thus we can simplify an economy with many state-contingent commodities into a simpler economy in which state-contingent income is a representative commodity.
Equilibrium prices of Önancial assets are not changed in the way.
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