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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




Contents

Introduction

1.1 Assets, portfolios and arbitrage opportunities

1.2 Absence of arbitrage and martingale measures

1.3 Derivative securities






Introduction

I Hans Föllmer, Alexander SchiedI Stochastic Finance - An Introduction in Discrete TimeI Chapter 1 - Arbitrage theory





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





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)





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





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)





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





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 .





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





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





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





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)





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 ).





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.





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





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)





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 )+





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





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.





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

].





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





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)





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





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





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 )).





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





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)





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





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


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