Lecture 15: Duality, Arbitrage, and
Asset Pricing
ISyE 6673: Financial Optimization
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Stewart School of Industrial and Systems Engineering Georgia Institute of Technology
October 24, 2022
Midterm survey recap
Asset Pricing
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Dr. ISyE 6673: Financial Optimization October 24, 2022 2 43
Asset Pricing
Duality Recap
• Lecture 10: Lagrangian duality Lagrangian relaxation
Dual problem is finding the best Lagrangian relaxation • Lecture 11: Sensitivity analysis
Origin of shadow prices
Generalized dependence on the right-hand side b
• Lecture 12: Application to A/B testing • Lecture 13: Geometry of duality
Normal cones Farkas lemma
• Lecture 14: Large-scale optimization Column generation and cutting planes
Applications of duality in asset pricing and relationship to arbitrage!
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Asset Pricing
• A market model
• What are desirable properties of a market?
Completeness
Law of one price
Fundamental theorem(s) of asset pricing
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Asset Pricing Two-period market model
Two-period market model
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Asset Pricing Two-period market model
Market setup
• We consider a portfolio of n assets (could be bonds, stocks, currencies, etc.)
• Two-period model:
At time 0 (today), we know the price of each asset, represented by a vector p
At time 1 (tomorrow), the asset prices are uncertain; the world could be in one of m scenarios
Scenario matrix:
S11 · · · S1n ..
p = . pn
S= . .. , Sij isthepriceofassetj inscenarioi Sm1 Smn
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Asset Pricing Two-period market model
Market setup
Consider the following market • Wehaven=3assets
1 • Prices at time 0 (today) are given by p = 2
• At time 1 (tomorrow) the world could be in one of 4 scenarios:
1 1 5 S = 1 5 10
Q: Are all these assets equally risky?
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Asset Pricing Two-period market model
Constructing portfolios
x1 .
A portfolio x = . counts shares purchased at time 0
xi > 0 means a long position in asset i
xi < 0 means a short position in asset i
• Price of portfolio at time 0 is p⊺x = nj=1 pjxj • Payoff of portfolio at time 1 is
n Sx n j=1 1j j
Sx=Sjxj = . ∈Rm
i=1 n S x j=1 mj j
One payoff value for each scenario
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Asset Pricing Two-period market model
Constructing portfolios
1 1 5
1 3 9 Market setup: p = 27, S = 1 5 10
1 • Portfolio x = −1
• Price: p⊺x=1−2+7=6
1 3 9 7
• 1 1 5 5 Payoff: Sx= ×1+ ×(−1)+ ×1= 1 5 10 6
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Asset Pricing Complete markets
Complete markets
Dr. ISyE 6673: Financial Optimization October 24, 2022 10 43
Asset Pricing Complete markets
Completeness
• A market is specified by prices p today and payoffs S tomorrow
Definition
A market is complete if any payoff z ∈ Rm can be achieved by some portfolio x.
• The asset span M is the set of payoffs achievable by some portfolio x:
M={z∈Rm :z=Sxforsomex∈Rn}
M is the column space of S; completeness means M = Rm
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Asset Pricing Complete markets
Completeness
Consider a market with 1 asset and 2 scenarios:
1 p=1, S= 1
• Canweachievethepayoffz= 2 ?
• We cannot! The payoff in scenario 1 must always be the same as the payoff in scenario 2
• This market is not complete!
• The matrix does not have full row rank
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Asset Pricing Complete markets
Complete markets: linear algebra review!
The following statements are equivalent:
• A market is complete: M = Rm
• The column space of S is Rm
• S has m linearly independent columns
• S has m linearly independent rows (full row rank) • Shasrankm
Number of assets
A complete market must have more assets than scenarios, i.e. m ≤ n. Otherwise, S has rank at most n < m
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Asset Pricing The Law of One Price
The Law of One Price
Dr. ISyE 6673: Financial Optimization October 24, 2022 14 43
Asset Pricing The Law of One Price
Law of One Price
If two portfolios x1 and x2 have the same payoff in all scenarios at time 1, they must have the same price at time 0.
Sx1 = Sx2 ⇒ p⊺x1 = p⊺x2
Equivalently
Sx = 0 ⇒ p⊺x = 0
(an asset with 0 payoff in all scenarios should have price 0)
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Implications of the Law of One Price
Asset Pricing The Law of One Price
• Sx=0⇔x∈null(S)
x is orthogonal to each row si of S x ⊥ span(s1,...,sm)
• p⊺x=0⇔p⊥x
Therefore p ∈ null(S)⊥ (orthogonal complement)
p ∈ span(s1,...,sm) = span(S⊺)
There exists some q ∈ Rm such that S⊺q = p
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Asset Pricing The Law of One Price
Example in an incomplete market
1 •p=1, S=1
• Lawofonepriceholdstrivially: S⊺x=0⇒x=0⇒p⊺x=0 • q⊺ =1 0orq⊺ =0 1bothsatisfyq⊺S=p⊺
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Asset Pricing The Law of One Price
Why do we care about the law of one price?
• Consider a market where the law doesn’t hold
• There exists a portfolio x such that
Sx = 0 (0 payoff in all scenarios) p⊺x = ε > 0 (nonzero price)
• Consider the portfolio αx (scale x by α)
Sαx = αSx = 0 (still 0 payoff in all scenarios) p⊺αx = αε (arbitrary price)
• Now take an arbitrary portfolio y ∈ Rn with payoff Sy = z ∈ Rm, and augment it by αx
S(y + αx) = Sy = z (same payoff as y) p⊺(y + αx) = p⊺y + αε (arbitrary price)
• Any portfolio can be purchased at any price!
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Asset Pricing The Law of One Price
Payoff pricing
Payoff pricing
How much do we need to pay at time 0 to secure a particular payoff z at time 1?
• Onlydefinedifz∈M=col(S)
• Same payoff can be achieved by multiple portfolios
• Set of prices determined by function q : Rm → P(R):
q(z) = {p⊺x : Sx = z,x ∈ Rn} • In general, q(z) is a set
• If the Law of One Price holds:
Sx1 = Sx2 ⇒ p⊺x1 = p⊺x2 ⇒ q(z) has a unique value
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Asset Pricing The Law of One Price
Payoff pricing function
• When q(z) is unique for all z ∈ M, we call q a payoff pricing function.
• q(z) unique ⇔ Law of One Price holds ⇔ p⊺ = q⊺S
• Putting both together for a portfolio x achieving payoff z:
q(z) = p⊺x = q⊺Sx = q⊺z
The Law of One Price holds in a market if and only if the payoff pricing function is linear.
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Asset Pricing The Law of One Price
Example in a complete market
1 1 1 •p=2,S=01
• CheckLOPholds: S⊺x=0⇒x=0⇒p⊺x=0 • q⊺ =1 1satisfiesq⊺S=p⊺ soq(z)=q⊺z
• Ifwewanttoachievepayoffz= 0 thenweneedtobuya
portfolio worth
⊺ 3 qz=110=3
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Asset Pricing The Law of One Price
Example in an incomplete market
1 •p=1, S=1
• CheckLOPholds: S⊺x=0⇒x=0⇒p⊺x=0
• q1⊺ =1 0orq2⊺ =0 1bothsatisfyq⊺S=p⊺
• We cannot achieve a payoff z = 0 at any price
• Toachieveapayoffz= 2 ,wemustpay
2 2 q1⊺z=10 2=2=01 2=q2⊺z
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Asset Pricing Arbitrage
Dr. ISyE 6673: Financial Optimization October 24, 2022 23 43
Strong arbitrage
Asset Pricing
• Sx≥0⇔x⊺si ≥0∀i∈[m]
• p⊺x < 0 ⇔ x⊺p < 0
• Equivalently,
p ∈/ cone(s1,...,sm)
Definition
In a market with price vector p and payoff matrix S, a strong arbitrage is a portfolio x that verifies:
• p⊺x < 0 (guaranteed gain at time 0)
• Sx ≥ 0 (guaranteed no losses at time 1)
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Asset Pricing Arbitrage
Strong arbitrage and the Farkas lemma
Theorem ( )
Let S ∈ Rm×n and p ∈ Rn. Exactly one of the following is true: 1. ThereexistssomexsuchthatSx≥0andp⊺x<0
2. There exists some q ≥ 0 such that S⊺q = p
Strong arbitrage: No strong arbitrage:
p ∈/ cone(s1,...,sm) p ∈ cone(s1,...,sm)
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Asset Pricing Arbitrage
Fundamental theorem of asset pricing
• Farkas’ lemma gives us two alternatives Either the market has a strong arbitrage Or there exists q ≥ 0 such that S⊺q = p
• Any portfolio x yielding payoff Sx = z will have price p⊺x = q⊺Sx = q⊺z
• We have a linear pricing rule again (so the LOP holds!) but this time it is non-negative.
Theorem (Fundamental theorem of asset pricing)
A market is strong-arbitrage-free if and only if there exists a linear and non-negative payoff pricing function.
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Asset Pricing Arbitrage
An edge case
• p ∈ cone(s1,...,sm) ⇔ no strong arbitrage
• Consider the portfolio x
p⊺x = 0 (no cost at time 0)
x⊺si ≥0∀i∈[m]⇔Sx≥0
(no losses at time 1)
s⊺1 x > 0 (at least one scenario has positive payoff)
• Portfolio x is called a weak arbitrage
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Asset Pricing
Strong vs. weak arbitrage
Strong arbitrage
Weak arbitrage
• ∃ i ∈ [ m ] , s ⊺i x > 0
• p⊺x<0 • Sx ≥ 0
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Asset Pricing Arbitrage
Ruling out weak arbitrage
How to rule out weak arbitrage?
• p ∈ cone(s1,...,sm) rules out strong arbitrage
• To rule out weak arbitrage, p cannot be on the boundary of the
• Equivalently, there exists q > 0 such that S⊺q = p
p ∈ int(cone(s1,…,sm))
• We need a stronger fundamental theorem of asset pricing to rule out weak arbitrage
• Ironic, I know!
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Asset Pricing Arbitrage
Fundamental theorem of asset pricing
Theorem (Weak form)
A market is strong-arbitrage-free if and only if there exists a linear and non-negative payoff pricing function.
Theorem (Strong form)
A market is arbitrage-free if and only if there exists a linear and positive payoff pricing function.
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Asset Pricing Arbitrage
Payoff pricing summary
Why do we care so much about linea pricing rules?
Because it gives us a way to price new assets (e.g. derivatives!)
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Asset Pricing Risk-neutral probabilities
Risk-neutral probabilities
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Asset Pricing Risk-neutral probabilities
Risk-neutral probabilities
• A market is arbitrage-free if and only if there exists a positive linear pricing function q > 0, i.e.
• For each asset j,
pj =S⊺jq=qiSij
• Letδ=mi=1qi:
pj = δ δSij = q ̃iδSij = Eq ̃[δSj], i=1 i=1
q ̃i ∈(0,1]
where we notice that q ̃ defines a valid probability distribution, which we call a risk-neutral probability distribution
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Asset Pricing Risk-neutral probabilities
The meaning of δ
• In most markets, there is at least one risk-free asset (e.g., US Treasury bonds); assume our first asset is risk-free with interest rate r
Price is p1
Future price is S1 = . for all scenarios
• Arbitrage-free market:
mm1 p 1 = q ̃ i δ S i 1 = q ̃ i δ ( 1 + r ) p 1 = δ ( 1 + r ) p 1 ⇒ δ = 1 + r
• Sj δ is a discount rate! Risk-neutral pricing is thus: pj = Eq ̃ 1+r
(1 + r)p1 .
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Asset Pricing Risk-neutral probabilities
Arbitrage-free and complete markets
• Arbitrage-free means there exists a positive linear pricing function, equivalently a risk-neutral probability distribution
• Complete means S has rank m ≤ n
• IfShasrankm,thensodoesS⊺,andS⊺q=phasaunique solution
Theorem (Second fundamental theorem of asset pricing)
An arbitrage-free market is complete if and only if the risk-neutral probability distribution is unique.
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Asset Pricing Derivative pricing
Derivative pricing
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Asset Pricing Derivative pricing
Derivatives
• Let’s say we have a new payoff vector z (also called a contingency claim)
• If z is a deterministic function of the existing n asset payoffs, it is called a derivative security:
z = f(S1,…,Sn)
Derivative
Forward contract Call option Put option
Outcome (at time 1) Must buy asset j at price K
Can buy asset j at price K Can sell asset j at price K
Sj − K max(Sj − K , 0) max(K − Sj , 0)
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Asset Pricing Derivative pricing
Put-call parity
zcall − zput = max(Sj − K , 0) − max(K − Sj , 0) = Sj − K
= zforward
In a complete and arbitrage-free market
q(zcall − zput) = q(zcall) − q(zput) = q(zforward)
Sj −K = Eq ̃ 1 + r
=pj− K 1+r
(unique linear pricing)
(risk-neutral probability)
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Asset Pricing Derivative pricing
Arbitrage-free derivative pricing
How to price derivatives?
Given a derivative with payoff/contingency claim z, how to determine the current price q(z)?
Basic idea
• Start with a market M1 = (p, S) with n assets p
• CreateanewmarketM2 = p
(the new contingency claim is the n + 1-th asset)
• Does there always exist an arbitrage-free price pn+1 for z? • Is the arbitrage-free price of z unique?
withn+1assets
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Asset Pricing Derivative pricing
Arbitrage-free derivative pricing
Suppose the derivative has non-negative payoff. If M1 is arbitrage-free and complete, there exists a unique price of the derivative such that M2 is also arbitrage-free and complete. Moreover, M1 and M2 have the same risk-neutral probability distribution.
• If M1 is complete, then M2 is also complete (adding an asset/column cannot decrease the column space)
• M1 is arbitrage-free and complete, so there is a unique risk-neutral probability distribution q ̃
• We can use this distribution to price z, i.e. z
q(z)=Eq ̃ (1+r)
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Asset Pricing Derivative pricing
Consider the following complete arbitrage-free market:
1 1 60 p= 50 ,S= 1 20
What is the price of a put option with strike price K = 40?
• First work out the risk-neutral probabilities (notice asset 1 is risk-free with 0 interest):
50 = q · 60 + (1 − q) · 20 ⇒ q = 3 4
• The risk-neutral probabilities are 3 1 44
•031 Thepayoffisz= 20 ,soq(z)=4 ·0+4 ·20=5.
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Asset Pricing Derivative pricing
Arbitrage-free pricing in incomplete markets
• As long as the market is arbitrage-free, we can always find arbitrage-free prices for derivatives
• If the market is not complete, the price may not be unique • This is the topic of Problem 3 on HW4!
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Asset Pricing Derivative pricing
• Farkas’ lemma underpins fundamental theorems in asset pricing • Markets that obey the law of one price have linear pricing rules • Complete markets have a unique pricing rule
• Arbitrage-free markets have positive pricing rules
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