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Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Stochastic Finance - Arbitrage theory
Tadeáš Horký
Department of Probability and Mathematical StatisticsFaculty of Mathematics and Physics
Charles University Prague
October 8th, 2007
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Contents
Introduction
1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures
1.3 Derivative securities
1.4 Complete market models
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Introduction
I Hans Föllmer, Alexander SchiedI Stochastic Finance - An Introduction in Discrete TimeI Chapter 1 - Arbitrage theory
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Introduction
I one-period model of financial market, trading at time t = 0I finite number of primary assets, their initial prices at time
t = 0 known, future prices at time t = 1 random variablesI arbitrage - trading opportunities which yield a profit without
any downside riskI characterization of financial market with absence of such
arbitrage - martingale measures, contingent claims, perfecthedge
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
One-period model
I financial market with d + 1 assets (equities, bonds,commodities, . . .)
I priced at initial time t = 0 and at the final time t = 1I we assume that the i th asset is available at time 0 for a price
πi ≥ 0, collection π = (π0, π1, . . . , πd) ∈ Rd+1+ is called a
price systemI prices at time 1 non-negative random variables S0, S1, . . . , Sd
on probability space (Ω,F , P)
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
One-period model
I riskless investment possibility in bonds: by assuming π0 = 1and S0 ≡ 1 + r , where r > −1 (enough, more natural wouldbe r ≥ 0)
I notation
S = (S0, S1, . . . , Sd) = (S0, S),
π = (π0, π),
ξ = (ξ0, ξ1, . . . , ξd) = (ξ0, ξ) ∈ Rd+1,
I where ξi represents the number of shares of the i th asset,which an investor choose at t = 0
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
One-period model
I price for buying the portfolio at time t = 1
π · ξ =d∑
i=0
πiξi
I at time t = 1 the portfolio will have the value
ξ · S(ω) =d∑
i=0
ξi S i (ω) = ξ0(1 + r) + ξ · S(ω)
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Loan and short sales
I ξ0 < 0 corresponds to taking out a loan, at t = 0 we receive|ξ0| and pay back (1 + r)|ξ0| at t = 1
I ξi < 0 for i ≥ 1 means short sale of the i th asset, receivedamount πi |ξi | can be used for buying quantities ξj ≥ 0, j 6= ioh the other assets
I price of the portfolio ξ is given by π · ξ = 0
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Definition 1.1. - arbitrage theory (page 5)
I A portfolio ξ ∈ Rd+1 is called an arbitrage opportunity ifπ · ξ ≤ 0 but ξ · S ≥ 0 P-a.s. and P[ξ · S > 0] > 0.
I with positive probability a positive profit without exposure toany downside risk
I market inefficiency, certain assets are not priced wellI absence of arbitrage implies that S i vanishes P-a.s. once
πi = 0, hence with no loss in generality we assume πi ≥ 0 fori = 1, . . . , d .
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Lemma 1.3. (page 5)
I The following statements are equivalent1. The market model admits an arbitrage opportunity2. There is a vector ξ ∈ Rd such that
ξ · S ≥ (1 + r)ξ · π P-a.s. and P[ξ · S > (1 + r)ξ · π] > 0
I no arbitrage opportunity =⇒ investment in risky assets whichyields with positive probability a better result than investingthe same amount in the risk-free asset must be open to somedownside risk
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Definition 1.4. - martingale measure (page 6)
I characterization of arbitrage-free market models which donot admit any arbitrage opportunities
I Definition 1.4. A probability measure P∗ is called arisk-neutral measure, or a martingale measure, if
πi = E ∗[
S i
1 + r
], i = 0, 1, . . . , d . (1)
I price πi is identified as the expectation of the discountedpayoff under the measure P∗
I pricing formula (1) does not take into account any riskaversion, therefore measure P∗ is called risk-neutral
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Fundamental theorem of asset pricing (page 6)
I let us define the set of risk-neutral measures which areequivalent to P
P := P∗|P∗is a risk-neutral measure with P∗ ≈ P
I Theorem 1.6. A market model is arbitrage-free ⇐⇒ P 6= ∅.In this case, there exists a P∗ ∈ P which has a boundeddensity dP∗/dP.
I Remark: in an infinite market model of tradable assetsS0, S1, S2, . . . is implication =⇒ no longer true
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Discounted net gains (page 7)
I in the proof is used random vector Y = (Y 1, . . . , Y d) ofdiscounted net gains
Y i :=S i
1 + r− πi , i = 0, 1, . . . , d . (2)
I with this notation Lemma 1.3. implies that arbitrage-freemodel is equivalent to the condition
for ξ ∈ Rd : ξ · Y ≥ 0 P-a.s. =⇒ ξ · Y = 0 P-a.s. (3)
I since Y i is bounded from below by πi , P∗ is a risk-neutralmeasure if and only if E ∗[Y ] = 0.
I discounted asset prices and time value of money
S i
1 + r, i = 0, 1, . . . , d . (4)
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Attainable payoff and its return (page 10)
I let V :=ξ · S | ξ ∈ Rd+1
denote the linear space of all
payoffs which can be generated by some portfolio, the set Vwill be called attainable payoff
I the portfolio that generates V ∈ V is not uniqueI therefore it is reasonable to define the price of V ∈ V as
π(V ) := π · ξ if V = π · S . (5)
I Definition 1.11. Suppose an arbitrage-free market model andV ∈ V an attainable payoff such that π(V ) 6= 0. Then thereturn of V is defined by
R(V ) :=V − π(V )
π(V ).
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Expected return of V in arbitrage-free model (page 11)
I Proposition 1.12. Suppose arbitrage-free market model andlet V ∈ V be an attainable payoff such that π(V ) 6= 0.
1. Under any risk-neutral measure P∗, the expected returnequals E∗[R(V )] = r .
2. Under any measure Q ≈ P such that EQ [|S |] < ∞
EQ [R(V )] = r − covQ
(dP∗
dQ, R(V )
),
where P∗ is an arbitrary risk-neutral measure in P and covQ
denotes the covariance with respect to Q.
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Derivative securities (page 13)
I in real financial markets not only the primary assets are tradedI securities whose payoff depends in non-linear way on the
primary assets S0, S1, . . . , Sd , and sometimes other factorsI derivative securities, options, contingent claimsI Example 1.15. - forward contract: one agent agrees to sell
another agent an asset S i at time 1 for a delivery price K attime 0, such contract corresponds to random payoffC fw = S i − K
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Call and put options (page 13-14)
I Example 1.16. - call/put option: the owner has the optionto buy/sell the i th asset at time 1 for a fixed strike price K
C call = (S i − K )+, C put = (K − S i )+,
C call − C put = S i − K
I relation between the price of call and put options: put-callparity
π(C call) = π(C put) + πi − K1 + r
. (6)
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Reverse convertible bond (page 15)
I Example 1.19. - reverse convertible bond: pays interest rwhich is higher than interest r of riskless bond
I at maturity t = 1 the issuer may convert the bond into apredetermined number of shares of asset S i instead of payingthe nominal value in cash
I equals purchase of standard bond and the sale of a put optionI suppose that 1 is the price of a reverse convertible bond at
t = 0, nominal value at maturity is 1 + r , and that it can beconverted into x shares of the i th asset
I conversion will happen if S i < K := (1 + r)/x and the payoffthe reverse convertible fond is 1 + r − x(K − S i )+
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Contingent claims (page 15)
I Definition 1.21. - A contingent claim is a random variableC on the underlying probability space (Ω,F , P) such that
0 ≤ C < ∞ P-a.s.
I contingent claim C is a derivative of S0, . . . , Sd if it ismeasurable with respect to the σ-field σ(S0, . . . , Sd)generated by the assets, i.e., if C = f (S0, . . . , Sd) for ameasurable function on Rd+1
I it is a contract which is sold at t = 0 and which pays arandom amount C (ω) ≥ 0 at time 1
I security with negative terminal value can be reduced tocombination of a non-negative contingent claim and a shortposition in some of S0, . . . , Sd
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Arbitrage-free price of a contingent claim (page 16)
I so far, we have fixed prices πi of standard assets S0, . . . , Sd
I for a contingent claim C it is not clear, what the correct priceshould be
I our goal: identify possible prices which do not generatearbitrage in the market
I trading C at time 0 for a price πC corresponds to introducinga new asset
πd+1 := πC and Sd+1 := C . (7)
I Definition 1.22.: A real number πC ≥ 0 is called anarbitrage-free price of a contingent claim C if the marketmodel extended according to (7) is arbitrage-free.
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
The set of arbitrage-free prices (page 16)
I the set of all arbitrage-free prices for C is denoted Π(C ),the respective lower and upper bounds of Π(C ) are
π↓(C ) := inf Π(C ) and π↑(C ) := sup Π(C )
I Theorem 1.23.: Suppose that the set P of equivalentrisk-neutral measures for the original market model isnon-empty. Then
Π(C ) =
[E ∗
C1 + r
]| P∗ ∈ P such that E ∗[C ] < ∞
. (8)
I Moreover, the lower and upper bounds are given byπ↓(C ) = inf
P∗∈P
[E ∗ C
1+r
]and π↑(C ) = sup
P∗∈P
[E ∗ C
1+r
].
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Superhedging duality (page 18)
I the definition of arbitrage bound π↑(C ) can be restated as:
π↑(C ) = inf
m ∈ R | ∃ξ ∈ Rd with m + ξ · Y ≥ C1 + r
P-a.s.
I π↑(C ) is the smallest amount of capital which, if investedrisk-free, yields a superhedge (superrepliacation) of C in thesense that
(m − π · ξ) +ξ · S1 + r
≥ C1 + r
P-a.s.
I in this view is π↑(C ) in Theorem 1.23. often called asuperhedging duality
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Lower and upper bounds of call and put option (page 18)
I consider arbitrage-free market model, and letC call = (S i − K )+ be a call option
I for any P∗ ∈ P is C call ≤ S i so that[E ∗ C call
1+r
]≤ πi
I from Jensen’s inequality, we obtain the lower bound:[
E ∗C call
1 + r
]≥
(E ∗
[S i
1 + r
]− K
1 + r
)+
=
(πi − K
1 + r
)+
I thus, we have following universal bound for any arbitrage-freemarket model:
(πi − K
1 + r
)+
≤ π↓(C call) ≤ π↑(C call) ≤ πi . (9)
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Lower and upper bounds of call and put option (page 18)
I for a put option C put = (K − S i )+, we could obtain in similarway
(K
1 + r− πi
)+
≤ π↓(C put) ≤ π↑(C put) ≤ K1 + r
. (10)
I in many situations, the universal bounds (9) and (10) are infact attained
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Attainable contingent claim and replicating portfolio(page 20)
I Definition 1.27. A contingent claim C is called attainable(replicable, redundant), if there exists a portfolio ξ ∈ Rd+1
such that
C = ξ · S = ξ0(1 + r) + ξ · S P-a.s.
Such strategy ξ is called a replicating portfolio for C.I if we can replicate a given contingent claim C by some
portfolio ξ, then the problem of determining a price for C hassimple solution, the price of C should be equal to the costξ · π of its replication
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Unique arbitrage-free price (page 20)
I Theorem 1.28. Suppose that the market model isarbitrage-free and that C is a contingent claim.
1. If C is attainable, then the set Π(C ) of arbitrage-free prices forC consists of the single element ξ · π, where ξ is any replicatingportfolio for C.
2. If C is not attainable, then either π↓(C ) = +∞, orπ↓(C ) < π↑(C ) and
Π(C ) = (π↓(C ), π↑(C )).
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Example 1.29. (page 21)
I Example 1.29. We consider a call option C call = (S i − K )+
traded in a arbitrage-free market model.I if the risk-free return r ≥ 0 and if C call is not deterministic,
then Jensen’s inequality yields
(πi−K )+ ≤(
πi − K1 + r
)+
< E ∗[
(S i − K )+
1 + r
]for all P∗ ∈ P.
I time value of a call option: the value of the right to buy thei th asset at t = 0 is strictly less than any arbitrage-free pricefor C call
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Complete arbitrage-free market model (page 22)
I Definition 1.30. An arbitrage-free market model is calledcomplete if every contingent claim is attainable.
I In every market model following inclusion holds for eachP∗ ∈ P:
V = ξ · S | ξ ∈ Rd+1 ⊆ L1(Ω, ω(S1, . . . , Sd), P∗)
⊆ L0(Ω,F , P∗) = L0(Ω,F , P)
I if the market is complete, these inclusions are in fact equalitiesI linear space V is finite dimensional and the model can be
reduced to a finite number of relevant scenarios - atoms of(Ω,F , P)
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Characterization of a complete arbitrage-free market model(page 23)
I Proposition 1.31. For any p ∈ [0,∞], the dimension of thelinear space is given by
dim Lp(Ω,F , P) = supn ∈ N | ∃ partition A1, . . . , An of Ω
with Ai ∈ F and P[Ai ] > 0.
Moreover, n := dim Lp(Ω,F , P) < ∞ if and only if thereexists a partition of Ω into n atoms of (Ω,F , P).
I we will use this result in following theorem
Tadeáš Horký Stochastic Finance - Arbitrage theory
Introduction1.1 Assets, portfolios and arbitrage opportunities
1.2 Absence of arbitrage and martingale measures1.3 Derivative securities
1.4 Complete market models
Characterization of a complete arbitrage-free market model(page 23)
I Theorem 1.32. An arbitrage-free market model is complete ifand only if there exists exactly one risk-neutral probabilitymeasure, i.e., if |P| = 1. In this case, dim L0(Ω,F , P) ≤ d + 1.
I application of this theorem in Example 1.33., pages 23-25
Tadeáš Horký Stochastic Finance - Arbitrage theory