4: SINGLE-PERIOD MARKET MODELS

4: SINGLE-PERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 1 / 87

General Single-Period Market Models The main differences between the elementary and general single period market models are: The investor is allowed to invest in several risky securities instead of only one. The sample set is bigger, that is, there are more possible states of the world at time t = 1. The sample space is Ω = {ω 1,ω 2,...,ω k } with F = 2 Ω. An investor s personal beliefs about the future behaviour of stock prices are represented by the probability measure P(ω i ) = p i > 0 for i = 1,2,...,k. The savings account B equals B 0 = 1 and B 1 = 1+r for some constant r > 1. The price of the jth stock at t = 1 is a random variable on Ω. It is denoted by S j t for t = 0,1 and j = 1,...,n. A contingent claim X = (X(ω 1 ),...,X(ω k )) is a random variable on the probability space (Ω,F,P). B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 2 / 87

Questions 1 Under which conditions a general single-period market model M = (B,S 1,...,S n ) is arbitrage-free? 2 How to define a risk-neutral probability measure for a model? 3 How to use a risk-neutral probability measure to analyse a general single-period market model? 4 Under which conditions a general single-period market model is complete? 5 Is completeness of a market model related to risk-neutral probability measures? 6 How to define an arbitrage price of an attainable claim? 7 Can we still apply the risk-neutral valuation formula to compute the price of an attainable claim? 8 How to deal with contingent claims that are not attainable? 9 How to use the class of risk-neutral probability measures to value non-attainable claims? B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 3 / 87

Outline We will examine the following issues: 1 Trading Strategies and Arbitrage-Free Models 2 Fundamental Theorem of Asset Pricing 3 Examples of Market Models 4 Risk-Neutral Valuation of Contingent Claims 5 Completeness of Market Models B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 4 / 87

PART 1 TRADING STRATEGIES AND ARBITRAGE-FREE MODELS B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 5 / 87

Trading Strategy Definition (Trading Strategy) A trading strategy (or a portfolio) in a general single-period market model is defined as the vector (x,φ 1,...,φ n ) R n+1 where x is the initial wealth of an investor and φ j stands for the number of shares of the jth stock purchased at time t = 0. If an investor adopts the trading strategy (x,φ 1,...,φ n ) at time t = 0 then the cash value of his portfolio at time t = 1 equals V 1 (x,φ 1,...,φ n ) := ( x n ) φ j S j 0 (1+r)+ j=1 n φ j S j 1. j=1 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 6 / 87

Wealth Process of a Trading Strategy Definition (Wealth Process) The wealth process (or the value process) of a trading strategy (x,φ 1,...,φ n ) is the pair (V 0 (x,φ 1,...,φ n ),V 1 (x,φ 1,...,φ n )). The real number V 0 (x,φ 1,...,φ n ) is the initial endowment V 0 (x,φ 1,...,φ n ) := x and the real-valued random variable V 1 (x,φ 1,...,φ n ) represents the cash value of the portfolio at time t = 1 V 1 (x,φ 1,...,φ n ) := ( x n ) φ j S j 0 (1+r)+ j=1 n φ j S j 1. j=1 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 7 / 87

Gains (Profits and Losses) Process The profits or losses an investor obtains from the investment can be calculated by subtracting V 0 ( ) from V 1 ( ). This is called the (undiscounted) gains process. The gain can be negative; hence it may also represent a loss. Definition (Gains Process) The gains process is defined as G 0 (x,φ 1,...,φ n ) = 0 and G 1 (x,φ 1,...,φ n ) := V 1 (x,φ 1,...,φ n ) V 0 (x,φ 1,...,φ n ) ( n ) n = x φ j S j 0 r + φ j S j 1 j=1 j=1 where the random variable S j 1 = Sj 1 Sj 0 in the price of the jth stock. represents the nominal change B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 8 / 87

Discounted Stock Price and Value Process To understand whether the jth stock appreciates in real terms, we consider the discounted stock prices of the jth stock Ŝ j 0 := Sj 0 = Sj 0 B 0, Ŝ j 1 := Sj 1 1+r = Sj 1 B 1. Similarly, we define the discounted wealth process as V 0 (x,φ 1,...,φ n ) := x, V1 (x,φ 1,...,φ n ) := V 1(x,φ 1,...,φ n ) B 1. It is easy to see that V 1 (x,φ 1,...,φ n ) = ( x ) n j=1 φj S j 0 + n j=1 φj Ŝ j 1 = x+ n j=1 φj (Ŝj 1 Ŝj 0 ). B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 9 / 87

Discounted Gains Process Definition (Discounted Gains Process) The discounted gains process for the investor is defined as Ĝ 0 (x,φ 1,...,φ n ) = 0 and Ĝ 1 (x,φ 1,...,φ n ) := V 1 (x,φ 1,...,φ n ) V 0 (x,φ 1,...,φ n ) n = φ j Ŝj 1 j=1 where Ŝj 1 = Ŝj 1 Ŝj 0 stock. is the change in the discounted price of the jth B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 10 / 87

Arbitrage: Definition The concept of an arbitrage in a general single-period market model is essentially the same as in the elementary market model. It is worth noting that the real-world probability P can be replaced here by any equivalent probability measure Q. Definition (Arbitrage) A trading strategy (x,φ 1,...,φ n ) in a general single-period market model is called an arbitrage opportunity if A.1. V 0 (x,φ 1,...,φ n ) = 0, A.2. V 1 (x,φ 1,...,φ n )(ω i ) 0 for i = 1,2,...,k, A.3. E P { V1 (x,φ 1,...,φ n ) } > 0, that is, k V 1 (x,φ 1,...,φ n )(ω i )P(ω i ) > 0. i=1 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 11 / 87

Arbitrage: Equivalent Conditions The following condition is equivalent to A.3. A.3. There exists ω Ω such that V 1 (x,φ 1,...,φ n )(ω) > 0. The definition of arbitrage can be formulated using the discounted value and gains processes. This is sometimes very helpful. Proposition (4.1) A trading strategy (x,φ 1,...,φ n ) in a general single-period market model is an arbitrage opportunity if and only if one of the following conditions holds: 1 Assumptions A.1-A.3 in the definition of arbitrage hold with V(x,φ 1,...,φ n ) instead of V(x,φ 1,...,φ n ). 2 x = 0 and A.2-A.3 in the definition of arbitrage are satisfied with Ĝ 1 (x,φ 1,...,φ n ) instead of V 1 (x,φ 1,...,φ n ). B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 12 / 87

Proof of Proposition 4.1 Proof of Proposition 4.1: First step. We will show that the following two statements are true: The definition of arbitrage and condition 1 in Proposition 4.1 are equivalent. In Proposition 4.1, condition 1 is equivalent to condition 2. To prove the first statement, we use the relationships between V(x,φ 1,...,φ n ) and V(x,φ 1,...,φ n ), that is, V 0 (x,φ 1,...,φ n ) = V 0 (x,φ 1,...,φ n ) = x, V 1 (x,φ 1,...,φ n 1 ) = 1+r V 1(x,φ 1,...,φ n ), E P { V1 (x,φ 1,...,φ n ) } 1 = 1+r E { P V1 (x,φ 1,...,φ n ) }. This shows that the first statement holds. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 13 / 87

Proof of Proposition 4.1 Proof of Proposition 4.1: Second step. To prove the second statement, we recall the relation between V(x,φ 1,...,φ n ) and Ĝ1(x,φ 1,...,φ n ) Ĝ 1 (x,φ 1,...,φ n ) = V 1 (x,φ 1,...,φ n ) V 0 (x,φ 1,...,φ n ) = V 1 (x,φ 1,...,φ n ) x. It is now clear that for x = 0 we have Ĝ 1 (x,φ 1,...,φ n ) = V 1 (x,φ 1,...,φ n ). Hence the second statement is true as well. In fact, one can also observe that Ĝ1(x,φ 1,...,φ n ) does not depend on x at all, since Ĝ1(x,φ 1,...,φ n ) = n j=1 φj Ŝj 1. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 14 / 87

Verification of the Arbitrage-Free Property It can be sometimes hard to check directly whether arbitrage opportunities exist in a given market model, especially when dealing with several risky assets or in the multi-period setup. We have introduced the risk-neutral probability measure in the elementary market model and we noticed that it can be used to compute the arbitrage price of any contingent claim. We will show that the concept of a risk-neutral probability measure is also a convenient tool for checking whether a general single-period market model is arbitrage-free or not. In addition, we will argue that a risk-neutral probability measure can also be used for the purpose of valuation of a contingent claim (either attainable or not). B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 15 / 87

Risk-Neutral Probability Measure Definition (Risk-Neutral Probability Measure) A probability measure Q on Ω is called a risk-neutral probability measure for a general single-period market model M if: R.1. Q(ω i ) > 0 for all ω i Ω, R.2. E Q ( Ŝ j 1) = 0 for j = 1,2,...,n. We denote by M the class of all risk-neutral probability measures for the market model M. Condition R.1 means that Q and P are equivalent probability measures. A risk-neutral probability measure is also known as an equivalent martingale measure. (Ŝj Note that condition R.2 is equivalent to E Q 1) = Ŝ j 0 or, more explicitly, ( E Q S j) 1 = (1+r)S j 0. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 16 / 87

Example: Stock Prices Example (4.1) We consider the following model featuring two stocks S 1 and S 2 on the sample space Ω = {ω 1,ω 2,ω 3 }. The interest rate r = 1 10 so that B 0 = 1 and B 1 = 1+ 1 10. We deal here with the market model M = (B,S 1,S 2 ). The stock prices at t = 0 are given by S 1 0 = 2 and S2 0 = 3. The stock prices at t = 1 are represented in the table: ω 1 ω 2 ω 3 S1 1 1 5 3 S1 2 3 1 6 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 17 / 87

Example: Value Process Example (4.1 Continued) For any trading strategy (x,φ 1,φ 2 ) R 3, we have V 1 (x,φ 1,φ 2 ) = ( x 2φ 1 3φ 2)( 1+ 1 ) +φ 1 S1 1 +φ 2 S 2 10 1. We set φ 0 := x 2φ 1 3φ 2. Then V 1 (x,φ 1,φ 2 ) equals ( V 1 (x,φ 1,φ 2 )(ω 1 ) = φ 0 1+ 1 ) +φ 1 +3φ 2, 10 ( V 1 (x,φ 1,φ 2 )(ω 2 ) = φ 0 1+ 1 ) +5φ 1 +φ 2, 10 ( V 1 (x,φ 1,φ 2 )(ω 3 ) = φ 0 1+ 1 ) +3φ 1 +6φ 2. 10 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 18 / 87

Example: Gains Process Example (4.1 Continued) The increments S j 1 are represented by the following table ω 1 ω 2 ω 3 S 1 1-1 3 1 S 2 1 0-2 3 The gains G 1 (x,φ 1,φ 2 ) are thus given by G 1 (x,φ 1,φ 2 )(ω 1 ) = 1 10 φ0 φ 1 +0φ 2, G 1 (x,φ 1,φ 2 )(ω 2 ) = 1 10 φ0 +3φ 1 2φ 2, G 1 (x,φ 1,φ 2 )(ω 3 ) = 1 10 φ0 +φ 1 +3φ 2. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 19 / 87

Example: Discounted Stock Prices Example (4.1 Continued) Out next goal is to compute the discounted wealth process V(x,φ 1,φ 2 ) and the discounted gains process Ĝ1(x,φ 1,φ 2 ). To this end, we first compute the discounted stock prices. Of course, Ŝj 0 = Sj 0 for j = 1,2. The following table represents the discounted stock prices Ŝj 1 for j = 1,2 at time t = 1 Ŝ1 1 Ŝ1 2 ω 1 ω 2 ω 3 10 11 30 11 50 11 10 11 30 11 60 11 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 20 / 87

Example: Discounted Value Process Example (4.1 Continued) The discounted value process V(x,φ 1,φ 2 ) is thus given by V 0 (x,φ 1,φ 2 ) = V 0 (x,φ 1,φ 2 ) = x and V 1 (x,φ 1,φ 2 )(ω 1 ) = φ 0 + 10 11 φ1 + 30 11 φ2, V 1 (x,φ 1,φ 2 )(ω 2 ) = φ 0 + 50 11 φ1 + 10 11 φ2, V 1 (x,φ 1,φ 2 )(ω 3 ) = φ 0 + 30 11 φ1 + 60 11 φ2, where φ 0 = x 2φ 1 3φ 2 is the amount of cash invested in B at time 0 (as opposed to the initial wealth given by x). B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 21 / 87

Example: Discounted Gains Process Example (4.1 Continued) The increments of the discounted stock prices equal Ŝ1 1 12 11 ω 1 ω 2 ω 3 28 11 Ŝ2 1 3 11 23 11 8 11 27 11 Hence the discounted gains Ĝ1(x,φ 1,φ 2 ) are given by Ĝ 1 (x,φ 1,φ 2 )(ω 1 ) = 12 11 φ1 3 11 φ2, Ĝ 1 (x,φ 1,φ 2 )(ω 2 ) = 28 11 φ1 23 11 φ2, Ĝ 1 (x,φ 1,φ 2 )(ω 3 ) = 8 11 φ1 + 27 11 φ2. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 22 / 87

Example: Arbitrage-Free Property Example (4.1 Continued) The condition Ĝ1(x,φ 1,φ 2 ) 0 is equivalent to 12φ 1 3φ 2 0 28φ 1 23φ 2 0 8φ 1 +27φ 2 0 Can we find (φ 1,φ 2 ) R 2 such that all inequalities are valid and at least one of them is strict? It appears that the answer is negative, since the unique vector satisfying all inequalities above is (φ 1,φ 2 ) = (0,0). Therefore, the single-period market model M = (B,S 1,S 2 ) is arbitrage-free. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 23 / 87

Example: Risk-Neutral Probability Measure Example (4.1 Continued) We will now show that this market model admits a unique risk-neutral probability measure on Ω = {ω 1,ω 2,ω 3 }. Let us denote q i = Q(ω i ) for i = 1,2,3. From the definition of a risk-neutral probability measure, we obtain the following linear system 12 11 q 1 + 28 11 q 2 + 8 11 q 3 = 0 3 11 q 1 23 11 q 2 + 27 11 q 3 = 0 q 1 +q 2 +q 3 = 1 The unique solution equals Q = (q 1,q 2,q 3 ) = ( 47 80, 15 80, 18 80). B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 24 / 87

PART 2 FUNDAMENTAL THEOREM OF ASSET PRICING B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 25 / 87

Fundamental Theorem of Asset Pricing (FTAP) In Example 4.1, we have checked directly that the market model M = (B,S 1,S 2 ) is arbitrage-free. In addition, we have shown that the unique risk-neutral probability measure exists in this model. Is there any relation between no arbitrage property of a market model and the existence of a risk-neutral probability measure? The following important result, known as the FTAP, gives a complete answer to this question within the present setup. The FTAP was first established by Harrison and Pliska (1981) and it was later extended to continuous-time market models. Theorem (FTAP) A general single-period model M = (B,S 1,...,S n ) is arbitrage-free if and only if there exists a risk-neutral probability measure for M, that is, M. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 26 / 87

Proof of ( ) in FTAP Proof of ( ) in FTAP. ( ) We first prove the if part. We assume that M, so that a risk-neutral probability measure Q exists. Let (0,φ) = (0,φ 1,...,φ n ) be any trading strategy with null initial endowment. Then for any Q M E Q ( V1 (0,φ) ) ( n ) = E Q φ j Ŝj 1 = j=1 n j=1 φ j ( E Q Ŝ j ) 1 = 0. } {{ } =0 If we assume that V 1 (0,φ) 0 then the last equation implies that the equality V 1 (0,φ)(ω) = 0 must hold for all ω Ω. Hence no trading strategy satisfying all conditions of an arbitrage opportunity may exist. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 27 / 87

Geometric Interpretation of X and Q The proof of the implication ( ) in the FTAP needs some preparation, since it is based on geometric arguments. Any random variable on Ω can be identified with a vector in R k, specifically, X = (X(ω 1 ),...,X(ω k )) T = (x 1,...,x k ) T R k. An arbitrary probability measure Q on Ω can also be interpreted as a vector in R k Q = (Q(ω1),...,Q(ωk)) = (q1,...,qk) R k. We note that k k E Q (X) = X(ω i )Q(ω i ) = x i q i = X,Q i=1 i=1 where, denotes the inner product of two vectors in R k. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 28 / 87

Auxiliary Subsets of R k We define the following classes: W = {X R k X = V1 (0,φ 1,...,φ n ) for some φ 1,...,φ n} { } W = Z R k X,Z = 0 for all X W The set W is the image of the map V 1 (0,,..., ) : R n R k. We note that W represents all discounted values at t = 1 of trading strategies with null initial endowment. The set W is the set of all vectors in R k orthogonal to W. We introduce the following sets of k-dimensional vectors: { A = X R k } X 0, xi 0 for i = 1,...,k { P + = Q R k k } q i = 1, q i > 0 i=1 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 29 / 87

W and W as Vector Spaces Corollary The sets W and W are vector (linear) subspaces of R k. Proof. It suffices to observe that the map V 1 (0,,..., ) : R n R k is linear. In other words, for any trading strategies (0,η 1,...,η n ) and (0,κ 1,...,κ n ) and arbitrary real numbers α,β (0,φ 1,...,φ n ) = α(0,η 1,...,η n )+β(0,κ 1,...,κ n ) is also a trading strategy. Hence W is a vector subspace of R k. In particular, the zero vector (0,0,...,0) belongs to W. It us easy to check that W, that is, the orthogonal complement of W is a vector subspace as well. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 30 / 87

Risk-Neutral Probability Measures Lemma (4.1) A single-period market model M = (B,S 1,...,S n ) is arbitrage free if and only if W A =. Proof. The proof hinges on an application of Proposition 4.1. Lemma (4.2) A probability measure Q is a risk-neutral probability measure for a single-period market model M = (B,S 1,...,S n ) if and only if Q W P +. Hence the set M of all risk-neutral probability measures for the model M satisfies M = W P + and thus M W P +. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 31 / 87

Proof of Lemma 4.2 Proof of ( ) in Lemma 4.2. ( ) We assume that Q is a risk-neutral probability measure. By the property R.1, it is obvious that Q belongs to P +. Using the property R.2, we obtain for any vector X = V 1 (0,φ) W X,Q = E Q ( V1 (0,φ) ) ( n ) = E Q φ j Ŝj 1 = n j=1 φ j ( E Q Ŝ j ) 1 = 0. } {{ } =0 We conclude that Q belongs to W as well. j=1 Consequently, Q W P + as was required to show. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 32 / 87

Proof of Lemma 4.2 Proof of ( ) in Lemma 4.2. ( ) We now assume that Q is an arbitrary vector in W P +. Since Q P +, we see that Q defines a probability measure satisfying condition R.1. It remains to show that Q satisfies condition R.2 as well. To this end, for a fixed (but arbitrary) j = 1,2,...,n, we consider the trading strategy (0,φ 1,...,φ n ) with (φ 1,...,φ n ) = (0,...,0,1,0,...,0) = e j. This trading strategy only invests in the savings account and the jth asset. The discounted wealth of this strategy is V 1 (0,e j ) = Ŝj 1. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 33 / 87

Proof of Lemma 4.2 Proof of ( ) in Lemma 4.2 (Continued). Since V 1 (0,e j ) W and Q W we obtain 0 = V 1 (0,e j ),Q = Ŝj 1,Q = E Q( Ŝ1) j. Since j was arbitrary, we see that Q satisfies condition R.2. Hence Q is a risk-neutral probability measure. From Lemmas 4.1 and 4.2, we get the following purely geometric reformulation of the FTAP: W A = W P +. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 34 / 87

Separating Hyperplane Theorem: Statement Theorem (Separating Hyperplane Theorem) Let B,C R k be nonempty, closed, convex sets such that B C =. Assume, in addition, that at least one of these sets is compact (that is, bounded and closed). Then there exist vectors a,y R k such that b a,y < 0 for all b B and c a,y > 0 for all c C. Proof of the Separating Hyperplane Theorem. The proof can be found in any textbook of convex analysis or functional analysis. It is sketched in the course notes. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 35 / 87

Separating Hyperplane Theorem: Interpretation Let the vectors a,y R k be as in the statement of the Separating Hyperplane Theorem It is clear that y R k is never a zero vector. We define the (k 1)-dimensional hyperplane H R k by setting { } H = a+ x R k x,y = 0 = a+{y}. Then we say that the hyperplane H strictly separates the convex sets B and C. Intuitively, the sets B and C lie on different sides of the hyperplane H and thus they can be seen as geometrically separated by H. Note that the compactness of at least one of the sets is a necessary condition for the strict separation of B and C. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 36 / 87

Separating Hyperplane Theorem: Corollary The following corollary is a consequence of the separating hyperplane theorem. It is more suitable for our purposes: it will be later applied to B = W and C = A + := {X A X,P = 1} A. Corollary (4.1) Assume that B R k is a vector subspace and set C is a compact convex set such that B C =. Then there exists a vector y R k such that that is, y B, and b,y = 0 for all b B c,y > 0 for all c C. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 37 / 87

Proof of Corollary 4.1 Proof of Corollary 4.1: First step. We note that any vector subspace of R k is a closed and convex set. From the separating hyperplane theorem, there exist a,y R k such that the inequality b,y < a,y is satisfied for all vectors b B. Since B is a vector subspace, the vector λb belongs to B for any λ R. Hence for any b B and λ R we have λb,y = λ b,y < a,y. This in turn implies that b,y = 0 for any vector b B, meaning that y B. Also, we have that a,y > 0. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 38 / 87

Proof of Corollary 4.1 Proof of Corollary 4.1: Second step. To establish the second inequality, we observe that from the separating hyperplane theorem, we obtain Consequently, for any c C c,y > a,y for all c C. c,y > a,y > 0. We conclude that c,y > 0 for all c C. We now are ready to establish the implication ( ) in the FTAP: W A = W P +. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 39 / 87

Proof of ( ) in FTAP: 1 Proof of ( ) in FTAP: First step. We assume that the model is arbitrage-free. From Lemma 4.1, this is equivalent to the condition W A =. Our goal is to show that the class M is non-empty. In view of Lemma 4.2, it thus suffices to show that W A = W P +. We define an auxiliary set A + = {X A X,P = 1}. Observe that A + is a closed, bounded (hence compact) and convex subset of R k. Since A + A, it is clear that W A = W A + =. Hence in the next step we may assume that W A + =. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 40 / 87

Proof of ( ) in FTAP: 2 Proof of ( ) in FTAP: Second step. By applying Corollary 4.1 to B = W and C = A +, we see that there exists a vector Y W such that X,Y > 0 for all X A +. (1) Our goal is to show that Y can be used to define a risk-neutral probability Q. We need first to show that y i > 0 for every i. For this purpose, for any fixed i = 1,2,...,k, we define X i = (P(ω i )) 1 (0,...,0,1,0...,0) = (P(ω i )) 1 e i so that X i A + since E P (X i ) = X i,p = 1. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 41 / 87

Proof of ( ) in FTAP: 3 Proof of ( ) in FTAP: Third step. Let y i be the ith component of Y. It follows from (1) that 0 < X i,y = (P(ω i )) 1 y i and thus y i > 0 for all i = 1,2,...,k. We set Q(ω i ) = q i where q i := y i y 1 + +y k = cy i > 0 It is clear that Q is a probability measure and Q P +. Since Y W, Q = cy for some scalar c and W is a vector space, we have that Q W. We conclude that Q W P + so that W P +. From Lemma 4.2, Q is a risk-neutral probability and thus M. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 42 / 87

PART 3 EXAMPLES OF MARKET MODELS B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 43 / 87

Example: Arbitrage-Free Market Model Example (4.1 Continued) We consider the market model M = (B,S 1,S 2 ) introduced in Example 4.1. The interest rate r = 1 10 so that B 0 = 1 and B 1 = 1+ 1 10. The stock prices at t = 0 are given by S 1 0 = 2 and S2 0 = 3. We have shown that the increments of the discounted stock prices Ŝ1 and Ŝ2 equal Ŝ1 1 12 11 ω 1 ω 2 ω 3 28 11 Ŝ2 1 3 11 23 11 8 11 27 11 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 44 / 87

Example: Arbitrage-Free Market Model Example (4.1 Continued) The vector spaces W and W are given by W = α +β and 12 28 8 W = γ 47 15 18 3 23 27 γ R. α,β R We first show the model is arbitrage-free using Lemma 4.1. It thus suffices to check that W A = B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 45 / 87

Example: Arbitrage-Free Market Model Example (4.1 Continued) If there exists a vector X W A then the following three inequalities are satisfied by a vector X = (x 1,x 2,x 3 ) R 3 x 1 = x 1 (α,β) = 12α 3β 0 x 2 = x 2 (α,β) = 28α 23β 0 x 3 = x 3 (α,β) = 8α+27β 0 with at least one strict inequality, where α,β R are arbitrary. It can be shown that such a vector X R 3 does not exist and thus W A =. This is left as an easy exercise. In view of Lemma 4.1, we conclude that the market model is arbitrage-free. In the next step, our goal is to show that M is non-empty. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 46 / 87

Example: Arbitrage-Free Market Model Example (4.1 Continued) Lemma 4.2 tells us that M = W P +. If Q W then 47 Q = γ 15 for some γ R. 18 If Q P + then 47γ +15γ +18γ = 1 so that γ = 1 80 > 0. We conclude that the unique risk-neutral probability measure Q is given by Q = 1 80 47 15 18. The FTAP confirms that the market model is arbitrage-free. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 47 / 87

Example: Market Model with Arbitrage Example (4.2) We consider the following model featuring two stocks S 1 and S 2 on the sample space Ω = {ω 1,ω 2,ω 3 }. The interest rate r = 1 10 so that B 0 = 1 and B 1 = 1+ 1 10. The stock prices at t = 0 are given by S0 1 = 1 and S2 0 = 2 and the stock prices at t = 1 are represented in the table: ω 1 ω 2 ω 3 S1 1 1 1 S 2 1 2 3 5 1 2 4 10 Does this market model admit an arbitrage opportunity? B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 48 / 87

Example: Market Model with Arbitrage Example (4.2 Continued) Once again, we will analyse this problem using Lemma 4.1, Lemma 4.2 and the FTAP. To tell whether a model is arbitrage-free it suffices to know the increments of discounted stock prices. The increments of discounted stock prices are represented in the following table ω 1 ω 2 ω 3 Ŝ1 1 1 Ŝ2 1 11 6 11 3 11 20 11 18 11 21 11 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 49 / 87

Example: Market Model with Arbitrage Example (4.2 Continued) Recall that Ĝ 1 (x,φ 1,φ 2 ) = φ 1 Ŝ1 1 +φ 2 Ŝ2 1 Hence, by the definition of W, we have W = α +β 1 6 20 3 18 21 α,β R. Let us take α = 3 and β = 1. Then we obtain the vector (0,0,39) T, which manifestly belongs to A. We conclude that W A and thus, by Lemma 4.1, the market model is not arbitrage-free. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 50 / 87

Example: Market Model with Arbitrage Example (4.2 Continued) We note that W = γ 6 1 0 γ R. If there exists a risk-neutral probability measure Q then Q W P +. Since Q W, we obtain Q(ω 1 ) = 6Q(ω 2 ). However, Q P + implies that Q(ω) > 0 for all ω Ω. We conclude that W P + = and thus, by Lemma 4.2, no risk-neutral probability measure exists, that is, M =. Hence the FTAP confirms that the model is not arbitrage-free. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 51 / 87

PART 4 RISK-NEUTRAL VALUATION OF CONTINGENT CLAIMS B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 52 / 87

Contingent Claims We now know how to check whether a given model is arbitrage-free. Hence the following question arises: What should be the fair price of a European call or put option in a general single-period market model? In a general single-period market model, the idea of pricing European options can be extended to any contingent claim. Definition (Contingent Claim) A contingent claim is a random variable X defined on Ω and representing the payoff at the maturity date. Derivatives nowadays are usually quite complicated and thus it makes sense to analyse valuation and hedging of a general contingent claim, and not only European call and put options. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 53 / 87

No-Arbitrage Principle Definition (Replication and Arbitrage Price) A trading strategy (x,φ 1,...,φ n ) is called a replicating strategy (a hedging strategy) for a claim X when V 1 (x,φ 1,...,φ n ) = X. Then the initial wealth is denoted as π 0 (X) and it is called the arbitrage price of X. Proposition (No-Arbitrage Principle) Assume that a contingent claim X can be replicated by means of a trading strategy (x,φ 1,...,φ n ). Then the unique price of X at 0 consistent with no-arbitrage principle equals V 0 (x,φ 1,...,φ n ) = x. Proof. If the price of X is higher (lower) than x, one can short sell (buy) X and buy (short sell) the replicating portfolio. This will yield an arbitrage opportunity in the extended market in which X is traded at time t = 0. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 54 / 87

Example: Stochastic Volatility Model In the elementary market model, a replicating strategy for any contingent claim always exists. However, in a general single-period market model, a replicating strategy may fail to exist for some claims. For instance, when there are more sources of randomness than there are stocks available for investment then replicating strategies do not exist for some claims. Example (4.3) Consider a market model consisting of bond B, stock S, and a random variable v called the volatility. The volatility determines whether the stock price can make either a big or a small jump. This is a simple example of a stochastic volatility model. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 55 / 87

Example: Stochastic Volatility Model Example (4.3 Continued) The sample space is given by Ω = {ω 1,ω 2,ω 3,ω 4 } and the volatility is defined as { h for i = 1,4, v(ω i ) = l for i = 2,3. We furthermore assume that 0 < l < h < 1. The stock price S 1 is given by { (1+v(ωi ))S S 1 (ω i ) = 0 for i = 1,2, (1 v(ω i ))S 0 for i = 3,4. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 56 / 87

Example: Stochastic Volatility Model Example (4.3 Continued) Unlike in examples we considered earlier, the amount by which the stock price in this market model jumps is random. It is easy to check that the model is arbitrage-free whenever 1 h < 1+r < 1+h. We claim that for some contingent claims a replicating strategy does not exist. In that case, we say that a claim is not attainable. To justify this claim, we consider the digital call option X with the payoff { 1 if S1 > K, X = 0 otherwise, where K > 0 is the strike price. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 57 / 87

Example: Stochastic Volatility Model Example (4.3 Continued) We assume that (1+l)S 0 < K < (1+h)S 0, so that and thus (1 h)s 0 < (1 l)s 0 < (1+l)S 0 < K < (1+h)S 0 X(ω i ) = { 1 for i = 1, 0 otherwise. Suppose that (x, φ) is a replicating strategy for X. Equality V 1 (x,φ) = X becomes (x φs 0 ) 1+r 1+r 1+r 1+r +φ (1+h)S 0 (1+l)S 0 (1 l)s 0 (1 h)s 0 = 1 0 0 0 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 58 / 87

Example: Stochastic Volatility Model Example (4.3 Continued) Upon setting β = φs 0 and α = (1+r)x φs 0 r, we see that the existence of a solution (x,φ) to this system is equivalent to the existence of a solution (α,β) to the system α 1 1 1 1 +β h l l h = It is easy to see that the above system of equations has no solution and thus a digital call is not an attainable contingent claim within the framework of the stochastic volatility model. Intuitively, the randomness generated by the volatility cannot be hedged, since the volatility is not a traded asset. 1 0 0 0 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 59 / 87

Valuation of Attainable Contingent Claims We first recall the definition of attainability of a contingent claim. Definition (Attainable Contingent Claim) A contingent claim X is called to be attainable if there exists a replicating strategy for X. Let us summarise the known properties of attainable claims: It is clear how to price attainable contingent claims by the replicating principle. There might be more than one replicating strategy, but no arbitrage principle leads the initial wealth x to be unique. In the two-state single-period market model, one can use the risk-neutral probability measure to price contingent claims. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 60 / 87

Risk-Neutral Valuation Formula Our next objective is to extend the risk-neutral valuation formula to any attainable contingent claim within the framework of a general single-period market model. Proposition (4.2) Let X be an attainable contingent claim and let Q M be any risk-neutral probability measure. Then the arbitrage price of X at t = 0 equals Proof of Proposition 4.2. π 0 (X) = 1 1+r E Q(X). Recall that a trading strategy (x,φ i,...,φ n ) is a replicating strategy for X whenever V 1 (x,φ 1,...,φ n ) = X B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 61 / 87

Proof of the Risk-Neutral Valuation Formula Proof of Proposition 4.2. We divide both sides by 1+r, to obtain Hence X 1+r = V 1(x,φ i,...,φ n ) = 1+r V 1 (x,φ 1,...,φ n ). 1 } 1+r E Q(X) = E Q { V1 (x,φ 1,...,φ n ) } = E Q {x+ĝ1(x,φ 1,...,φ n ) { n } = x+e Q φ j Ŝj 1 = x+ j=1 n φ j ( E Q Ŝ1) j = x. (from R.2.) j=1 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 62 / 87

Example: Stochastic Volatility Model Example (4.3 Continued) Proposition 4.2 shows that risk-neutral probability measures can be used to price attainable contingent claims. Consider the market model introduced in Example 4.3 with the interest rate r = 0. Recall that in this case the model is arbitrage-free since 1 h < 1+r = 1 < 1+h. The increments of the discounted stock price Ŝ are represented in the following table ω 1 ω 2 ω 3 ω 4 Ŝ1 hs 0 ls 0 ls 0 hs 0 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 63 / 87

Example: Stochastic Volatility Model Example (4.3 Continued) By the definition of the linear subspace W R 4, we have h W = γ l l γ R. h The orthogonal complement of W is thus the three-dimensional subspace of R 4 given by z 1 z 1 h W = z 2 z 3 R4 z 2 z 3, l l = 0. z 4 z 4 h B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 64 / 87

Example: Stochastic Volatility Model Example (4.3 Continued) Recall that a vector (q 1,q 2,q 3,q 4 ) belongs to P + if and only if the equality 4 i=1 q i = 1 holds and q i > 0 for i = 1,2,3,4. Since the set of risk-neutral probability measures is given by M = W P +, we find that q 1 q 2 q 3 q 4 M {(q 1,q 2,q 3,q 4 ) qi > 0, 4 q i = 1 i=1 } and h(q 1 q 4 )+l(q 2 q 3 ) = 0. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 65 / 87

Example: Stochastic Volatility Model Example (4.3 Continued) The class M of all risk-neutral probability measures in our stochastic volatility model is therefore given by q 1 M = q 2 q 1 > 0, q 2 > 0, q 3 > 0, q 3 q 1 +q 2 +q 3 < 1,. l(q 1 q 1 q 2 q 2 q 3 ) = h(1 2q 1 q 2 q 3 ) 3 This set appears to be non-empty and thus we conclude that our stochastic volatility model is arbitrage-free. Recall that we have already shown that the digital call option is not attainable if (1+l)S 0 < K < (1+h)S 0. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 66 / 87

Example: Stochastic Volatility Model Example (4.3 Continued) It is not difficult to check that for every 0 < q 1 < 1 2 probability measure Q M such that Q(ω 1 ) = q 1. Indeed, it suffices to take q 1 (0, 1 2 ) and to set there exists a q 4 = q 1, q 2 = q 3 = 1 2 q 1. We apply the risk-neutral valuation formula to the digital call X = (1,0,0,0). For Q = (q 1,q 2,q 3,q 4 ) M, we obtain E Q (X) = q 1 1+q 2 0+q 3 0+q 4 0 = q 1. Since q 1 is any number from (0, 1 2 ), we see that every value from the open interval (0, 1 2 ) can be achieved. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 67 / 87

Extended Market Model and No-Arbitrage Principle We no longer assume that a contingent claim X is attainable. Definition A price π 0 (X) of a contingent claim X is said to be consistent with the no-arbitrage principle if the extended model, which consists of B, the original stocks S 1,...,S n, as well as an additional asset S n+1 satisfying S0 n+1 = π 0 (X) and S1 n+1 = X, is arbitrage-free. The interpretation of Definition 4.1 is as follows: We assume that the model M = (B,S 1,...,S n ) is arbitrage-free. We regard the additional asset as a tradable risky asset in the extended market model M = (B,S 1,...,S n+1 ). We postulate its price at time 0 should be selected in such a way that the extended market model M is still arbitrage-free. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 68 / 87

Valuation of Non-Attainable Claims We already know that the risk-neutral valuation formula returns the arbitrage price for any attainable claim. The next result shows that it also yields a price consistent with the no-arbitrage principle when it is applied to any non-attainable claim. The price obtained in this way is not unique, however. Proposition (4.3) Let X be a possibly non-attainable contingent claim and Q is an arbitrary risk-neutral probability measure. Then π 0 (X) given by π 0 (X) := 1 1+r E Q(X) (2) defines a price at t = 0 for the claim X that is consistent with the no-arbitrage principle. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 69 / 87

Proof of Proposition 4.3 Proof of Proposition 4.3. Let Q M be an arbitrary risk-neutral probability for M. We will show that Q is also a risk-neutral probability measure for the extended model M = (B,S 1,...,S n+1 ) in which S0 n+1 = π 0 (X) and S1 n+1 = X. For this purpose, we check that ( ) { } X E Q Ŝn+1 1 = E Q 1+r π 0(X) = 0 and thus Q M is indeed a risk-neutral probability in the extended market model. By the FTAP, the extended model M is arbitrage-free. Hence the price π 0 (X) given by (2) complies with the no-arbitrage principle. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 70 / 87

PART 5 COMPLETENESS OF MARKET MODELS B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 71 / 87

Complete and Incomplete Models The non-uniqueness of arbitrage prices is a serious theoretical problem, which is still not completely resolved. We categorise market models into two classes: complete and incomplete models. Definition (Completeness) A financial market model is called complete if for any contingent claim X there exists a replicating strategy (x,φ) R n+1. A model is incomplete when there exists a claim X for which a replicating strategy does not exist. Given an arbitrage-free and complete model, the issue of pricing all contingent claims is completely solved. How can we tell whether a given model is complete? B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 72 / 87

Algebraic Criterion for Market Completeness Proposition (4.4) Assume that a single-period market model M = (B,S 1,...,S n ) defined on the sample space Ω = {ω 1,...,ω k } is arbitrage-free. Then M is complete if and only if the k (n+1) matrix A A = 1+r S 1 1 (ω 1) S n 1 (ω 1) 1+r S 1 1 (ω 2) S n 1 (ω 2) 1+r S 1 1(ω k ) S n 1(ω k ) = (A 0,A 1,...,A n ) has a full row rank, that is, rank(a) = k. Equivalently, M is complete whenever the linear subspace spanned by the vectors A 0,A 1,...,A n coincides with the full space R k. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 73 / 87

Proof of Proposition 4.4 Proof of Proposition 4.4. By the linear algebra, A has a full row rank if and only if for every X R k the equation AZ = X has a solution Z R n+1. If we set φ 0 = x n j=1 φj S j 0 then we have 1+r S 1 1(ω 1 ) S n 1(ω 1 ) 1+r S 1 1 (ω 2) S n 1 (ω 2) 1+r S 1 1 (ω k) S n 1 (ω k) where V 1 (ω i ) = V 1 (x,φ)(ω i ). φ 0 φ 1 φ n = V 1 (ω 1 ) V 1 (ω 2 ) V 1 (ω k ) This shows that computing a replicating strategy for X is equivalent to solving the equation AZ = X. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 74 / 87

Example: Incomplete Model Example (4.3 Continued) Consider the stochastic volatility model from Example 4.3. We already know that this model is incomplete, since the digital call is not an attainable claim. The matrix A is given by A = 1+r S 1 1 (ω 1) 1+r S 1 1 (ω 2) 1+r S 1 1 (ω 3) 1+r S 1 1 (ω 4) The rank of A is 2, and thus it is not equal to k = 4. In view of Proposition 4.4, this confirms that this market model is incomplete. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 75 / 87

Probabilistic Criterion for Attainability Proposition 4.4 yields a method for determining whether a market model is complete. Given an incomplete model, how to recognize an attainable claim? Recall that if a model M is arbitrage-free then the class M is non-empty. Proposition (4.5) Assume that a single-period model M = (B,S 1,...,S n ) is arbitrage-free. Then a contingent claim X is attainable if and only if the expected value E Q ( (1+r) 1 X ) has the same value for all risk-neutral probability measures Q M. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 76 / 87

Proof of Proposition 4.5: 1 Proof of Proposition 4.5. ( ) It is immediate from Proposition 4.2 that if a contingent claim X is attainable then the expected value E Q ( (1+r) 1 X ) has the same value for all Q M. ( ) (MATH3975) We prove this implication by contrapositive. Let us thus assume that the contingent claim X is not attainable. Our goal is to find two risk-neutral probabilities, say Q and Q, for which E Q ( (1+r) 1 X ) E Q( (1+r) 1 X ). (3) B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 77 / 87

Proof of Proposition 4.5: 2 Proof of Proposition 4.5. Consider the matrix A introduced in Proposition 4.4. Since the claim X is not attainable, there is no solution Z R n+1 to the linear system AZ = X. We define the following subsets of R k and C = {X}. B = image(a) = { AZ Z R n+1} R k Then B is a proper subspace of R k and, obviously, the set C is convex and compact. Moreover, B C =. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 78 / 87

Proof of Proposition 4.5: 3 Proof of Proposition 4.5. In view of Corollary 4.1, there exists a non-zero vector Y = (y 1,...,y k ) R k such that b,y = 0 for all b B, c,y > 0 for all c C. In view of the definition of B and C, this means that for every j = 0,1,...,n A j,y = 0 and X,Y > 0 (4) where A j is the jth column of the matrix A. It is worth noting that the vector Y depends on X. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 79 / 87

Proof of Proposition 4.5: 4 Proof of Proposition 4.5. We assumed that the market model is arbitrage-free and thus, by the FTAP, the class M is non-empty. Let Q M be an arbitrary risk-neutral probability measure. We may choose a real number λ > 0 to be small enough in order to ensure that for every i = 1,2,...,k Q(ω i ) := Q(ω i )+λ(1+r)y i > 0. (5) In the next step, our next goal is to show that Q is also a risk-neutral probability measure and it is different from Q. In the last step, we will show that inequality (3) is valid. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 80 / 87

Proof of Proposition 4.5: 5 Proof of Proposition 4.5. From the definition of A in Proposition 4.4 and the first equality in (4) with j = 0, we obtain k λ(1+r)y i = λ A 0,Y = 0. i=1 It then follows from (5) that k Q(ω i ) = i=1 k k Q(ω i )+ λ(1+r)y i = 1 i=1 i=1 and thus Q is a probability measure on the space Ω. In view of (5), it is clear that Q satisfies condition R.1. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 81 / 87

Proof of Proposition 4.5: 6 Proof of Proposition 4.5. It remains to check that Q satisfies also condition R.2. We examine the behaviour under Q of the discounted stock price Ŝj 1. For every j = 1,2,...,n, we have (Ŝj ) E Q 1 = = k Q(ω i )Ŝj 1 (ω i) i=1 k Q(ω i )Ŝj 1 (ω i)+λ i=1 = E Q (Ŝj 1) +λ Aj,Y } {{ } =0 k Ŝ j 1 (ω i)(1+r)y i i=1 (in view of (4)) = Ŝj 0 (since Q M) B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 82 / 87

Proof of Proposition 4.5: 7 Proof of Proposition 4.5. We conclude that E Q( Ŝ j 1) = 0 and thus Q M, that is, Q is a risk-neutral probability measure for the market model M. From (5), it is clear that Q Q. We have thus proven that if M is arbitrage-free and incomplete then there exists more than one risk-neutral probability measure. To complete the proof, it remains to show that inequality (3) is satisfied for a contingent claim X. Recall that X was a fixed non-attainable contingent claim and we constructed a risk-neutral probability measure Q corresponding to X. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 83 / 87

Proof of Proposition 4.5: 8 Proof of Proposition 4.5. We observe that E Q ( ) X 1+r = = > k i=1 Q(ω i ) X(ω i) 1+r k Q(ω i ) X(ω i) k 1+r +λ y i X(ω i ) i=1 } {{ } i=1 k i=1 >0 Q(ω i ) X(ω ( ) i) X 1+r = E Q 1+r since the inequalities X,Y > 0 and λ > 0 imply that the braced expression is strictly positive. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 84 / 87

Probabilistic Criterion for Market Completeness Theorem (4.1) Assume that a single-period model M = (B,S 1,...,S n ) is arbitrage-free. Then M is complete if and only if the class M consists of a single element, that is, there exists a unique risk-neutral probability measure. Proof of ( ) in Theorem 4.1. Since M is assumed to be arbitrage-free, from the FTAP it follows that there exists at least one risk-neutral probability measure, that is, the class M is non-empty. ( ) Assume first that a risk-neutral probability measure for M is unique. Then the condition of Proposition 4.5 is trivially satisfied for any claim X. Hence any claim X is attainable and thus the model M is complete. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 85 / 87

Proof of ( ) in Theorem 4.1 Proof of ( ) Theorem 4.1. ( ) Assume M is complete and consider any two risk-neutral probability measures Q and Q from M. For a fixed, but arbitrary, i = 1,2,...,k, let the contingent claim X i be given by { X i 1+r if ω = ωi, (ω) = 0 otherwise. Since M is now assumed to be complete, the contingent claim X i is attainable. From Proposition 4.2, it thus follows that ( X i ) ( X Q(ω i ) = E Q = π 0 (X i i ) ) = = 1+r E Q 1+r Q(ω i ). Since i was arbitrary, we see that the equality Q = Q holds. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 86 / 87

Summary Let us summarise the properties of single-period market models: 1 A single-period market model is arbitrage-free if and only if it admits at least one risk-neutral probability measure. 2 An arbitrage-free single-period market model is complete if and only if the risk-neutral probability measure is unique. 3 Under the assumption that the model is arbitrage-free: An arbitrary attainable contingent claim X (that is, any claim that can be replicated by means of some trading strategy) has the unique arbitrage price π 0 (X). The arbitrage price π 0 (X) of any attainable contingent claim X can be computed from the risk-neutral valuation formula using any risk-neutral probability Q. If X is not attainable then we may define a price of X consistent with the no-arbitrage principle. It can be computed using the risk-neutral valuation formula, but it will depend on the choice of a risk-neutral probability Q. B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market Models 87 / 87