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MATH0060 Lecture Notes Carlo Marinelli This version: 9.III.2021 Contents Preface 5 Chapter 1. Preliminaries 7 Chapter 2. A market model on a finite probability space 9 2.1. Introduction 9 2.2. Price processes and trading strategies 9 2.3. Contingent claims, completeness and arbitrage 11 2.4. Pricing by no-arbitrage I 15 2.5. Pricing by no-arbitrage II 17 2.6. Single period models 19 2.7. Notes 20 2.8. Exercises 20 Chapter 3. The binomial model 23 3.1. The one-period model 23 3.2. The multiperiod model 24 3.3. Continuous-time limit 26 3.4. The formula of Black and Scholes 28 3.5. Complements 30 3.6. Notes 31 3.7. Exercises 31 Chapter 4. Finance in continuous time 33 4.1. Notions of no-arbitrage and the fundamental theorem of asset pricing 33 4.2. Superreplication 35 4.3. Attainable claims 37 4.4. Brownian market model 37 4.5. Exercises 39 Chapter 5. Portfolio optimization 43 5.1. Utility functions 43 5.2. Maximization of the expected utility of final wealth 43 5.3. Optimization and no-arbitrage 44 5.4. A closer look 45 5.5. The case of incomplete markets 47 5.6. Notes 48 5.7. Exercises 48 Bibliography 49 3 Preface These lecture notes are preliminary and will be corrected and/or expanded when needed. They are meant to serve as a summary of the topics covered in class, and should not be redistributed. Please report any error or misprint to the author. 5 CHAPTER 1 Preliminaries Let ( ,F ,P) be a probability space endowed with a filtration F = (Fn)n∈N, where Fn is a sub-σ-algebra of F for every n ∈ N. Given a measurable space (E, E), an E-valued stochastic process X is a family of E-valued random variables (Xn)n∈N. In other words, Xn : → E is an (F , E)-measurable function for every n ∈ N. The following notions of measurability of discrete-time stochastic processes are standard. 1.0.1. Definition. A stochastic process X with values on a measurable space E is called F-adapted if Xn is Fn-measurable for every n ∈ N. The process X is called F-predictable if Xn is Fn 1-measurable for all n > 1. Let us introduce some notation. For any stochastic process Y with values in a vector space, let us set Yt := Yt Yt 1, t > 1, Let E be a vector space endowed with a scalar product 〈·, ·〉. If X and Y are E-valued processes, the real stochastic process X · Y defined by (X · Y )t := t∑ u=1 Xu, Yu = t∑ u=1 Xu, (Yu Yu 1) , t > 1, is called the (discrete-time) stochastic integral of X with respect to Y . In order to simplify notation, note that any element f of E can be identified with the linear form 〈f, ·〉, so that we can write, with a minor abuse of notation, 〈f, g〉 = fg for any g ∈ E. We shall use this notational convention whenever confusion cannot arise. The simple identity (XY )t = Xt Yt + XtYt 1, t > 1, is quite useful. With the convention that X denotes the process (Xn 1)n>1, this can be written as (XY ) = X Y + Y X. This identity implies (XY )t = X0Y0 + t∑ u=1 (XY )u = X0Y0 + t∑ u=1 Xu Yu + t∑ u=1 Yu 1 Xu = X0Y0 + (X · Y )t + (Y ·X)t, which can also be written in the equivalent more symmetric form XY = X0Y0 +X · Y + Y ·X + [X,Y ], where the process [X,Y ] is defined by [X,Y ]t := t∑ u=1 Xu Yu, t > 1. 7 CHAPTER 2 A market model on a finite probability space Kann man aus Nichts Etwas machen fragte er lauernd, worauf ich mi trauisch den Kopf schüttelte. Er nickte traurig. Friedrich Dürrenmatt, Das Bild des Sisyphos. 2.1. Introduction In this chapter we deal with a discrete-time market model with finite horizon T ∈ N. In particular, the market can be described as follows: at each time t = 1, . . . , T agents (market participants) can trade (buy and sell) 1 + d financial assets. One may think, for instance, to a market where is a risk-free cash account and d risky assets, such as stocks, are traded. More generality, however, is allowed, as will be soon clear. The uncertainty underlying the market will be modeled by a finite probability space ( ,F ,P), where = {ω1, ω2, . . . , ωN}, N ∈ N, F is the power set of ,, and P has full support, i.e. P({ωn}) > 0 for all n ∈ {1, . . . , N}. The flow of information available to all agents will be modeled by a filtration F = (Ft)t=0,1,…,T such that FT F . The informal economic interpretation of this setup is that Ft contains all information available to market participants up to and including time t, and that no information gets lost with time, i.e. all information contained in Ft 1 is also contained in Ft. 2.2. Price processes and trading strategies We consider a financial market where agents can trade 1 + d assets at times t ∈ {1, . . . , T 1}. The prices of the tradeable assets are assumed to be exogenous, i.e. not influenced by trading. We shall denote the price at time t ∈ {0, . . . , T} of asset k ∈ {0, 1, . . . , d} by S kt . 2.2.1. Assumption. The stochastic process S : × {0, 1, . . . , T} → R1+d defined by S = (S t)t∈{0,1,…,T} = (S 0t , S 1 t , . . . , S d t )t∈{0,1,…,T}, is adapted, with S 0t > 0 for all t and S 0 0 = 1. The 0-th asset is assumed to have a strictly positive price at all times because it will be used as a numéraire. Although it is common to choose as numéraire the risk-free cash account, there is no compelling reason to do so: for the developments to come, the strict positivity of its price process is enough. Concerning trading by market participants, it is natural to assume that at time t their investment decisions can only be based on the information available prior to time t: between time t 1 and time t an agent holds H kt units of the k-th asset, and the decision is taken at time t 1. Let us formalize this observation. 2.2.2. Definition. A trading (or portfolio) strategy H is an R1+d-valued predictable process H = (H t)t∈{1,…,T} = (H 0t , H 1 t , . . . , H d t )t∈{1,…,T}. The portfolio value V associated to the trading strategy H is the real-valued process V t := 〈H t, S t〉 := d∑ k=0 H kt S k t , t = 1, . . . , T. 9 10 2. A MARKET MODEL ON A FINITE PROBABILITY SPACE We shall use only trading strategies of the type defined next. 2.2.3. Definition. A strategy H is said to be self-financing if it satisfies 〈H t, S t〉 = 〈H t+1, S t〉 t = 1, . . . , T 1. (2.2.1) Note that (2.2.1) is equivalent to d∑ k=0 H kt S k t = d∑ k=0 H kt+1S k t . The meaning of (2.2.1) is that by changing the porfolio from H t to H t+1 there is neither input not outflow of funds, i.e. the investor only “rearranges” his holdings without adding or subtracting funds to the investment. Note also that the initial investment required for a self-financing strategy H can be expressed, thanks to (2.2.1) and to the assumption S 00 = 1, as V 0 = 〈H 1, S 0〉 = H 01 + d∑ k=1 H k1 S k 0 . 2.2.4. Exercise. Show that the set of self-financing trading strategies is a vector space. 2.2.5. Proposition. Let H be a trading strategy. The following assertions are equivalent: (a) H is self-financing; (b) 〈 H t+1, S t〉 = 0 for all t ∈ {1, . . . , T 1}; (c) V t = V 0 + (H · S )t for all t ∈ {1, . . . , T}. Proof. The equivalence of (a) and (b) is just a reformulation of the definition of self-financing strategy. Let us show that (b) and (c) are equivalent: the integration by parts formula yields V = V 0+H ·S +S ·H , hence (c) is equivalent to (S · H )t = 0 for every t > 1, where (S · H )t = t∑ u=1 〈S u 1, H u〉, t > 1, from which it immediately follows that (b) implies (c). Conversely, (S · H )t = 0 for every t > 1 implies that 〈 H t+1, S t〉 = (S · H )t+1 (S · H )t = 0. We are now going to discount the price processes S 1, . . . , S d in terms of the numéraire, i.e. we introduce the adapted Rd-valued stochastic process S defined by St = (S 1 t , . . . , S d t ) := ( S 1t S 0t , · · · , S d t S 0t ) , t = 0, . . . , T, to which we shall refer as the process of discounted prices. Analogously, the discounted portfolio value process V is defined as Vt := V t S 0t , t = 0, . . . , T. Note that V0 = V 0 by the convention that S 0 0 = 1. Moreover, the integration by parts formula yields V = V0 + H · S S 0 + S S 0 · H , where ( S S 0 · H ) t = t∑ u=1 1 S 0u 1 〈 H u, S u 1〉 = 0 and, by the obvious identity (S 0/S 0) = 0, H · S S 0 = H · S, 2.3. CONTINGENT CLAIMS, COMPLETENESS AND ARBITRAGE 11 where H = (H 1, . . . , H d) is the Rd-valued predictable process of portfolio holding in the assets 1 to d. We thus have the representation of the discounted portfolio value process V as V = V0 +H · S. Unsurprisingly, the discounted portfolio value does not depend on H 0, simply because the price of the 0-th asset remains constant in discounted terms (that is, in terms of itself!). The random variable (H · S)t can be interpreted as the discounted cumulative gain (or loss) up to time t corresponding to the trading strategy H . Let H and SF denote the vector space of Rd-valued predictable processes and the vector space of self-financing trading strategies, respectively. 2.2.6. Lemma. The vector space of self-financing trading strategies SF is isomorphic to the vector space H× L0( ,F0.P). Proof. Recall that the initial value of a portfolio strategy H is given by V0 = V 0 = 〈H 1, S 0〉 = H 01 + d∑ k=1 H k1 S k 0 = H 0 1 + d∑ k=1 Hk1S k 0 and note that the map SF → L0( ,F0,P) H 7 → H 01 + d∑ k=1 H k1 S k 0 = H 0 1 + 〈H1, S0〉 is a homomorphism. Since the map H 7→ H is clearly a homomorphism from SF to H, it follows that the map Φ: SF → H× L0( ,F0,P) H 7 → (H, H 01 + 〈H1, S0〉) is a homomorphism of vector spaces, which is easily seen to be injective. Let us show that it is also surjective: let H ∈ H and a ∈ L0( ,F0,P), and set H k := Hk for all k = 1, . . . , d, H 01 := a 〈H1, S0〉. Moreover, let H 0t , t > 2, be defined through the self-financing condition H 0t S 0 t + d∑ k=1 H kt S k t = H 0 t+1S 0 t + d∑ k=1 H kt+1S k t . We have thus shown that Φ is an isomorphism of vector spaces. The lemma implies that for any predictable process H modeling the portfolio holdings in the securities 1 to d and an initial endowment a, there exists a unique self-financing trading strategy with initial value equal to a. In other words, given H and a as said, H can be complemented in a unique way by a real predictable process H 0 modeling the holding in the 0-th security such that (H 0, H) is a self-financing trading strategy with initial value equal to a. Thanks to the isomorphism established by the lemma, from now on we shall work, whenever it is convenient, in discounted terms, i.e. representing discounted portfolio value processes of self-financing strategies with initial value equal to a by processes of the type a+H · S. 2.3. Contingent claims, completeness and arbitrage One of our main interests is the pricing of so-called financial derivatives (contingent claims). These are contracts that guarantee a flow of payments (X t) T t=1 which depend on the price evolution of the basic underlying assets, e.g. X t = φ(S 0, . . . , S t) for a given function φ. For the time being, we are going to content ourselves with contingent claims entitling to a single payoff at time T , so that X t = 0 for all t < T , X := X T . 2.3.1. Definition. A contingent claim is a random variable on the probability space ( ,F ,P). 12 2. A MARKET MODEL ON A FINITE PROBABILITY SPACE A central question is to determine what a fair price for a contingent claim would be. Let us start with the following observation: suppose that we can find an initial investment (or endowment) V0 and a self-financing strategy such that X(ω) = V0 + (H · S)T (ω) for all ω ∈ , where X := X /S 0T . It is then clear that there is no difference between holding the contingent claim or holding the self-financing portfolio. In fact, independently of the evolution of asset prices, at time T the value of the contingent claim and the value of the portfolio associated to H will be identical. Thus V0 is a very good candidate to be considered as fair price of X (or equivalently of X ). Let us formalize this reasoning. 2.3.2. Definition. A contingent claim X is called attainable if there exist V0 ∈ R and an Rd-valued predictable process H = (Ht) T t=1 such that V0 + (H · S)T = X. In this case we shall say that V0 is a fair price of X. Moreover, a market is called complete if every contingent claim is attainable. The above reasoning does not guarantee that the fair price of an attainable claim is unique. In order to make sure that this is actually the case (a quite natural requirement indeed), we have first to introduce another very important concept, which will keep us busy for a while. 2.3.3. Definition. We say that arbitrage possibilities exist if there exists a predictable process H with values in Rd such that (H · S)T (ω) > 0 for all ω ∈ , and (H · S)T (ω) > 0 for at least one ω ∈ . A market is free of arbitrage if it does not admit any arbitrage possibility. In other words, we want to rule out the following situation: starting from zero initial capital, there exists a trading strategy such that the resulting portfolio value at time T is non-negative and not identically equal to zero. A market where such possibilities exist would clearly make no (economic) sense. If a market does not admit any arbitrage possibility, then we shall say that it satisfies the (NA) condition. The first important consequence of the absence of arbitrage is a uniqueness principle for the price of attainable contingent claims. 2.3.4. Proposition (Law of one price). If the market satisfies the (NA) condition, then every attainable contingent claim admits a unique fair price. Proof. We proceed by contradiction. Suppose that X = V0 + (H · S)T = Vˉ0 + (Hˉ · S)T , and assume that V0 < Vˉ0. Then ( (H Hˉ) · S) T = Vˉ0 V0 > 0, i.e. investing according toH Hˉ gives rise to an arbitrage opportunity, which contradicts the hypotheses. The case V0 > Vˉ0 is completely similar. Let us also prove that H1 ·S = H2 ·S (identity of processes!): we have (H1 ·S)T = H2 ·S)T , define A = { (H1 · S)t < (H2 · S)t } ∈ Ft and H := (H1 H2)1A1]t,T ], which is a predictable process. Here t is the first t for which (H1 · S)t 6= (H2 · S)t. Since ((H1 H2) · S)T = 0 implies 0 = 1A((H 1 H2) · S)T = 1A((H1 H2) · S)t + (H · S)T , where, by definition of A, 1A((H 1 H2) · S)t < 0, hence (H · S)T > 0, which contradicts the absence of arbitrage. We shall denote by K the subset of L0( ,FT ,P) defined as K := { (H · S)T : H ∈ H } , i.e. K is the set of contingent claims that are attainable with zero initial capital. 2.3. CONTINGENT CLAIMS, COMPLETENESS AND ARBITRAGE 13 We shall also use the convex cones C0 L0( ,FT ,P) and C L∞( ,FT ,P) of super-replicalble claims at price zero, defined as C0 := { g ∈ L0( ,FT ,P) : f ∈ K, f > g } = K L0+ and C := { g ∈ L∞( ,FT ,P) : f ∈ K, f > g } = C0 ∩ L∞( ,FT ,P). Note that C0 C L∞ . We can now give a geometric definition of the no-arbitrage property. 2.3.5. Definition. A financial market S is free of arbitrage if K ∩ L0+ = {0}. (2.3.1) The following characterization of the no-arbitrage property is often useful. 2.3.6. Lemma. Condition (2.3.1) is equivalent to C ∩ L∞+ = {0}. Proof. Assume that C ∩ L∞+ = {0}. If f ∈ K ∩ L0+, then f ∧ n converges to f pointwise as n → +∞ and f ∧ n ∈ C ∩ L∞+ , hence f ∧ n = 0 for every n ∈ N, thus also f = 0. The converse implication is even more immediate. 2.3.7. Proposition. Assume that (NA) holds. Then C0 ∩ ( C0) = K. In particular, C ∩ ( C) = K ∩ L∞. Proof. The inclusion K C0∩( C0) is obvious: if g ∈ K then g ∈ C0, and, recalling that K is a vector space, also g ∈ K, hence g ∈ C0. We have to prove the inclusion K C0∩( C0): if g ∈ C0∩( C0), then there exist f1 ∈ K and h1 ∈ L0+ such that g = f1 h1, as well as f2 ∈ K and h2 ∈ L0+ such that g = f2 h2. This implies K f1 + f2 = h1 + h2 ∈ L0+, hence, by the (NA) condition, h1 = h2 = 0, which yields g = f1 ∈ K. 2.3.8. Proposition. Assume that (NA) holds. The set C of bounded contingent claims superreplicable at price zero is closed in L∞( ,FT ,P). Proof. We shall first prove that C0 = K + L 0 is closed in L 0. Let (kn + hn)n∈N be a sequence in C0, with kn ∈ K and hn ∈ L∞ for every n ∈ N, converging in L0. Identifying L0 with RN , let us assume, by contradiction, that kn is unbounded, i.e. that ‖kn‖ → ∞ as n→∞. Setting k n := kn ‖kn‖ ∈ K, h n := hn ‖kn‖ ∈ L 0 , z n := kn + hn ‖kn‖ , one has z n → 0 as n→∞ and ‖k n‖ = 1 for every n ∈ N, which implies that (k n) admits a subsequence,

denoted by the same symbol, converging to k of norm one. Therefore (h n) also admits a subsequence converging to h with k + h = 0. Since both K and L∞ are closed, one has k ∈ K and h ∈ L0 , hence k = h ∈ K ∩L0+ = {0}. This is a contradiction because ‖k ‖ = 1. Let us now show that C = C0 ∩L∞ is closed in L∞. If (gn) is a sequence in C converging to g ∈ L∞, then, a fortiori gn converges to g in L0, and g ∈ C0 by the previous reasoning, hence g ∈ C. 2.3.9. Definition. A probability measure Q on ( ,F) is called a martingale measure for S if S is a martingale with respect to Q. We shall denote the sets of martingale measures for S that are absolutely continuous with respect to P by Ma, and the subset of Ma of measures that are equivalent to P by Me. Both sets are convex. Since we have assumed that is finite and P has full support (i.e. P(ω) > 0 for all ω ∈ ), then Q is equivalent to P if and only if it has full support. In particular, in this setting one simply has dQ dP (ω) = Q(ω) P(ω) ω ∈ . 14 2. A MARKET MODEL ON A FINITE PROBABILITY SPACE It will sometimes be convenient to identify the probability measure Q with its Radon-Nikodym derivative dQ/dP ∈ L1( ,F ,P). Recall that S is a martingale with respect to Q if one has St ∈ L1( ,Ft,Q) for all t ∈ {0, . . . , T} and EQ [ St+1|Ft ] = St t = 0, 1, . . . , T 1. The following result is known as the Fundamental Theorem of Asset Pricing (FTAP). 2.3.10. Theorem. A model S satisfies the no-arbitrage condition if and only if there exists an equivalent martingale measure for S. For the proof of the FTAP we need results on discrete-time stochastic integrals and on separation of convex sets. 2.3.11. Proposition. A stochastic process M on a filtered probability space ( ,F ,F,P), adapted to F , is a P-martingale if and only if EP(H ·M)T = 0 for any predictable process H. Proof. (a) If M is a martingale and H is predictable, then H ·M is a martingale by an application of conditional expectation. (b) Conversely, assume that EP(H ·M)T = 0 for any predictable processH. In particular, let k ∈ 1, . . . , T be arbitrary, and take A ∈ Fk. Then the process H defined by Ht(ω) = { 0, t 6= k + 1, 1A(ω), t = k + 1 is predictable, because Hk+1 is Fk-measurable. In particular, by (a), one has 0 = EP(H ·M)T = EP ( (Mk+1 Mk)1A ) . Since A was arbitrary, this is equivalent to EP[Mk+1|Fk] =Mk. The process M is hence a martingale, because k was also arbitrary. The following corollary is not directly needed in the proof of Theorem 2.3.10, but it will be very useful soon. 2.3.12. Corollary. Assume that S is free of arbitrage. Let Q be a probability measure on FT . The following assertions are equivalent: (i) Q is a martingale measure for S; (ii) EQ f = 0 for all f ∈ K; (iii) EQ g 6 0 for all g ∈ C. Proof. The equivalence of (i) and (ii) is just the statement of the previous proposition. If g ∈ C, then there exist f ∈ K such that f > g, hence EQ g 6 0 for every martingale measure Q, that is (ii) implies (iii). The converse implication, the only one that needs the NA hypothesis, follows immediately by the identity K = C ∩ ( C). 2.3.13. Theorem (Hahn-Banach, geometric version). Let E be a Banach space, A E and B E non-empty convex disjoint subsets, with A compact and B closed. Then there exists a closed hyperplane that strictly separates A and B. For a proof see e.g. [2]. Proof of Theorem 2.3.10. Let us assume that Q is an equivalent martingale measure for S. We are going to show that K ∩ L0+ = {0}. Let f ∈ K ∩ L0+. Then f = (H · S)T , with H predictable, hence EQ f = 0 by Proposition 2.3.11. Moreover, since f > 0 and Q(ω) > 0 for all ω > 0, EQ f = 0 implies f = 0. Let us now prove that the converse implication is true. Let us introduce the space P := { N∑ k=1 μk1{ωk}, μk > 0, ∑ μk = 1 } , 2.4. PRICING BY NO-ARBITRAGE I 15 that is, P is the convex hull of the so-called Arrow-Debreu securities. Note that P L∞+ {0} and is compact. Invoking the Hahn-Banach theorem, we can strictly separate the closed convex set K and the compact convex set P . In particular, there exists a linear functional Q ∈ L1 and real numbers α < β such that 〈Q , f〉 6 α f ∈ K and 〈Q , f〉 > β f ∈ P. Since K is a vector space, it can be easily seen that one can take α = 0, and that the first inequality in the previous display is actually an equality. Moreover, since β > 0, one has η := 〈Q , e〉 > 0, where e := (1, 1, . . . , 1), hence Q := Q /η is a probability measure. Since f ∈ K is arbitrary, Proposition 2.3.11 implies that Q is an equivalent martingale measure for S. From now on we shall always assume that (NA) holds. We are now going to prove the equivalence of single-period and multi-period no-arbitrage. 2.3.14. Theorem. Let S be a discounted price process. The following are equivalent: (a) S satisfies the no-arbitrage property; (b) for any t ∈ {1, . . . , T}, there does not exist η ∈ L0( ,Ft 1,P;Rd) such that 〈η, St〉 ∈ L0+ {0}. Proof. (b) (a): assume, by contradiction, that S admits arbitrage. In particular, there exists a predictable process H such that, setting t := min { s ∈ {1, . . . , T} : Vs := (H · S)s ∈ L+0 {0} } , one has t 6 T . For such t, it must hold Vt 1 = 0 or P ( Vt 1 < 0 ) > 0. In the former case one has 〈Ht, St〉 = Vt Vt 1 = Vt ∈ L+0 {0}, therefore η := Ht contradicts (b). In the latter case, setting η := Ht1{Vt 1<0}, one has 〈η, St St 1〉 = 〈Ht, St St 1〉1{Vt 1<0} = (Vt Vt 1)1{Vt 1<0} > Vt 11{Vt 1<0} > 0, again contradicting (b). (a) (b): let us assume, by contradiction, that there exist such t and η. One can then define the predictable process H by Hs := η if s = t, and Hs = 0 for all s 6= t, thus obtaining (H · S)T = 〈η, St〉 ∈ L+0 {0}, which contradicts (a). 2.4. Pricing by no-arbitrage I Given a random variable f and a number a, we are going to use the notation Kf,a = span ( K, f a). 2.4.1. Definition. Let X ∈ L∞( ,F ,P) be a bounded contingent claim. A number pi is called an arbitrage-free price of X if KX,pi satisfies the NA condition, i.e. if KX,pi ∩ L∞+ = {0}. The set of all arbitrage-free prices of X will be denoted by Π(X). The infimum and the supremum of this set will be denoted by pi(X) and pi(X), respectively. 2.4.2. Remark. Arbitrage-free prices can also be introduced as follows: a contingent claimX ∈ L∞( ,F ,P) has arbitrage-free price pi if there exists an adapted process Sd+1 such that Sd+10 = pi, S d+1 T = X, and (S, Sd+1) satisfies the NA property. In fact, simply setting Sd+1t := EQ[X|Ft], it is easy to show that the extended market (S, Sd+1) satisfies the above conditions. Conversely, if (S, Sd+1) has the NA property, the FTAP yields an equivalent measure Q such that Sk, k = 1, . . . , d, d + 1 are Q-martingale, hence Q ∈ Q and EQX = EQ Sd+1T = pi. We now establish a description of the set Π(X) in terms of the set Q of equivalent martingale measures. 16 2. A MARKET MODEL ON A FINITE PROBABILITY SPACE 2.4.3. Theorem. Let X ∈ L∞( ,F ,P). One has Π(X) = { EQX : Q ∈ Q } . In particular, Π(X) is a bounded interval. Proof. Let a = EQX, with Q ∈ Q. Then EQ(X a) = 0, hence KX,a ∩ L∞+ = {0}. In fact, f ∈ KX,a means that there exist α, β ∈ R and a predictable process H such that f = α(H ·S)T +β(X a), which in turn implies EQ f = 0. If, moreover, f ∈ L∞+ , then it must be f = 0. We have thus proved that{ EQX : Q ∈ Q } Π(X). Let us now show that the reverse inclusion also holds: let pi be such that KX,pi∩L∞+ = {0}. By the Hahn-Banach argument used in the proof of FTAP, one obtains a probability measure Q ≈ P such that EQ g = 0 for all g ∈ KX,pi K. In particular, EQ g = 0 for all g ∈ K, hence Q is a martingale measure, and EQ(X pi) = 0, i.e. pi = EQX. Recall that, if X is attainable, then Π(X) reduces to a singleton, i.e. EQX is independent of Q ∈ Q. The next theorem implies, among other things, that this property characterizes attainable claims, i.e. X ∈ K if and only if EQX = 0 for all Q ∈ Q. 2.4.4. Theorem. Let X ∈ L∞( ,F ,P). The following “dichotomy” holds: (a) if X is attainable, then Π(X) is a singleton; (b) if X is not attainable, then Π(X) is an open interval. Proof. It is enough to prove (b). We are going to show that, for any pi ∈ Π(X), there exist pi0 and pi1 ∈ Π(X) such that pi0 < pi < pi1. Let pi = EQX and set Xt := EQ[X|Ft], so that X = X0 + T∑ t=1 Xt = X0 + T∑ t=1 (Xt Xt 1). By hypothesis X is not reachable, hence there exists t 6 T such that Xt Xt 1 6∈ Kt, where Kt := {〈η, St〉 : η Ft-measurable}. Note that Kt is a convex closed linear subspace of L ∞( ,Ft,P), hence also of L∞( ,Ft,Q). By the Hahn-Banach theorem, we can strictly separate Kt and the compact convex set { Xt}, thus obtaining that there exists ζ ∈ L1( ,Ft,Q) such that 〈ζ, f〉 < 〈ζ, Xt〉 f ∈ Kt, or, equivalently, EQ ζf < EQ ζ Xt f ∈ Kt. Since Kt is linear, we deduce that EQ ζf = 0 for all f ∈ Kt, hence EQ ζ Xt > 0. Let us assume, Without loss of generality, that |ζ| < 1/3, and define the measure Q by dQ dQ = ζ , ζ := 1 + ζ EQ[ζ|Ft 1]. It is immediate to check that ζ > 0 and EQ ζ = 1, so that Q is a probability measure equivalent to Q. Let us show that pi1 := EQ X > EQX = pi: one has EQ X = EQ ζ X = EQX + EQ ζX EQX EQ[ζ|Ft 1], where, recalling that ζ is Ft-measurable, EQ ζX = EQ ζ EQ[X|Ft] = EQ ζXt, and EQX EQ[ζ|Ft 1] = EQ EQ[X|Ft 1]EQ[ζ|Ft 1] = EQXt 1 EQ[ζ|Ft 1] = EQ EQ[ζXt 1|Ft 1] = EQ ζXt 1, hence EQ X = EQX + EQ ζ(Xt Xt 1) > EQX, because EQ ζ(Xt Xt 1) by the Hahn-Banach separation argument above. In order to prove that pi1 ∈ Π(X), we need to show that Q is a martingale measure, i.e. that EQ [Sk Sk 1|Fk 1] = 0 for all k. 2.5. PRICING BY NO-ARBITRAGE II 17 To this purpose, we distinguish three cases: (i) if k < t, then Q = Q on Fk, hence the claim is verified. In fact, since k 6 t 1 implies Fk Ft 1, one has, for any A ∈ Fk, Q (A) = EQ ζ 1A = EQ 1A + EQ ζ1A EQ 1A EQ[ζ|Ft 1] = Q(A) + EQ ζ1A EQ EQ[ζ1A|Ft 1] = Q(A). (ii) If k = t, we recall that, by the Hahn-Banach argument above, one has EQ ζf = 0 for all f ∈ Kt. Therefore, for any A ∈ Ft 1, so that 1A is Ft 1-measurable, one has EQ 1A(St St 1)ζ = 0, which is equivalent to EQ[(St St 1)ζ|Ft 1] = 0. Since EQ[ζ |Ft 1] = 1, it holds EQ [(St St 1)|Ft 1] = EQ[(St St 1)ζ |Ft 1] = EQ[(St St 1)|Ft 1] + EQ[(St St 1)ζ|Ft 1] EQ[(St St 1)|Ft 1] EQ[ζ|Ft 1] = 0. (iii) Let k > t. One has EQ [Sk Sk 1|Fk 1] = 1 EQ[ζ |Fk 1] EQ[(Sk Sk 1)ζ |Fk 1], where EQ[ζ |Fk 1] = ζ because ζ is Ft-measurable and k > t implies Ft Fk 1. For the same reason one also has EQ[(Sk Sk 1)ζ |Fk 1] = ζ EQ[(Sk Sk 1)|Fk 1], hence EQ [Sk Sk 1|Fk 1] = EQ[(Sk Sk 1)|Fk 1] = 0. This concludes the proof that there exists pi1 ∈ Π(X) such that pi1 > pi. The proof that there also exists pi0 ∈ Π(X) such that pi0 < pi, one simply sets pi0 := EQˉX, with dQˉ dQ := 2 dQ dQ . In particular, Qˉ is equivalent to Q because 1/3 < |dQˉ/dQ| < 5/3. All other properties can be easily verified. 2.4.5. Corollary. An arbitrage-free market is complete if and only if the set of equivalent martingale measures is a singleton. Proof. Assume that Q = {Q}. Then X ∈ L∞( ,F ,P) must be attainable because, if it were not, the set {EQX}Q∈Q would be an open interval, which is clearly impossible. If the market is complete, then, for any A ∈ F , 1A is attainable. Therefore, for any Q1, Q2 ∈ Q, one has Q1(A) = EQ1 1A = EQ2 1A = Q2(A), i.e. Q1 = Q2. 2.5. Pricing by no-arbitrage II Let us consider the duality between L∞( ,FT ,P) and L1( ,FT ,P) induced by the bilinear form L∞ × L1 → R (f, g) 7 → 〈f, g〉 := E fg. Recall that all Lp( ,FT ,P) spaces are isomorphic with RN if | | = N , as we assume in this section. The set C L∞ of contingent claims that are superreplicable at price zero is a closed convex cone, as it follows by the identity C = K +L∞ . In particular, the bipolar theorem implies that the bipolar C of C coincides with C. The following result gives a fundamental duality relation between superreplicable claims at price zero and equivalent martingale measures. 18 2. A MARKET MODEL ON A FINITE PROBABILITY SPACE 2.5.1. Proposition. Let S satisfy NA. Then C is the cone generated by the set of martingale measures absolutely continuous with respect to P. Therefore, for any g ∈ L∞, the following facts are equivalent: (i) g ∈ C; (ii) EQ g 6 0 for every martingale measure Q absolutely continuous with respect to P. (iii) EQ g 6 0 for every martingale measure Q equivalent to P. Proof. That (i) implies (ii) has already been proved in Corollary 2.3.12, and obviously (ii) implies (iii). Since equivalent martingale measures are dense in the set of martingale measures, (iii) implies (ii). It thus remains only to show that (ii) implies (i). To this purpose, note that, since C is a cone, one has C = { Q ∈ L1 : 〈Q, g〉 6 0 g ∈ C}. Although C is not a set of probability measures, it is certainly a subset of L1+ because C L∞ . From this it easily follows that C is the cone generated by Ma. Since C = { g ∈ L∞ : 〈Q, g〉 6 0 Q ∈ C }, the bipolar theorem yields C = C = { g ∈ L∞ : EQ g 6 0 Q ∈Ma } . Recalling that K = C ∩ ( C), one immediately obtains the 2.5.2. Corollary. Let S satisfy NA and f ∈ L∞. The following facts are equivalent: (i) f ∈ K; (ii) EQ f = 0 for all Q ∈Ma; (iii) EQ f = 0 for all Q ∈Me. It immediately follows that if f ∈ L∞ is such that EQ f = a for all Q ∈ Me, then there exists a predictable process H such that f = a+ (H · S)T . Another consequence of the above developments is the following characterization of complete mar- kets, under the assumptions of NA. 2.5.3. Theorem. Assume that S satisfies NA. The market is complete if and only if there exists a unique equivalent martingale measure. Proof. SinceMe contains only one element Q, it follows by the above observation that f = a+(H ·S)T . The reverse implication is trivial. With the results just obtained we can prove that the set of no-arbitrage prices of a non-attainable claim is an open interval. Proof of Theorem 2.4.4. Let X be a bounded non-attainable contingent claim and Π := {EQX; Q ∈ Me} its set of no-arbitrage prices, whose infimum and supremum we shall denote by pi0 and pi1, re- spectively. Recall that Π is an interval, bounded because X is bounded. Let us show that pi1 6∈ Π: one obviouly has EQ(X pi1) = EQX sup Q∈Me EQX 6 0 Q ∈Me, hence X pi1 ∈ C by Proposition 2.5.1. Then, by definition of C, there exists f ∈ K such that f > X pi1. Assuming, by contradiction, that pi1 ∈ Π, i.e. that there exists Q1 ∈ Me such that pi1 = EQ1 X, one has EQ1 [f (X pi1)] = 0, hence f = X pi1 because f > X pi1 and Q1 is equivalent to P. But this implies that X is attainable at price pi1, which is absurd, hence Π is open on the right. The same reasoning applied to X shows that Π is also open on the left. 2.6. SINGLE PERIOD MODELS 19 2.6. Single period models In this subsection we specialize some of the results obtained above to answer basic questions of no- arbitrage and completeness in a one-period model (i.e. with T = 1). We still assume that = {ω1, . . . , ωN}, with N <∞. Let us begin with a motivativating example: assume N = 3 and d = 1, with S10 = 5, S 1 1(ω) = 3, ω = ω1, 4, ω = ω2, 6, ω = ω3, and let us look for an arbitrage strategy H ∈ R. By definition, H is an arbitrage strategy if it satisfies (H · S)1 = 2H > 0, H > 0, H > 0, with at least one strict inequality. This system of inequalities admits only the trivial solution H = 0, which is clearly not an arbitrage strategy, thus the market model is free of arbitrage. This implies, by the fundamental theorem of asset pricing, that an equivalent risk neutral measure exists, i.e. that EQ S1 = S0, or equivalently 0 = EQ S = 2q1 q2 + q3, where we have set, as usual, qi = Q({ωi}), i = 1, 2, 3. We are thus led to the system{ 2q1 q2 + q3 = 0 q1 + q2 + q3 = 1, q1 > 0, q2 > 0, q3 > 0, which admits the family of solutions (q1, q2, q3) = (2a 1, 2 3a, a), 1 2 < a < 2 3 , i.e. there exist (infinitely) many equivalent risk neutral measures. Let us now check whether the model is complete: again by definition, we need to find V0 ∈ R and H ∈ R that satisfy V0 + (H · S)1 = X V0 2H = x1, V0 H = x2, V0 +H = x3, where X is identified with the vector in R3 with components x1, x2 and x3. Simple computations reveal immediately that the system of three equations with two unknowns does not ad