Discrete-Time Market Models

Transcription

1 Discrete-Time Market Models Stochastic Calculus I Geneviève Gauthier HEC Montréal

2 I We will study in depth Section 2: The Finite Theory in the article Martingales, Stochastic Integrals and Continuous Trading by J. M. Harrison and S. R. Pliska. The model here is rather general. Indeed, a nite number (which may however be enormous) of securities are modelled during a nite number (which, again, may however be very large) of time periods. The only restriction on the distributions of the unit prices of the securities at each time is that such distributions must be discrete and positive, i.e. for each security and at each time, the unit price of the security at that time can only take a nite number of strictly positive values.

3 II The rst important result in that Section is Proposal 2.6 on page 227. It is proven that, if there exists a probability measure under which all discounted price processes are martingales, then we can build, based on such a measure, a price system for attainable contingent claims that is consistent with the model. On the other hand, if there exists such a price system, then we can build, based on the latter, a probability measure that transforms our discounted price processes into martingales. Such a proposal thus establishes a bijective correspondence between the set of martingale measures and the consistent price systems.

4 III It should be noted that such a proposal is silent on whether at least one martingale measure exists, or whether at least one price system exists. It only states that, if either one exists, then both exist, and it makes the link between them explicit.

5 IV on page 228 sets the necessary and su cient condition for at least one martingale measure, and as a consequence, at least one consistent price system, to exist. That condition is that the model contains no arbitrage opportunities. Since there may be several price systems that are consistent with the, we need to ensure that, whatever the price system that is used, the price associated with any given attainable conditional claim X is always the same, i.e. if π 1 and π 2 are two consistent price systems, then for any attainable contingent claim X, π 1 (X ) = π 2 (X ). Such a result is provided by a corollary on page 228.

6 V How can it be veri ed that our model contains no arbitrage opportunities? A necessary and su cient condition for no arbitrage is established by a lemma on page 228. Proposal 2.8, on page 230, proves that the discounted value of an admissible strategy is a martingale under all martingale measures in a model. Such a result is used in proving Proposal 2.9 (page 230), which shows how to determine the value of an attainable contingent claim at any time. The last part addresses the notion of completeness.

7 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function The probability space We are working with a ltered probability space (Ω, F, F, P). The sample space Ω has a nite number of elements, each of which is interpreted as a possible state of the world. We assume that the participants agree on the fact that Ω represents all the possible states of the world. Thus, since every state of the world ω 2 Ω is possible, the probability measure P applying on the measurable space (Ω, F) must be such that 8ω 2 Ω, P (ω) > 0. The measure P, that associates a probability to each state of the world, represents a particular investor s vision, i.e. two investors may associate di erent probabilities to the same state of the world ω.

8 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function The ltration Information structure We choose to observe the system until a time horizon T, which is a terminal date for all economic activity under consideration. Since time is considered discretely, the ltration F = ff t : t 2 f0, 1,..., T gg represents information available at each time. We set F 0 = f, Ωg, the sigma-algebra that contains no information, and F T = the set of all events in Ω, i.e. F T allows us to distinguish each possible state of the world. Since Card (Ω) <, for all t 2 f0, 1,..., T g, there exists a nite partition P t = A t 1,..., A t n t that generates the sub-sigma-algebra F t.

9 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function We model K + 1 securities. The securities I De nition The multidimensional stochastic process! S = n!s t : t 2 f0, 1,..., T go representing the evolution of the individual unit prices of all securities is made up of the random column vectors! S t = St 0, St 1,...St K 0 where S k t = the unit price of the security k at time t.

10 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function The securities II Each of the components S k = S k t : t 2 f0, 1,..., T g of the price process! S is a stochastic process adapted to the ltration F and it takes strictly positive values. Since Card (Ω) <, we have that 8t 2 f0, 1,..., T g and 8k 2 f0, 1,..., K g, St k discretely distributed random variable, i.e. it can only take a nite number of values.

11 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function The securities III De nition The 0 th security plays a part a bit peculiar, in that it is a riskless security (we can think, for example, of a bank account or a bond). It implies that 8ω 2 Ω and 8t 2 f0, 1,..., T 1g, S 0 t (ω) S 0 t+1 (ω). It can also be assumed, without loss of generality, that 8ω 2 Ω, S 0 0 (ω) = 1. The unidimensional stochastic process β t = 1 S 0 t will be used as discount factor.

12 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function The securities IV The ltration F describes how information is revealed to the investors at each time. The fact that F 0 = f, Ωg implies that the components of! S 0 are constants, i.e. at time t = 0, all security prices are known with certainty. Since F t is generated by the nite partition P t = A t 1,..., At n t, then, at time t, every investor knows with certainty which of the cells of P t has occurred, but the investor is not able to distinguish between the elements in that cell. Since F T is the sigma-algebra made up of all possible events of Ω, investors can, at time T, determine with certainty which state of the world ω has occurred. This means that we group under a single name, say ω i, all the states of the world that have the same e ect on the. This is one the reasons why it is warranted to restrict the sample space to a nite set (Card (Ω) < ).

13 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function Trading strategy I De nition A trading strategy is a predictable vector stochastic process! n o φ = t : t 2 f1,..., T g where the random row vector! φ t = φ 0 t, φ1 t,...φk t represents the investor s portfolio at time t : φ k t = the number of shares of security k held at time t.

14 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function Trading strategy II By requiring that the strategy be a predictable process, we are allowing the investor to select his time t portfolio right after the share prices at time t 1 are announced. As a result, the random variable φ k t represents the number of shares of security k held during the time period (t 1, t] (except for φ k 1 that represents the number of shares of security k held during the time period [0, 1]). The value of portfolio! φ t, right after the security prices at time t 1 are available, is! φ t! S t 1, whereas the value of the same portfolio, but at time t, is! φ t! S t.

15 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function Trading strategy III De nition We say a strategy is self- nancing if no funds are added to or withdrawn from the value of the portfolio after time t = 0, i.e. 8t 2 f1,..., T 1g,! φ t! S t =! φ t+1! S t. So,! φ t! S t represents the amount we receive at time t + ε (ε represents a very small positive quantity) when the portfolio is liquidated! φ t and! φ t+1! S t is the amount we must pay, again at time t + ε, to purchase portfolio! φ t+1. Note that there are no transaction costs.

16 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function Trading strategy IV o The process V = nv t : t 2 f0, 1,..., T g represents the value of the strategy at any time. V t = ( 1! S 0 if t = 0! φ t! S t if t 2 f1,..., T g.

17 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function Trading strategy V De nition A trading strategy is called admissible if it is self- nancing and its value is never negative. So, an admissible strategy is such that an investor is never put into a situation of debt. This doesn t mean however that short sales are prohibited. Φ = the set of admissible strategies 8 each of the components φ k 9 ><! of! φ is predictable, >= = φ! φ is self- nancing. >: and 8t 2 f0,..., T g, V t 0 >;

18 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function De nition Arbitrage opportunity I An admissible strategy! φ is an arbitrage opportunity if V 0 = 0 and if E P h V T i > 0. Note that the latter condition implies that there exists ω 2 Ω for which V T, ω > 0. Indeed, E P h V T i = ω2ω V T, ω P (ω) {z } {z } 0 >0 So, in order to have E P h V T i > 0, there must be at least one ω for which V T, ω > 0.

19 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function Arbitrage opportunity II So, the strategy! φ is an arbitrage opportunity when, while investing nothing V 0 = 0, we are assured not to lose any money (V T, ω 0 since! φ is admissible) and we have a positive probability to make a gain (9ω 2 Ω for which V T, ω > 0).

20 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function De nition Contingent claims I A contingent claim X is a (Ω, F T ) non-negative random variable. It can be thought of as a contract between two parties, such that X (ω) will be paid by one party to the other if state ω pertains. De nition X = the set of contingent claims = X X is a (Ω, F T ) random variable such that 8ω 2 Ω, X (ω) 0.

21 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function Contingent claims II De nition A contingent claim X is said to be attainable if there exists an admissible trading strategy! φ that can replicate the cash ow generated by X, i.e. V T = X.

22 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function Price systems I De nition A price system π is a linear operator on X that returns non-negative values π : X! [0, ) and that satis es and π (X ) = 0, X = 0 8a, b 0 and 8X 1 X 2 2 X, π (ax 1 + bx 2 ) = aπ (X 1 ) + bπ (X 2 ). As its name indicates, a price system is intended to associate with each of the contingent claims, a price. Since, for any admissible strategy! φ, V T is a (Ω, F T ) V T 2 X. random variable taking non-negative values,

23 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function Price systems II De nition A price system is said to be consistent with the model if the price associated with the contingent claim V T is its value at time t = 0, V 0, i.e. π V T = V 0.

24 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function De nition Π Price systems III = the set of price systems consistent with the model 8 9 π (X ) = 0, X = 0 >< 8a, b 0 and 8X 1, X 2 2 X, >= = π : X! [0, ) π (ax 1 + bx 2 ) = aπ (X 1 ) + bπ (X 2 ). >: 8! φ 2 Φ, π V T = V 0 >;

25 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function Risk-neutral measures I We will build, if possible, a set of probability measures on (Ω, F) that we call equivalent martingale measures (also know as risk-neutral measures). Such measures bear, a priori, no relationship to the actual probability that an event occurs, no more than they re ect the knowledge of an investor. They are nothing more than a very convenient calculation tool in pricing contingent claims. De nition P = the set of martingale measures equivalent to P 8 9 < = : Q Q is a probability measure on (Ω, F), = 8ω 2 Ω, Q (ω) > 0 and 8k 2 f0, 1,..., K g, βs k ; is a Q martingale.

26 Probability space Filtration Securities Trading strategy Arbitrage Contingent claim Price system Risk-neutral measure Indicator function Indicator functions I De nition Throughout this text, indicator functions will be denoted by I, i.e. 8A 2 F, I A : Ω! f0, 1g is de ned by 1 if ω 2 A I A (ω) = 0 if ω /2 A. To become familiar with such functions, the reader can verify that: (i) If A 2 F t then I A is F t measurable. (ii) If A, B 2 F are disjoint, then I A[B = I A + I B. (iii) If Y is a random variable, then Y = ω2ω Y (ω) I ω.

27 Interpretation Part I Part II Proposal 2.6 Theorem Proposal. There is bijective correspondence between the set Π of price systems that are consistent with the model and the set P of martingale measures equivalent to P. Such as correspondence is (i) π (X ) = E Q [β T X ], X 2 X (ii) Q (A) = π S 0 T I A, A 2 F.

28 Interpretation Part I Part II Interpretation I Proposal 2.6 If we know a martingale measure Q equivalent to P, then we can build a consistent price system π by setting 8X 2 X, π (X ) = E Q [β T X ]. On the other hand, if a price system π consistent with the model is available, then we can build a martingale measure Q equivalent to P by de ning 8A 2 F, Q (A) = π S 0 T I A. Such a proposal tells us that, if there exists a martingale measure equivalent to P or if there exists a price system consistent with the model, then both exist, and the proposal establishes the link between them. However, nothing allows us to show that either of such two objects exist.

29 Interpretation Part I Part II Interpretation II Proposal 2.6 The necessary and su cient condition for a martingale measure, and, as a consequence, for the price system to exist, is addressed by.

30 Interpretation Part I Part II De nition De nition Let P and Q be two probability measures that exist on the measurable space (Ω, F). The measures P and Q are said to be equivalent if and only if the impossible events are the same under both measures, i.e. 8A 2 F, P (A) = 0, Q (A) = 0. In our case, since 8ω 2 Ω, P (ω) > 0, all measures Q equivalent to P shall satisfy the condition 8ω 2 Ω, Q (ω) > 0.

31 Interpretation Part I Part II First part I To be shown. Let Q be a martingale measure equivalent to P. Then, the function π de ned on the set of contingent claims X by π (X ) = E Q [β T X ] is a price system consistent with the model. Since Q 2 P then 1 Q is equivalent to P, i.e. 8ω 2 Ω, Q (ω) > 0, 2 and the components of the discounted security prices, β! S, are martingale under the measure Q.

32 Interpretation Part I Part II First part II We want to show that π is a consistent price system. We therefore need to verify that (i) π (X ) = 0, X = 0, (ii) 8a, b 0 and 8X 1, X 2 2 X, π (ax 1 + bx 2 ) = aπ (X 1 ) + bπ (X 2 ), (iii) 8! φ 2 Φ, π V T = V 0.

33 Interpretation Part I Part II Starting from the de nition of π, we get π (X ) = E Q [β T X ] = First part - Point (i) ω2ω β T (ω) {z } >0 X (ω) {z } 0 Q (ω). {z } >0 Note that β T (ω) > 0 since β T = 1 and the model ST 0 assumes that security prices can only be strictly positive. Therefore, π (X ) = 0, β T (ω) X (ω) Q (ω) ω2ω {z } {z } {z } >0 0 >0, 8ω 2 Ω, X (ω) = 0. = 0

34 Interpretation Part I Part II 8a, b 0 and 8X 1, X 2 2 X, First part - Point (ii) π (ax 1 + bx 2 ) = E Q [β T (ax 1 + bx 2 )] = ae Q [β T X 1 ] + be Q [β T X 2 ] = aπ (X 1 ) + bπ (X 2 ).

35 Interpretation Part I Part II We want to show that 8! φ 2 Φ, First part - Point (iii) I π V T = V 0. But, from the de nition of π, π V h i T = E Q β T V T, (1) so we determine, in a rst step, β T V T :

36 Interpretation Part I Part II β T V T First part - Point (iii) II!! = β T φ T S T from the de nition of V T, (ref: eq (2.4), p. 226). T 1!!!!! = β T φ T S T + β i i S i φ i+1 S i since! φ is self- nancing. i=1 {z } =0 T 1!! = β T φ T S T + i=1 T 1!!!! = β T φ T S T + β i φ i S i i=1 T 1!!!! β i φ i S i β i φ i+1 S i i=1 T!! β i 1 φ i S i 1 i=2 T 1!!!!!! = β T φ T S T + β 1 φ 1 S 1 + φ i β i S i β i! 1 S i 1 i=2 T!!!! = β 1 φ 1 S 1 + φ i β i S i β i! 1 S i 1 i=2 β T 1! φ T! S T 1

37 Interpretation Part I Part II First part - Point (iii) III Therefore, substituting the expression above into the equation (1), π V T h = E Q i β T V T = E "β Q T!! 1 φ 1 S 1 + i=2! φ i β i! S i β i 1! S i 1 # = h E Q!! i T β 1 φ 1 S 1 + E Q!! φ i β i S i β i! 1 S i i 1 i=2 = h h E Q E Q!! ii T β 1 φ 1 S 1 jf 0 + E Q E Q!! φ i β i S i β i! 1 S i 1 jf i i=2 = h E Q! h φ 1 E Q! ii β 1 S 1 jf 0 + T h E Q! h φ i E Q! β i S i β i! 1 S i 1 jf i since! φ is predictable, i=2 1 ii 1 ii

38 Interpretation Part I Part II 2 = E Q! h 6 φ 1 E Q! i T β 1 S 1 jf {z } 5 i=2 β 0! S 0 First part - Point (iii) IV 3 2 E Q! h φ 6 i E Q!! i β i S i β i 1 S i 1 jf i 1 4 {z } 7 h 5 =E Q! i! β i S i jf i 1 β i 1 S i 1 =! 0 since the components of β! S are martingales under the measure Q, 3 = E Q h! φ 1 β 0! S 0 i = β 0! φ 1! S 0 since β 0,! φ 1 and! S 0 are F 0 measurable, therefore constant, =!! φ 1 S 0 since β 0 = 1 S0 0 = 1 = V 0 from the de nition of V 0 (ref: eq (2.4), p. 226).

39 Interpretation Part I Part II I Second part To be shown. Let π be a price system consistent with the model. Then, the function Q, de ned on the set of events F by Q (A) = π S 0 T I A martingale measure equivalent to P. Since π 2 then 1 π (X ) = 0, X = 0; 2 8a, b 0 and 8X 1, X 2 2 X, π (ax 1 + bx 2 ) = aπ (X 1 ) + bπ (X 2 ) ; 3 8! φ 2 Φ, π V T = V 0.

40 Interpretation Part I Part II We want to show that Q 2 P, i.e. (i) Q is a probability measure; II Second part (ii) Q is equivalent to P, i.e. 8ω 2 Ω, Q (ω) > 0; (iii) The components of the discount price process for the securities β! S, are martingales under the measure Q.

41 Interpretation Part I Part II Second part - Point (i) I Let s start by building a strategy that will be subsequently of use : 8t 2 f1,..., T g, let! φ t = (1, 0,..., 0). This trading strategy is such that, at any time, we hold nothing more than a single share of the riskless security. Note that at any time! φ t is constant, i.e. 8ω 2 Ω,! φ t (ω) = (1, 0,..., 0). A a consequence,! φ t is F 0 measurable, therefore F t 1 measurable, which implies that! φ is predictable. It is also easy to show that! φ is self- nancing since we keep the same portfolio.

42 Interpretation Part I Part II Now, V T and V 0 = Second part - Point (i) II K φ k T S T k from the de nition of V k=0 = S 0 T from the de nition of! φ. (2) = K φ k 1 S 0 k from the de nition of V k=0 = S 0 0 from the de nition of! φ. = 1 since, by hypothesis, S 0 0 = 1. (3)

43 Interpretation Part I Part II Let s now show that Q (Ω) = 1. Second part - Point (i) III Starting from the de nition Q, Q (Ω) = π ST 0 I Ω = π S 0 T = π V T from Equality (2). = V 0 since π is consistent with the model. = 1 from Equality (3).

44 Interpretation Part I Part II Second part - Point (i) IV We must also ensure that for all disjoint events A 1,..., A n 2 F,! n[ n Q A i = Q (A i ). i=1 i=1

45 Interpretation Part I Part II But! n[ Q A i i=1 Second part - Point (i) V = π S 0 T IS n i=1 A i = π S 0 T = = n i=1! n I Ai i=1 π S 0 T I A i from Property (2.5b), p n Q (A i ) from the very de nition of Q. i=1

46 Interpretation Part I Part II Second part - Point (ii) We want to prove that 8ω 2 Ω, Q (ω) > 0. Let s assume there exists at least one ω 2 Ω for which Q (ω ) = 0 and let s show this leads to a contradiction. From the equivalence relation π (X ) = 0, X = 0, we have 0 = Q (ω ) = π S 0 T I ω, 8ω 2 Ω, S 0 T (ω) I ω (ω) = 0 ) S 0 T (ω ) = S 0 T (ω ) I ω (ω ) = 0. Since the prices are strictly positive, we have that 8ω 2 Ω, ST 0 (ω) > 0. As a consequence S 0 T (ω ) I ω (ω ) = S 0 T (ω ) > 0. Contradiction! So, there cannot exist ω 2 Ω for which Q (ω) = 0.

47 Interpretation Part I Part II Second part - Point (iii) I The objective is to show that each of the components in the vector of discounted security prices, β! S, is a Q martingale. Let s choose one of the components arbitrarily: let k 2 f0, 1,..., K g. By the Optional Stopping Theorem, it will be su cient to show that h i h i E Q β τ Sτ k = E Q β 0 S0 k for all (Ω, F, F) bounded stopping time τ (8ω 2 Ω, τ (ω) T ). So let s choose arbitrarily a bounded stopping time τ.

48 Interpretation Part I Part II Second part - Point (iii) II Let s consider the trading strategy de ned as follows: 8 I >< ftτg if j = k φ j t = Sτ k I Sτ >: 0 ft>τg = Sτ k β τ I ft>τg if j = 0 0 if j 2 f1,..., K g, j 6= k. Such a strategy consists in holding a share of security k until the random time τ and then in exchanging it for a random quantity S k τ β τ of bonds (riskless security) until maturity (t = T ).

49 Interpretation Part I Part II Second part - Point (iii) III Let s verify that such as strategy is admissible, i.e.! φ is a predictable process, that strategy is self- nancing and its value, V t is non-negative at all times.

50 Interpretation Part I Part II Second part - Point (iii) IV Let s begin by verifying that! φ is a predictable process : 8t 2 f1,..., T g, 1 φ 0 t = S τ k β τ I fτ<tg = t i=0 1 S τ k β τ I fτ=ig = t i=0 1 S i k β i I fτ=ig is F t 1 measurable, {z } F i measurable 2 φ k t = I fτtg is F t 1 measurable since 0 1 fτ tg = fτ < tg c = S t 1 C i=0 fτ = iga {z } 2F i F t 1 c 2 F t 1 and 3 φ j t = 0 is F 0 measurable therefore F t 1 measurable.

51 Interpretation Part I Part II But Second part - Point (iii) V For! φ to be self- nancing, it must be that 8t 2 f1,..., T 1g,! φ t! S t =! φ t+1! S t. 8!! S k < φ t S t = t if t τ Sτ k β τ St 0 if t > τ = : and since 8t 2 f0,..., T g, β t St 0 = 1 S 0 St 0 t! φ t+1! S t = S k t if t + 1 τ S k τ β τ S 0 t if t + 1 > τ = St k St k S k τ β τ S 0 t = 1, then 8 < : St k St k S k τ β τ S 0 t t < τ t = τ t > τ t < τ t = τ t > τ therefore! φ truly is self- nancing.

52 Interpretation Part I Part II Second part - Point (iii) VI Now, let s verify that 8t 2 f0,..., T g, V t 0. V 0 = = and 8t 2 f1,..., T g, S k 0 if τ > 0 Sτ k β τ S0 0 if τ = 0 S k 0 if τ > 0 S0 k β 0 S 0 0 if τ = 0 = S 0 k > 0 V t =!! S k φ t S t = t if t τ Sτ k β τ St 0 if t > τ > 0. Therefore the trading strategy! φ truly is admissible.

53 Interpretation Part I Part II Second part - Point (iii) VII Let s now show that βs k is a martingale under the measure Q. From its de nition, we already know that the process βs k is adapted to the ltration F. Recall that we want to show that h i h i E Q β τ Sτ k = E Q β 0 S0 k. Recall also that and V T = S k τ β τ S 0 T V 0 = S k 0.

54 Interpretation Part I Part II Then E Q h β τ S k τ i Second part - Point (iii) VIII = β τ(ω) (ω) Sτ(ω) k (ω) Q (ω) = ω2ω β τ(ω) (ω) Sτ(ω) k (ω) π ST 0 ω2ω {z } ω from the de nition of Q. >0! = π β τ(ω) (ω) Sτ(ω) k (ω) S T 0 I ω ω2ω (ref : l eq (2.5b), p. 226). = π β τ Sτ k ST 0

55 Interpretation Part I Part II Second part - Point (iii) IX = π V T see equality as recalled above. = V 0 because π is consistent and! φ is admissible = S k 0 see equality as recalled above. = β 0 S0 k h simce β 0 = 1. i = E Q β 0 S0 k since β 0 S0 k is F 0 The proof is now complete. measurable, hence constant.

56 Interpretation Theorem Lemma on page 228. If there exists a self- nancing strategy! φ (not necessarily admissible) such that V0 = 0, h i V T 0 and E P V T > 0, then there exists an arbitrage opportunity.

57 Interpretation I Interpretation This lemma provides us with a necessary and su cient condition for the existence of an arbitrage opportunity. The advantage of this result is that, from now on, we don t have to verify any more whether the strategy that is a candidate for an arbitrage opportunity is admissible.

58 Interpretation II Interpretation We also want to emphasize a comment made on page 229 in the article : a close look at the proof of this lemma reveals that the de nition of admissibility can be relaxed so that self- nancing strategies for which the value may turn negative become admissible, provided that the value at maturity, i.e. at time t = T, be non-negative.

59 Interpretation Interpretation III Based on such a new de nition of admissibility, we can rede ne the arbitrage opportunity as being a self- nancing trading strategy the value of which at time t = 0 is nil, V 0 = 0, the value at maturity is non-negative, V T 0, and the expected value, again at maturity, is positive, E P h V T i > 0. Under these new de nitions of admissibility and arbitrage, remains valid.

60 I Interpretation If 8t 2 f0, 1,..., T g, V t 0, then the strategy! φ is admissible the hypotheses of the lemma are such that it is itself an arbitrage opportunity.

61 II Interpretation So let s assume there exists a t 2 f0, 1,..., T g and a ω 2 Ω for which Let a = V t, ω < 0. (4) min t2f0,1,...,t g ω2ω V t, ω, (5) be the smallest value that can ever be reached by the value process of strategy! φ (note that due to inequality (4), a < 0);

62 III Interpretation The rst time when the value process of strategy! φ, V, may ever reach that quantity a is n o t 0 = min t 2 f0, 1,..., T g 9ω 2 Ω such that V t, ω = a, n o A = ω 2 Ω. V t0, ω = a Note that 8ω 2 Ω, P (ω) > 0 implies that P (A) > 0.

63 IV We build, from strategy! φ, a strategy! ψ which is, for its part, admissible: Interpretation 8 >< ψ k t (ω) = >: 0 if ω 2 A c 0 if ω 2 A and t 2 f0,..., t 0 g φ 0 t (ω) if ω 2 A, t 2 ft 0 + 1,,..., T g and k = 0 a S 0 t 0 (ω) φ k t (ω) if ω 2 A, t 2 ft 0 + 1,,..., T g and k 2 f1,..., K g. We probability 1 gain, no loss! P (A), we are holding a portfolio empty at all times: no But, with probability P (A), the trading strategy! ψ consists in holding nothing until time t 0, then in acquiring portfolio! φ t0 +1 plus some quantity of bonds (riskless security). It is due to the latter that the value of the strategy! ψ is non-negative at all times. Recall that, on the set A, V t0, ω = a < 0, which implies that, to acquire the portfolio! φ t0 +1 (ω), we will receive an amount amount we invest into buying bonds. a. That is the

64 V Interpretation To ensure that! ψ is an admissible strategy, we must verify that : (i)! ψ is predictable, (ii)! ψ is self- nancing, (iii) 8t 2 f0, 1,..., T g and 8ω 2 Ω, V t!ψ, ω 0.

65 Point (i) I Interpretation By construction,! ψ is predictable and we leave it to the reader to convince himself that this is so.

66 Point (ii) I Interpretation We want to prove that strategy! ψ is self- nancing, i.e. 8t 2 f1,..., T 1g,!! ψ t S t =!! ψ t+1 S t. If ω 2 A c, then 8t 2 f1,..., T 1g,! ψ t (ω)! S t (ω) = 0 =! ψ t+1 (ω)! S t (ω). So let s assume that ω 2 A. 8t 2 f1,..., t 0 1g,! ψ t (ω)! S t (ω) = 0 =! ψ t+1 (ω)! S t (ω).

67 Interpretation Point (ii) II Since! φ is self- nancing, then 8t 2 ft 0,..., T 1g, = =! ψ t+1 (ω)! S t (ω) K k=0 ψ k t+1 (ω) S k t (ω) φ 0 t+1 (ω) a St 0 0 (ω) S 0 t (ω) + = a S t 0 K (ω) St 0 0 (ω) + φ k t+1 (ω) S t k (ω) k=0 = a S t 0 K (ω) St 0 0 (ω) + φ k t (ω) S t k (ω). k=0 K k=1 φ k t+1 (ω) S k t (ω)

68 Interpretation So if t = t 0,! ψ t0 +1 (ω)! S t0 (ω) Point (ii) III = a S t 0 0 (ω) K St 0 0 (ω) + φ k t 0 (ω) St k 0 (ω) k=0 {z } = 0 =! ψ t0 (ω)! S t0 (ω) =V t0 (! φ,ω)=a

69 Interpretation and if t > t 0,! ψ t+1 (ω)! S t (ω) = a S t 0 K (ω) St 0 0 (ω) + φ k t (ω) S t k (ω) k=0 = φ 0 t (ω) St 0 (ω) + a S 0 t 0 (ω) =! ψ t (ω)! S t (ω). Point (ii) IV K k=1 φ k t (ω) S k t (ω) So, as we had announced, strategy! ψ is self- nancing.

70 Interpretation Point (iii) I The goal, this time, is to prove that for all time t 2 f0, 1,..., T g and for all ω 2 Ω, V t!ψ, ω 0. If ω 2 A c, then V 0!ψ, ω 8t 2 f1,..., T g, V t!ψ, ω So, let s assume ω 2 A. V 0!ψ, ω 8t 2 f1,..., t 0 g, V t!ψ, ω =! ψ 1 (ω)! S 0 (ω) = 0 =! ψ t (ω)! S t (ω) = 0. =! ψ 1 (ω)! S 0 (ω) = 0 =! ψ t (ω)! S t (ω) = 0 Obviously, if t 0 = 0, the line above does not apply.

71 Interpretation Point (iii) II 8ω 2 A and 8t 2 ft 0 + 1,..., T g, V t!ψ, ω =! ψ t (ω)! S t (ω)! = φ 0 t (ω) a St 0 0 (ω) S 0 t (ω) + K k=1 φ k t (ω) S k t (ω) = a S t 0 K (ω) St 0 0 (ω) + φ k t (ω) S t k (ω) k=0 = a S t 0 (ω) St 0 0 (ω) + V t, ω {z } (ref : eq (5)). a from the choice of a! St 0 (ω) a St (ω) {z } 0 0 since the 0 th security is the riskless investment, therefore 8t 2 f0,..., T 1g and 8ω 2 Ω, St+1 0 (ω) S t 0 (ω).

72 Point (iii) III Interpretation According to the arbitrage de nition provided on page 226!ψ in the article, we must show that V 0 = 0 and that h!ψ i E P V T > 0. But, as it has been previously established, V 0!ψ =! ψ 1! S 0 = 0

73 Interpretation and E P h V T!ψ i!ψ = V T, ω P (ω) ω2ω!ψ = V T, ω P (ω) + ω2a = > 0. 0 V T, ω ω2a {z } 0 (by hypothesis) a {z} >0 ω2a ST 0 (ω) St 0 P (ω) 0 (ω) {z } {z } >0 >0 Point (iii) IV!ψ V T, ω P (ω) ω2a c {z } =0 1 a S T 0 (ω) St 0 C 0 (ω) A P (ω) The proof is therefore complete.

74 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) Theorem. The model contains no arbitrage opportunities if and only if there exists at least one martingale measure equivalent to P.

75 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) The proof is broken down into two parts: 1 To be shown. if there exists at least one martingale measure equivalent to P, then the model contains no arbitrage opportunities. 2 To be shown. If the model contains no arbitrage opportunities, then there exists at least one martingale measure equivalent to P.

76 First part I Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) To be shown. If there exists at least one martingale measure equivalent to P, then the model contains no arbitrage opportunities.

77 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) First part II In order to show that the model contains no arbitrage opportunities, we must ensure that all admissible strategy! φ 2 Φ satisfying V0 = 0 also veri es the equality h i E P V T = 0 (ref.: de nition of arbitrage, page 226). Let! φ 2 Φ, be an admissible strategy such that V 0 = 0. Wehwant to show that, in such as case, i E P V T = 0.

78 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) Indeed, First part III We claim that it is su cient to show that E Q h β T V T i = 0 since h i h i E P V T = 0, E Q β T V T = 0. 0 = E P h V T i = ω2ω, 8ω 2 Ω, V T, ω = 0, E Q h β T V T i = ω2ω V T, ω P (ω) {z } {z } >0 0 β T (ω) {z } >0 V T, ω {z } 0 Q (ω) = 0. {z } >0

79 First part IV Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) On the other hand, using Proposal 2.6, we know there exists a price system π consistent with the model: π (X ) = E Q [β T X ]. Thus h i E Q β T V T = π V T = V 0 = 0. (6) where the second equality is warranted by the consistency of the price system. The last equality, that derives from the choice of the strategy! φ, completes the proof.

80 Second part of Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) To be shown. If the model contains no arbitrage opportunities, then there exists at least one martingale measure equivalent to P.

81 Remarks about random variables I of Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) Since Card (Ω) <, we can represent the random variables as points in the space R Card (Ω), i.e. if Y is a (Ω, F) random variable and y i = Y (ω i ), i 2 f1,..., Card (Ω)g then Y (ω 1 ),...Y ω Card (Ω) = y 1,..., y Card (Ω) 2 R Card (Ω).

82 Remarks about random variables II of Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) An alternative representation of the random variable Y will be most useful throughout the proof: since we can write 8ω 2 Ω, Y (ω) = Y = Card (Ω) y i I ωi (ω) i=1 Card (Ω) y i I ωi. i=1

83 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) Let and de nitions I of 1 Y = the set of random variables built on (Ω, F) ; 2 the set of random variables on (Ω, F) (not necessarily with non-negative values) such that there exists a self- nancing strategy! φ (not necessarily admissible) for which V T = X and V 0 = 0 : ( X 0 = X 2 Y 9! φ self- nancing such that V T = X and V 0 = 0 3 and the set of contingent claims with positive values the expectation of which, under the measure P, is strictly positive n o X + = X 2 X 8ω 2 Ω, X (ω) 0 and E P [X ] > 0. ),

84 Idea of the proof I of theorem 2.7 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) The proof of this theorem is based on a corollary of the Hahn-Banach Theorem, in functional analysis, which is called the separating hyperplane theorem. We will state that corollary, and using a simple example, illustrate its purpose. We will establish the links between such a result and the topic of interest in order to nally complete the proof.

85 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) The separating hyperplane I Idea of the proof of Theorem Separating hyperplane theorem. Let V, a topological vector space, C V, a non-empty convex cone S V, a vector subspace such that C \ S = 0. Then there exists a continuous linear form L on V such that L (v) = 0 8v 2 S and L (v) > 0 8v 2 C, v 6= 0. De nition V is a vector space if, among other things, 8a, b 2 R and 8v 1, v 2 2 V, av 1 + bv 2 2 V (see the appendix for the other characteristics). For example, the space R n is a vector space.

86 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) The separating hyperplane II Idea of the proof of De nition A subset C of V is a cone if for all point v 2 C, cv 2 C for all c 2 [0, ). De nition A subset A of a vector space is said to be convex if, for any two points v 1, v 2 2 A, every point αv 1 + (1 α) v 2, 0 < α < 1, also belongs to the set A. De nition S is a vector subspace of V if S is a vector space included in V.

87 The separating hyperplane III Idea of the proof of Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) De nition L : V! R is a linear form on V if 8a, b 2 R and 8v 1, v 2 2 V, L (av 1 + bv 2 ) = al (v 1 ) + bl (v 2 ).

88 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) The separating hyperplane IV Idea of the proof of We will be able to show that the set Y of the random variables built on the measurable space (Ω, F) is a vector space. We will prove that the set X 0 is a vector subspace of Y, whereas X = X + [ f0g is a convex cone in the same space. We will use the hypothesis of absence of arbitrage to show that X 0 \ X + =?. We will then be able to apply the Hahn-Banach theorem, hence the existence of a linear form L on the space Y such that L (X ) = 0 8X 2 X 0 and L (X ) > 0 8X 2 X +. (7)

89 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) The separating hyperplane V Idea of the proof of Let s de ne for all contingent claim X 2 X, π (X ) = L (X ) L (ST 0 (8) ). We will show that the application π : X!R, so conveniently chosen, is a price system consistent with the model. Using Proposal 2.6, we will then be able to build a martingale measure Q equivalent to P from the price system.

90 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) We must show that Structure of the proof I of (i) the set Y of the random variables built on (Ω, F) is a vector space, (ii) X 0 is a vector subspace of Y, (iii) X = X + [ f0g is a convex cone of Y, (iv) X 0 \ X + =?, (v) π, de ned on line (8), is a price system consistent with the model. We will prove the points (iv) and (v) before the rst three ones.

91 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) We want to show that X 0 \ X + =?. Point (iv) I of Let s assume there exists a random variable X 2 X 0 \ X + and let s show that, in such as case, there exist arbitrage opportunities, which directly contradicts our basic hypothesis. If X 2 X 0, then there exists a self- nancing strategy! φ (not necessarily admissible) for which V T = X and V 0 = 0. On the other hand, if X 2 X +, then X is a contingent claim (X 0) such that E P [X ] > 0.

92 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) So, if X 2 X 0 \ X +, then V 0 V T E P h V T i Point (iv) II of = 0; = X 0 and = E P [X ] > 0. From the lemma on page 228, we know that, in such a case, it is possible to build an arbitrage opportunity. Since, by hypothesis, the model contains no arbitrage opportunities, then the set X 0 \ X + must be empty.

93 Point (v) I of Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) We want to show that π (X ) = L (X ) /L S 0 T is a price system consistent with our model. Since S 0 T is a random variable strictly positive, S 0 T 2 X+ then from choosing the linear form (7), L S 0 T > 0. It implies that, when de ning π, on line (8), we did not make the silly mistake of dividing by zero. Moreover, 8ω 2 Ω, I ω 2 X + implies that L (I ω ) > 0.

94 Point (v) II of Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) Theorem π is a price system.

95 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii). Point (v) III of First, we will show that 8X 2 X, π (X ) 0. Card (Ω) Indeed, X 2 X ) X = i=1 x i I ωi where 8i 2 f1,..., Card (Ω)g, x i = X (ω i ) 0. As a consequence, π (X ) = L (X ) L (ST 0 from the very de nition of π. ) = L Card (Ω) i=1 x i I ωi L (ST 0 ) = 1 L (S 0 T ) {z } >0 Card (Ω) i=1 x i {z} 0 L (I ωi ) {z } >0 0.

96 Point (v) IV of Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) Indeed, Secondly, π (X ) = 0, X = 0. 0 = π (X ) = 1 L (ST 0 {z ) } >0, 8i 2 f1,..., Card (Ω)g, x i = 0, X = 0. Card (Ω) i=1 x i {z} 0 L (I ωi ) {z } >0

97 Point (v) V of Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) Thirdly, since L is a linear form, we have that 8a, b 0 and 8X 1, X 2 2 X, π (ax 1 + bx 2 ) = L (ax 1 + bx 2 ) L (S 0 T ) = al (X 1) + bl (X 2 ) L (S 0 T ) = aπ (X 1 ) + bπ (X 2 ).

98 Point (v) VI of Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) We now need to ensure that our price system is consistent with the model, i.e. for all admissible strategy! φ 2 Φ, we must establish the equality π V T = V 0.

99 Point (v) VII of Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) Let! φ 2 Φ be any admissible strategy. We build a new trading strategy! ψ from the! φ : ψ k t = ( φ 0 t φ k t V 0 if k = 0 if k 2 f1,...k g i.e. at time t = 0, we borrow an amount of V 0 by selling short V 0 shares of the riskless security in order to acquire the portfolio! φ 0.

100 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) Point (v) VIII of Since! φ is self- nancing, then! ψ is also self- nancing. Indeed, 8t 2 f1,..., T 1g! ψ t! S t = = = = φ 0 t V 0 St 0 + K φ k t S t k V 0 St 0 k=0 K k=1 K φ k t+1 S t k V 0 St 0 k=0 φ 0 t+1 V 0 =! ψ t+1! S t. S 0 t + φ k t S k t K k=1 φ k t+1 S k t

101 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) Point (v) IX of However, it is possible that! ψ is not admissible. We now want to show that π is consistent with the model. Since V 0!ψ =! ψ 1! S 0 = then V T!ψ 2 X 0. φ 0 1 V 0 S0 0 + K k=1 K = φ k 1 S 0 k V 0 S0 0 k=0 = V 0 V 0 S0 0 = 0, φ k 1 S k 0

102 Point (v) X of Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) As a consequence, since L (X ) = 0 8X 2 X 0, π V T!ψ = L V T!ψ L (S 0 T ) = 0 L (S 0 T ) = 0.

103 Point (v) XI of Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) On the other hand, since V T!ψ =! ψ T! S T = φ 0 T V 0 ST 0 + K = φ k T S T k V 0 ST 0 k=0 = V T V 0 ST 0 K k=1 φ k T S k T

104 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) then hence!ψ 0 = π V T = π V T = π V T = π V T Point (v) XII of V 0 ST 0 V 0 π ST 0 {z } = L(S0 T ) L(S T 0 ) =1 V 0 π V T = V 0. We have just nished building a price system consistent with the model.

105 Point (i) of Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) Let Y be the set of all random variables built on the probability space (Ω, F, P). We want to prove that Y is a vector space. Y is a vector space In order to show that Y is a vector space, we must verify that 8a, b 2 R and 8Y 1, Y 2 2 Y, ay 1 + by 2 2 Y. This is indeed the case, since all nite linear combination of (Ω, F) random variables is a (Ω, F) random variable. We must also verify the other conditions stated in the appendix.

106 Point (ii) I of Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) X 0 is a vector subspace of Y where ( X 0 = X 2 Y We must show that 9! φ self- nancing satisfying V T = X and V 0 = 0 8a, b 2 R and 8X 1, X 2 2 X 0, ax 1 + bx 2 2 X 0. )

107 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) Point (ii) II of Let X 1 and X 2, be two elements of X 0 arbitrarily chosen. Since X 1 2 X 0, then there exists a self- nancing strategy! φ satisfying V T = X 1 and V 0 = 0. Similarly, we know there exists a self- nancing strategy! ϕ such that V T! ϕ = X2 and V 0! ϕ = 0. Let a and b, be any two real numbers. Since a! φ + b! ϕ is also a self- nancing strategy satisfying V T a! φ + b! ϕ = av T + bv! T ϕ = ax1 + bx 2 and V 0 a! φ + b! ϕ = av 0 + bv 0! ϕ = 0, then ax 1 + bx 2 belongs to the set X 0. Again we must verify the other conditions stated in the appendix.

108 Part 1 Part 2 Separating hyperplane Structure of the proof Point (iv) Point (v) Points (i), (ii) and (iii) X = X + [ f0g is a cone de Y. Point (iii) of. Let any X 2 X. Since, for any real number c 0, cx is a non-negative random variable, then cx is a contingent claim, which implies that cx 2 X. X + is convex.. Let 0 < α < 1 and X 1, X 2 2 X. 8ω 2 Ω, αx 1 (ω) + (1 α) X 2 (ω) 0, hence αx 1 + (1 α) X 2 2 X. The proof of is complete.

109 Corollary p 228 Interpretation Corollary Corollary on page 228. If the model contains no arbitrage, then there is a single price associated with any attainable contingent claim X and it satis es π = E Q [β T X ] for each martingale measure Q 2 P.

110 Interpretation I Corollary p 228 Interpretation Proposal 2.6 associates a price system to each of the martingale measures. But it is also possible that there exists an in nity of such measures. So, it is also possible that there is an in nity of price systems. The corollary states that, if X is any attainable contingent claim, and π and eπ are two di erent prices systems, then π (X ) = eπ (X ), i.e. the price of the attainable contingent claim is the same, whatever price system is used to price it.

111 I Corollary p 228 Interpretation Let X be any attainable contingent claim. Since it is attainable, there exists at least one admissible strategy! ϕ such that V T! ϕ = X. Let! ϕ and! φ 2 Φ be two admissible strategies such that V T! ϕ = X and VT = X. (9) Note that it is possible that! ϕ =! φ, in particular if there is only a single admissible strategy allowing to attain X.

112 II Corollary p 228 Interpretation Since our model contains no arbitrage opportunities (by hypothesis), then there exists at least one price system consistent with the model (by followed by Proposal 2.6). If there only exists one such system, then there will only be a single price associated with the attainable contingent claim X.

113 Interpretation III Corollary p 228 We want to verify that, if there exist several price systems consistent with the model, then there will always only be a single price associated with X, i.e. if π and eπ are two di erent price systems, then π (X ) = eπ (X ). (10) From De nition (2.5a), page 226, of price systems, we have that X = 0, π (X ) = 0 = eπ (X ). As a consequence, the equation (10) is trivially satis ed when X = 0. So let s assume there exists an ω 2 Ω for which X (ω ) > 0.

114 Interpretation IV Corollary p 228 Let π, eπ 2 Π, π 6= eπ be two price systems consistent with the model. Since they are consistent, we have 8! φ 2 Φ, π V T = V 0 and eπ V T = V 0. (11) In particular, because of equalities (9), we can write π (X ) = π V! T ϕ = V0! ϕ =!! ϕ 1 S 0 (12) eπ (X ) = eπ V T = V 0 =!! φ 1 S 0 We want to show that π (X ) = eπ (X ). Let s assume the contrary and let s show that, in such a case, the contains arbitrage opportunities, which contradicts the premise of the corollary.

115 Interpretation V Corollary p 228 So, let s assume that π (X ) 6= eπ (X ) (since π (X ) and eπ (X ) are real numbers, we can, without loss of generality, assume that π (X ) < eπ (X )). We are going to build ourselves a strategy that generates arbitrage opportunities. To begin with, note that, due to the inequality π (X ) < eπ (X ) the two strategies chosen on line (9) are di erent. Indeed, from the equations on line (12), 0 < eπ (X ) π (X ) =!! φ 1 S!! 0 ϕ 1 S 0 =!!S 1 ϕ 1 0 hence! φ 1! ϕ 1 6=! 0.

116 VI Corollary p 228 Interpretation Let s build a new strategy! ψ consisting of a fortunate mix of the rst two:! ψ =! π (X )! ϕ φ. eπ (X ) Recall that X 6= 0 implies, from Property (2.5a), p. 226, that eπ (X ) > 0.

117 VII Corollary p 228 Interpretation We must verify that! ψ is self- nancing, i.e. 8t 2 f0,..., T 1g,! ψ t! S t =! ψ t+1! S t then we will show that! ψ allows us to create an arbitrage opportunity.

118 Interpretation VIII Corollary p 228 Since strategies! ϕ and! φ are self- nancing, then 8t 2 f0,..., T 1g!! ψ t S t =!ϕ π (X )!!S t φ t t eπ (X ) =!! π (X )!! ϕ t S t φ t S t eπ (X ) =!! π (X )!! ϕ t+1 S t φ t+1 S t eπ (X ) =!ϕ π (X )!!S t+1 φ t+1 t eπ (X ) =! ψ t+1! S t thus proving that! ψ is also self- nancing.

119 IX Corollary p 228 Interpretation In order to show that! ψ allows us to create an arbitrage opportunity, we must verify that: (i) V 0!ψ = 0, (ii) V T!ψ 0, (iii) E P h V T!ψ i > 0. Indeed, using the lemma on page 228, we can conclude from (i), (ii) and (iii) that there exists an arbitrage opportunity. Note that we don t need to verify that strategy! ψ is admissible.

120 Interpretation - Point (i) Corollary p 228 Using the relations established on line number (12),!ψ V 0 =!! ψ 1 S 0 =!ϕ 1 π (X )!!S φ 1 0 eπ (X ) =!! π (X )!! ϕ 1 S 0 φ 1 S 0 eπ (X ) = π (X ) π (X ) eπ (X ) eπ (X ) = 0.

121 Interpretation - Point (ii) Corollary p 228 From the choice of strategies! ϕ and! φ (ref.: line (9)), V T!ψ =! ψ T! S T =!ϕ π (X )!!S T φ T T eπ (X ) =!! π (X )!! ϕ T S T φ T S T eπ (X ) = V! π (X ) T ϕ eπ (X ) V T π (X ) = X eπ (X ) X π (X ) = 1 X 0. eπ (X ) {z } >0 since π(x )<eπ(x )

122 Interpretation - Point (iii) I Corollary p 228 Since X is such that X (ω ) > 0 for some ω 2 Ω, E P h V T!ψ i = E 1 P π (X ) X eπ (X ) π (X ) = 1 X (ω) P (ω) eπ ω2ω (X ) {z } {z } {z } 1 > 0 >0 0 >0 π (X ) X (ω ) P (ω ) eπ (X ) The proof of the corollary is thus completed.

123 Proposal 2.8 Theorem Proposal 2.8. If! φ is an admissible strategy, then the process βv, representing its discounted value, is a Q martingale for each measure Q 2 P.

124 I Proposal 2.8 Let! φ 2 Φ be any admissible strategy and Q 2 P, a martingale measure arbitrarily chosen. In order to show that the adapted process βv is a Q martingale, it is su cient to verify that 8t 2 f1,..., T g, E Q h β t V t β t 1 V t 1 jf t 1 i = 0.

125 II Proposal 2.8 Since 8!! >< β 0 φ 1 S 0 if t = 0 β t V t = >:!! β t φ t S t if t 2 f1,..., T g then, by setting! φ 0 =! φ 1 we have 8t 2 f0,..., T g, β t V t = β t! φ t! S t

126 III Proposal 2.8 As a consequence, 8t 2 f1,..., T g, h i E Q β t V t β t 1 V t 1 jf t 1 h = E Q!!!! i β t φ t S t β t 1 φ t 1 S t 1 jf t 1 = E Q h β t! φ t! S t β t 1! φ t! S t 1 jf t 1 i car! φ is self- nancing =! φ t E Q h β t! S t β t 1! S t 1 jf t 1 i {z } =! 0 since the components of β! S are Q since! φ is predictable = 0. martingales

127 Proposal 2.9 Interpretation Theorem Proposal 2.9. If X 2 X is an attainable contingent claim, then β t V t = E Q [β T X jf t ] for any trading strategy! φ which generates X and for each measure Q 2 P.

128 Interpretation Interpretation I Proposal 2.9 If the contingent claim X can be replicated with strategy! φ, VT = X, then the value of such a contingent claim must be, at any time, equal to the value of the strategy, otherwise, there is a way to create an arbitrage opportunity. Indeed, if at time t, the value of the contingent claim X, let s denote it X t, is less than V t, then let s sell short portfolio! φ t, and, with part of the proceeds from such a sale, let s purchase for an amount X t the contingent claim X. With the other part of the proceeds, V t X t, let s purchase V t X t /St 0 shares of the riskless security. Our initial investment is nil. On the time horizon t = T, we use the contingent claim to pay back the portfolio that had been sold short, since, at that time, their respective values are equal. We now only have to sell the shares of the riskless security to make a pro t of Vt X t ST 0. S 0 t