6: MULTI-PERIOD MARKET MODELS

Transcription

1 6: MULTI-PERIOD MARKET MODELS Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2016 M. Rutkowski (USydney) 6: Multi-Period Market Models 1 / 55

2 Outline We will examine the following issues: 1 Trading Strategies and Arbitrage-Free Models 2 Risk-Neutral Probability Measures and Martingales 3 Fundamental Theorem of Asset Pricing 4 Arbitrage Pricing of Attainable Claims 5 Risk-Neutral Valuation of Non-Attainable Claims 6 Completeness of Multi-Period Market Models M. Rutkowski (USydney) 6: Multi-Period Market Models 2 / 55

3 PART 1 SELF-FINANCING TRADING STRATEGIES M. Rutkowski (USydney) 6: Multi-Period Market Models 3 / 55

4 Primary Traded Assets In a multi-period market model M = (B,S 1,...,S n ), we need to examine the concept of a dynamic trading strategy φ and the associated wealth process V(φ). We first define primary traded assets Let r be the interest rate. The money market account is denoted by B t for t = 0,1,...,T where B t = (1+r) t. There are n risky assets, called stocks, with price processes denoted by S j t for t = 0,1,...,T. We are given a probability space (Ω,F,P) endowed with a filtration F generated by price processes of stocks. M. Rutkowski (USydney) 6: Multi-Period Market Models 4 / 55

5 Dynamic Trading Strategy and Wealth Process A dynamic trading strategy in a multi-period market model is defined as a stochastic process φ t = ( φ 0 ) t,φ1 t,...,φn t for t = 0,1,...,T where: φ 0 t is the number of shares of the money market account B held at time t. φ j t is the number of shares of the jth stock held at time t. Definition (Value Process) The wealth process (also known as the value process) of a trading strategy φ = ( φ 0,φ 1,...,φ n) is the stochastic process V(φ) given by V t (φ) := φ 0 t B t + n φ j t Sj t. j=1 M. Rutkowski (USydney) 6: Multi-Period Market Models 5 / 55

6 Self-Financing Trading Strategy Definition (Self-Financing Trading Strategy) A trading strategy φ is said to be self-financing strategy if for every t = 0,1,...,T 1, φ 0 tb t+1 + n φ j t Sj t+1 = φ0 t+1b t+1 + j=1 n φ j t+1 Sj t+1. (1) j=1 The LHS of (1) represents the value of the portfolio at time t+1 before its revision, whereas the RHS represents the value at time t+1 after the portfolio was revised. Condition (1) says that these two values must be equal and this means that no cash was withdrawn or added. For t = T 1, both sides of (1) represent the wealth at time T, that is, V T (φ). We do not revise the portfolio at time T. M. Rutkowski (USydney) 6: Multi-Period Market Models 6 / 55

7 Gains Process Lemma (6.1) For any self-financing trading strategy φ, the wealth process can be alternatively computed by, for t = 1,...,T, Proof. V t (φ) = φ 0 t 1B t + n φ j t 1 Sj t. j=1 The statement is an immediate consequence of formula (1). Definition (Gains Process) The gains process G(φ) = (G t (φ)) 0 t T of a trading strategy φ is given by G t (φ) := V t (φ) V 0 (φ). M. Rutkowski (USydney) 6: Multi-Period Market Models 7 / 55

8 Multi-Period Market Model The concept of a multi-period market model is a natural extension of the notion of a single-period market model with only two dates: t = 0 and t = T = 1. Definition (Market Model) A multi-period market model M = (B,S 1,...,S n ) is given by the following data: 1 A probability space (Ω,F,P) with a filtration F = (F t ) 0 t T. 2 The money market account B given by B t = (1+r) t. 3 A number of risky financial assets with prices S 1,...,S n, which are assumed to be F-adapted stochastic processes. 4 The class Φ of all self-financing trading strategies. M. Rutkowski (USydney) 6: Multi-Period Market Models 8 / 55

9 Increment Processes As in the single-period model, we define the discounted value process, gains process and discounted gains process. It will be convenient to use the increment processes for traded assets. Definition (Increment Processes) The increment process S j corresponding to the jth stock is defined by S j t+1 := Sj t+1 Sj t for t = 0,...,T 1. The increment process B of the money market account is given by B t+1 := B t+1 B t = (1+r) t r = B t r for t = 0,...,T 1. M. Rutkowski (USydney) 6: Multi-Period Market Models 9 / 55

10 Discounted Processes Definition (Discounted Processes) The discounted stock prices are given by Ŝ j t := Sj t B t and the increments of discounted prices are Ŝj t+1 := Ŝj t+1 Ŝj t. The discounted wealth process V(φ) of φ is given by V t (φ) := V t(φ) B t. The discounted gains process Ĝ(φ) of φ equals Ĝ t (φ) := V t (φ) V 0 (φ). M. Rutkowski (USydney) 6: Multi-Period Market Models 10 / 55

11 Discounted Processes Proposition (6.1) An F-adapted trading strategy φ = (φ t ) 0 t T is self-financing if and only if any of the two equivalent statements hold: 1 for every t = 1,...,T 2 for every t = 1,...,T t 1 G t (φ) = φ 0 u B t 1 u+1 + u=0 Ĝ t (φ) = t 1 u=0 j=1 u=0 j=1 n φ j u Ŝj u+1. n φ j u Sj u+1. M. Rutkowski (USydney) 6: Multi-Period Market Models 11 / 55

12 Properties of Discounted Gains and Wealth Proof of Proposition 6.1. The proof is elementary and thus it is left as an exercise. It is important to note that the process Ĝ(φ) given by condition 2) does not depend on the component φ 0 of a trading strategy φ Φ. In view of Proposition 6.1, the discounted wealth process of any φ Φ satisfies V t (φ) = V t 1 0 (φ)+ Hence for every t = 0,...,T 1 u=0 j=1 V t+1 (φ) = Ĝt+1(φ) = n φ j u Ŝj u+1. n φ j t Ŝj t+1. j=1 M. Rutkowski (USydney) 6: Multi-Period Market Models 12 / 55

13 PART 2 RISK-NEUTRAL PROBABILITY MEASURES AND MARTINGALES M. Rutkowski (USydney) 6: Multi-Period Market Models 13 / 55

14 Arbitrage Opportunity As usual, we work under the standing assumption that the sample space Ω is finite (or at most countable). Definition (Arbitrage Opportunity) A trading strategy φ Φ is said to be an arbitrage opportunity if 1 V 0 (φ) = 0, 2 V T (φ)(ω) 0 for all ω Ω, 3 V T (φ)(ω) > 0 for some ω Ω or, equivalently, E P (V T (φ)) > 0. We say that a multi-period market model M is arbitrage-free if no arbitrage opportunities exist in the class Φ of all self-financing trading strategies. M. Rutkowski (USydney) 6: Multi-Period Market Models 14 / 55

15 Arbitrage Conditions Observe that in the arbitrage conditions one can use either the discounted wealth process V or the discounted gains process Ĝ (instead of the wealth V). It is also important to note that conditions 1) 3) hold under P whenever they are satisfied under some probability measure Q equivalent to P. The next step is to introduce the concept of a risk-neutral probability measure for a multi-period market model. The risk-neutral probability measures (also known as the martingale measures) are very closely related to the question of arbitrage-free property and completeness of a multi-period market model. M. Rutkowski (USydney) 6: Multi-Period Market Models 15 / 55

16 Risk-Neutral Probability Measure Definition (Risk-Neutral Probability Measure) A probability measure Q on (Ω,F T ) is called a risk-neutral probability measure for a multi-period market model M = (B,S 1,...,S n ) if 1 Q(ω) > 0 for all ω Ω, 2 E Q ( Ŝj t+1 F t) = 0 for all j = 1,...,n and t = 0,...,T 1. We denote by M the class of all risk-neutral probability measures for the market model M. Observe that condition 2) is equivalent to the following equality for every t = 0,...,T 1 E Q (Ŝj t+1 F t) = Ŝ j t. M. Rutkowski (USydney) 6: Multi-Period Market Models 16 / 55

17 Martingales (MATH3975) Martingales are stochastic processes representing fair games. Definition (Martingale) An F-adapted process X = (X t ) 0 t T on a finite probability space (Ω,F,P) is called a martingale whenever for all s < t E P (X t F s ) = X s. To establish the martingale property, it suffices to check that for every t = 0,1,...,T 1 E P (X t+1 F t ) = X t. In particular, this means that the discounted stock price Ŝj is a martingale under any risk-neutral probability measure Q M. M. Rutkowski (USydney) 6: Multi-Period Market Models 17 / 55

18 Discounted Wealth as a Martingale (MATH3975) Proposition (6.2) Let φ Φ be a trading strategy. Then the discounted wealth process V(φ) and the discounted gains process Ĝ(φ) are martingales under any risk-neutral probability measure Q M. Proof. Recall that V t (φ) = V 0 (φ)+ĝt(φ) for every t = 0,...,T, Since V 0 (φ) (the initial endowment) is a constant, it suffices to show that the process Ĝ(φ) is a martingale under any Q M. From Proposition 6.1, we obtain Ĝ t+1 (φ) = Ĝt(φ)+ n φ j t Ŝj t+1. j=1 M. Rutkowski (USydney) 6: Multi-Period Market Models 18 / 55

19 Proof of Proposition 6.2 (MATH3975) Proof of Proposition 6.2 (Continued). Hence E Q (Ĝt+1(φ) F t ) = Ĝt(φ)+ = Ĝt(φ)+ = Ĝt(φ) n ) E Q (φ j t Ŝj t+1 F t j=1 n j=1 φ j t E Q( Ŝ j ) t+1 Ft }{{} =0 We used the fact that φ j t is F t-measurable and the take out what is known property of the conditional expectation. We conclude that Ĝ(φ) is a martingale under any Q M. M. Rutkowski (USydney) 6: Multi-Period Market Models 19 / 55

20 PART 3 FUNDAMENTAL THEOREM OF ASSET PRICING M. Rutkowski (USydney) 6: Multi-Period Market Models 20 / 55

21 Fundamental Theorem of Asset Pricing We will show that the Fundamental Theorem of Asset Pricing (FTAP) can be extended to a multi-period market model. Recall that the class of admissible trading strategies Φ in a multi-period market model is assumed to be the full set of all self-financing and F-adapted trading strategies. It possible to show that in that case, the relationship between the existence of a martingale measure Q and no arbitrage for the model M is if and only if. We will only prove here the following implication: Existence of Q M Model M is arbitrage-free M. Rutkowski (USydney) 6: Multi-Period Market Models 21 / 55

22 Fundamental Theorem of Asset Pricing Theorem (FTAP) Consider a multi-period market model M = (B,S 1,...,S n ). The following statements hold: 1 if the class M of martingale measures for M is non-empty, then there are no arbitrage opportunities in the class Φ of all self-financing trading strategies and thus the model M is arbitrage-free, 2 if there are no arbitrage opportunities in the class Φ of all self-financing trading strategies, then there exists a martingale measure for M so that the class M is non-empty. To sum up: Class M is non-empty Market model M is arbitrage-free M. Rutkowski (USydney) 6: Multi-Period Market Models 22 / 55

23 Proof of the FTAP ( ) Proof of the FTAP ( ). Let us assume that a martingale measure Q for M exists. Our goal it to show that the model M is arbitrage-free. To this end, we argue by contradiction. Let us thus assume that there exists an arbitrage opportunity φ Φ. Such a strategy would satisfy the following conditions: 1 the initial endowment V 0 (φ) = 0, 2 the discounted gains process ĜT(φ) 0, 3 there exists at least one ω Ω such that ĜT(φ)(ω) > 0. On the one hand, from conditions 2. and 3. above, we deduce easily that E Q (ĜT (φ) ) > 0. M. Rutkowski (USydney) 6: Multi-Period Market Models 23 / 55

24 Proof of the FTAP ( ) Proof of the FTAP ( ). On the other hand, using Proposition 6.1, we obtain E Q (ĜT (φ) ) ( n T 1 ) = E Q φ j u Ŝj u+1 = j=1 u=0 n T 1 ( = E Q EQ (φ j u Ŝj u+1 F u) ) = j=1 u=0 n T 1 j=1 u=0 ( E Q φ j u E Q ( Ŝj u+1 F ) u) = 0. }{{} =0 n T 1 ( ) E Q φ j u Ŝj u+1 j=1 u=0 This clearly contradicts the inequality obtained in the first step. Hence the market model M is arbitrage-free. M. Rutkowski (USydney) 6: Multi-Period Market Models 24 / 55

25 Martingale Argument (MATH3975) Let us now present a simple martingale argument: We already know from Proposition 6.2 that the discounted gains process Ĝ(φ) is a martingale under Q. Hence E Q (ĜT(φ)) = E Q (ĜT(φ) F 0 ) = Ĝ0(φ) = 0 where we used the properties of the conditional expectation. This observation shortens the argument in the proof of the implication ( ) in the FTAP. The proof of the implication ( ) in the FTAP requires a good familiarity with the theory of martingales, although its idea is the same as in the single-period market case. M. Rutkowski (USydney) 6: Multi-Period Market Models 25 / 55

26 PART 4 ARBITRAGE PRICING OF ATTAINABLE CLAIMS M. Rutkowski (USydney) 6: Multi-Period Market Models 26 / 55

27 Replicating Strategy Note that a contingent claim of European style can only be exercised at its maturity date T (as opposed to contingent claims of American style). A European contingent claim in a multi-period market model is an F T -measurable random variable X on Ω to be interpreted as the payoff at the terminal date T. For brevity, European contingent claims will also be referred to as contingent claims or simply claims. Definition (Replicating Strategy) A replicating strategy (or a hedging strategy) for a contingent claim X is a trading strategy φ Φ such that V T (φ) = X, that is, the terminal wealth of the trading strategy matches the claim s payoff for all ω. M. Rutkowski (USydney) 6: Multi-Period Market Models 27 / 55

28 Principle of No-Arbitrage Definition (Principle of No-Arbitrage) An F-adapted stochastic process (π t (X)) 0 t T is a price process for the contingent claim X that complies with the principle of no-arbitrage if there is no F-adapted and self-financing arbitrage strategy in the extended model M = (B,S 1,...,S n,s n+1 ) with an additional asset S n+1 given by St n+1 = π t (X) for 0 t T 1 and S n+1 T = X. The standard method to price a contingent claim is to employ the replication principle, if it can be applied. The price will now depend on time t and thus one has to specify a whole price process π(x), rather than just an initial price, as in the single-period market model. Obviously, π T (X) = X for any claim X. M. Rutkowski (USydney) 6: Multi-Period Market Models 28 / 55

29 Arbitrage Pricing of Attainable Claims In the next result, we deal with an attainable claim, meaning that we assume a priori that a replicating strategy for X exists. Proposition (6.3) Let X be a contingent claim in an arbitrage-free multi-period market model M and let φ Φ be any replicating strategy for X. Then the only price process of X that complies with the principle of no-arbitrage is the wealth process V(φ). The arbitrage price at time t of an attainable claim X is unique and it is also denoted as π t (X). Hence the equality π t (X) = V t (φ) holds for any replicating strategy φ Φ for X. In particular, the price at time t = 0 is the initial endowment of any replicating strategy for X, that is, π 0 (X) = V 0 (φ) for any strategy φ Φ such that V T (φ) = X. M. Rutkowski (USydney) 6: Multi-Period Market Models 29 / 55

30 Example: Replication of a Digital Call Option Example (6.1) We will now examine replication of a contingent claim in a two-period market model. Consider a two-period market model consisting of the savings account and one risky stock. The interest rate equals r = 1 9 so that B t = (1+r) t = ( ) 10 t. 9 The price of the stock is represented in the following exhibit in which the real-world probability P is also specified. It is easy to check that this model is arbitrage-free. Our goal is to price a particular contingent claim X using replication. M. Rutkowski (USydney) 6: Multi-Period Market Models 30 / 55

31 Example: Replication of a Digital Call Option Example (6.1 Continued) 0.3 S 2 = 3 ω 1 S 1 = S 2 = 1 ω 2 S 0 = S 2 = 1.5 ω S 1 = S 2 = 0.5 ω 4 Figure: Stock Price Dynamics M. Rutkowski (USydney) 6: Multi-Period Market Models 31 / 55

32 Example: Replication of a Digital Call Option Example (6.1 Continued) Consider a digital call option with the payoff function { 1 if S2 (ω) > 2 X(ω) = g(s 2 (ω)) = 0 otherwise X = ( X(ω 1 ),X(ω 2 ),X(ω 3 ),X(ω 4 ) ) = (1,0,0,0). By the definition of replication, we have V 2 (φ) = X or, more explicitly, V 2 (φ) = φ 0 1 B 2 +φ 1 1 S 2 = g(s 2 ) = (1,0,0,0). Observe that φ i 1 (ω 1) = φ i 1 (ω 2) and φ i 1 (ω 3) = φ i 1 (ω 4) for i = 0,1 since φ is an F-adapted process. M. Rutkowski (USydney) 6: Multi-Period Market Models 32 / 55

33 Example: Replication of a Digital Call Option Example (6.1 Continued) At time t = 1 we obtain two linear systems for φ 1 = (φ 0 1,φ1 1 ) For ω {ω 1,ω 2 } For ω {ω 3,ω 4 } ( 10 9 ( 10 9 ( 10 9 ( 10 9 ) 2 φ φ 1 1 = 1 ) 2 φ 0 1 +φ1 1 = 0 ) 2 φ φ 1 1 = 0 ) 2 φ φ1 1 = 0 M. Rutkowski (USydney) 6: Multi-Period Market Models 33 / 55

34 Example: Replication of a Digital Call Option Example (6.1 Continued) We obtain the replicating strategy at t = 1: ( φ 0 1,φ 1 ) 1 = ( , 1 ) if ω {ω 1,ω 2 } 2 ( φ 0 1,φ 1 ) 1 = (0,0) if ω {ω3,ω 4 } If S 1 = 1.5 then the price of the digital call at t = 1 equals V 1 (φ) = φ 0 1B 1 +φ 1 1S 1 = = 0.3 If S 1 = 0.75 then the price of the digital call at t = 1 equals 0. The price of X at time 1 equals π 1 (X) = {S1 =1.5}. M. Rutkowski (USydney) 6: Multi-Period Market Models 34 / 55

35 Example: Replication of a Digital Call Option Example (6.1 Continued) We now compute the price of the digital call at t = 0. The replicating strategy at time t = 0 satisfies φ 0 0 B 1 +φ 1 0 S 1 = π 1 (X), that is, 10 9 φ φ 1 0 = φ φ1 0 = 0 Then ( φ 0 0,φ1 0) = ( 0.27,0.4). The price of the digital call at time t = 0 (recall that S 0 = B 0 = 1) thus equals V 0 (φ) = φ 0 0B 0 +φ 1 0S 0 = = 0.13 Hence the arbitrage price of X at time 0 equals π 1 (X) = M. Rutkowski (USydney) 6: Multi-Period Market Models 35 / 55

36 Example: Replication of a Digital Call Option Example (6.1 Continued) Summary of pricing and hedging results for a digital call option. Recall that the price at time t = 2 equals π 2 (X) = X. Replicating strategy φ satisfies V 2 (φ) = X. The arbitrage price process of X equals π(x) = V(φ). ( {ω 1,ω 2 } {ω 3,ω 4 } t = 0 φ 0 t,φ 1 ) ( t = ( 0.27,0.4) φ 0 t,φ 1 ) t = ( 0.27,0.4) t = 0 ( π 0 (X) = 0.13 π 0 (X) = 0.13 t = 1 φ 0 t,φ 1 ) ( t = , 2) 1 ( φ 0 t,φ 1 t) = (0,0) t = 1 π 1 (X) = 0.3 π 1 (X) = 0 M. Rutkowski (USydney) 6: Multi-Period Market Models 36 / 55

37 PART 5 RISK-NEUTRAL VALUATION OF NON-ATTAINABLE CONTINGENT CLAIMS M. Rutkowski (USydney) 6: Multi-Period Market Models 37 / 55

38 Attainability of Contingent Claims and Completeness Definition (Attainable Contingent Claim) A contingent claim X is called to be attainable if there exists a trading strategy φ Φ, which replicates X, i.e., V T (φ) = X. For attainable contingent claims, it is clear how to price them by the initial investment needed for a replicating strategy. As in single period market models, for some contingent claims a hedging strategy may fail to exist. Definition (Completeness) A multi period market model is said to be complete if and only if all contingent claims have replicating strategies. If a multi period market model is not complete, it is said to be incomplete. M. Rutkowski (USydney) 6: Multi-Period Market Models 38 / 55

39 Risk-Neutral Valuation Formula Proposition (6.4) Let X be a contingent claim (possibly non-attainable) and Q any risk-neutral probability measure for the multi-period market model M. Then the risk-neutral valuation formula π t (X) = B t E Q ( X B T Ft ) defines a price process π(x) = (π t (X)) 0 t T for X that complies with the principle of no-arbitrage. Proof. The proof hinges the same arguments as in the single-period case and thus it is left as an exercise. If X is attainable then can also observe that V(φ) is a martingale under Q and apply the definition of a martingale. M. Rutkowski (USydney) 6: Multi-Period Market Models 39 / 55

40 Example: Risk-Neutral Valuation Example (6.2) 0.3 S 2 = 3 ω 1 S 1 = S 2 = 1 ω 2 S 0 = S 2 = 1.5 ω S 1 = S 2 = 0.5 ω 4 Figure: Stock Price Dynamics M. Rutkowski (USydney) 6: Multi-Period Market Models 40 / 55

41 Example: Risk-Neutral Valuation Example (6.2 Continued) Consider again the market model M = (B,S) introduced in Example 6.1. Recall that the conditional probabilities describe the mouvements under the real-world probability P. Let Q be a risk-neutral probability measure, that is, Q M. We denote q i = Q(ω i ) for i = 1,2,3,4. By the definition of the risk-neutral probability, we have q 1 +q 2 +q 3 +q 4 = r E Q(S 2 F 1 ) = S r E Q(S 1 F 0 ) = S 0 M. Rutkowski (USydney) 6: Multi-Period Market Models 41 / 55

42 Example: Risk-Neutral Valuation Example (6.2 Continued) The conditional probabilities under Q at time t = 1 are and Q(S 2 = 3 {ω 1,ω 2 }) = q 1 q 1 +q 2 Q(S 2 = 1 {ω 1,ω 2 }) = q 2 q 1 +q 2 Q(S 2 = 1.5 {ω 3,ω 4 }) = q 3 q 3 +q 4 Q(S 2 = 0.5 {ω 3,ω 4 }) = q 4 q 3 +q 4 Q(S 2 = 1.5 {ω 1,ω 2 }) = 0, Q(S 2 = 0.5 {ω 1,ω 2 }) = 0 Q(S 2 = 3 {ω 3,ω 4 }) = 0, Q(S 2 = 1 {ω 3,ω 4 }) = 0 M. Rutkowski (USydney) 6: Multi-Period Market Models 42 / 55

43 Example: Risk-Neutral Probability Measure Example (6.2 Continued) Also, the probability distribution of S 1 reads Q(S 1 = 1.5) = q 1 +q 2 Q(S 1 = 0.75) = q 3 +q 4 Hence we obtain the following linear system: q 1 +q 2 +q 3 +q 4 = 1 ( 9 3q1 + q ) 2 = 3 10 q 1 +q 2 q 1 +q 2 2 = S 1(ω i ) for i = 1,2 ( 9 3 q ) q 4 = q 3 +q 4 2q 3 +q 4 4 = S 1(ω i ) for i = 3,4 [ (q 1 +q 2 )+ 3 ] 4 (q 3 +q 4 ) = 1 = S 0 M. Rutkowski (USydney) 6: Multi-Period Market Models 43 / 55

44 Example: Risk-Neutral Valuation Example (6.2 Continued) Equivalently, The unique solution reads q 1 +q 2 +q 3 +q 4 = 1 2q 1 q 2 = 0 2q 3 q 4 = q q q q 4 = 10 9 q 1 = 13 81, q 2 = 26 81, q 3 = 14 81, q 4 = M. Rutkowski (USydney) 6: Multi-Period Market Models 44 / 55

45 Example: Risk-Neutral Valuation Example (6.2 Continued) The price of the digital call option considered in Example 6.1 can be computed as follows. For ω {ω 1,ω 2 }, we obtain π 1 (X)(ω) = 1 1+r E Q(X {ω 1,ω 2 }) ( = = For ω {ω 3,ω 4 }, we have π 1 (X)(ω) = 0 since the payoff is 0 at t = 2 if the stock price at time t = 1 equals S 1 = ) M. Rutkowski (USydney) 6: Multi-Period Market Models 45 / 55

46 Example: Risk-Neutral Valuation Example (6.2 Continued) The price of the digital call option at time 0 equals 1 π 0 (X) = (1+r) 2 E 1 Q(X F 0 ) = (1+r) 2 E Q(X) = 81 ( 1 13 ) = 0.13 These pricing results coincide with those obtained in Example 6.1, where we computed directly the wealth process V(φ) of the replicating strategy φ for X. As was indicated earlier, a multi-period market model M can be decomposed into several single-period market models. M. Rutkowski (USydney) 6: Multi-Period Market Models 46 / 55

47 Example: Backward Induction Example (6.3) 0.3 S 2 = 3 ω 1 S 1 = S 2 = 1 ω 2 S 0 = S 2 = 1.5 ω S 1 = S 2 = 0.5 ω 4 Figure: Stock Price Dynamics M. Rutkowski (USydney) 6: Multi-Period Market Models 47 / 55

48 Example: Backward Induction Example (6.3 Continued) We consider the market model in Example 6.1 once again. The two-period market model is composed of the following single-period market models: 1 S 1 = 1.5, S 2 = 3 and S 2 = 1. 2 S 1 = 0.75, S 2 = 1.5 and S 2 = S 0 = 1, S 1 = 1.5 and S 1 = Note that these models are elementary market models. Hence the unique martingale measure can be computed using the formula p = 1+r d u d u (1+r), q = 1 p =. u d M. Rutkowski (USydney) 6: Multi-Period Market Models 48 / 55

49 Example: Backward Induction Example (6.3 Continued) Recall that if the elementary market model is arbitrage-free then it is also complete and all contingent claims can be priced using the risk-neutral probability. In the first model, the martingale measure is p 1 = 1+r d 1 u 1 d 1 = q 1 = 1 p 1 = 2 3 = 1 3 The price of the digital call option in the considered model equals π1(x) u = 1 ( ) = M. Rutkowski (USydney) 6: Multi-Period Market Models 49 / 55

50 Example: Backward Induction Example (6.3 Continued) In the second model, the martingale measure is p 2 = 1+r d 2 u 2 d 2 = q 2 = 1 p 2 = 2 3 = 1 3 The price of the digital call option in this model equals 0 since its payoff is 0. Formally, π1(x) d = 1 ( ) = M. Rutkowski (USydney) 6: Multi-Period Market Models 50 / 55

51 Example: Backward Induction Example (6.3 Continued) In the last model, the martingale measure is p 3 = 1+r d 3 = u 3 d q 3 = 1 p 3 = = We consider the contingent claim π 1 (X) with the payoff at time t = 1 given by { π u π 1 (X) = 1 (X) = 0.3 if S 1 = u 3 S 0 = 1.5 π1 d(x) = 0 if S 1 = d 3 S 0 = 0.75 M. Rutkowski (USydney) 6: Multi-Period Market Models 51 / 55

52 Example: Backward Induction Example (6.3 Continued) The price of X at time 0 equals π 0(X) = 1 1+r E Q(π 1(X)) = ( ) = Hence π 0 (X) = 0.13, π 1 (X)(ω) = 0.3 if ω {ω 1,ω 2 } and π 1 (X)(ω) = 0 if ω {ω 3,ω 4 }. The unique martingale measure Q in the two-period market model can be recomputed as follows: Q(ω 1) = p 3 p 1 = = Q(ω 2) = p 3(1 p 1) = = Q(ω 3) = (1 p 3) p 2 = = Q(ω 4) = (1 p 3)(1 p 2) = = M. Rutkowski (USydney) 6: Multi-Period Market Models 52 / 55

53 PART 6 COMPLETENESS OF MULTI-PERIOD MARKET MODELS M. Rutkowski (USydney) 6: Multi-Period Market Models 53 / 55

54 Completeness As a handy criterion for the market completeness, we have the theorem, which extends the known result for the single-period case. Theorem (6.1) Assume that a multi-period market model M = (B,S 1,...,S n ) is arbitrage-free. Then M is complete if and only if there is only one martingale measure, that is, M = { P} is a singleton. In the context of a model decomposition, the following statements are known to hold: If all single-period models which compose a multi-period model are arbitrage-free then the multi-period model is also arbitrage-free. If they are also complete then the multi-period model is also complete. The converse of the above statement is also correct. M. Rutkowski (USydney) 6: Multi-Period Market Models 54 / 55

55 Summary of Pricing and Hedging Approaches We examined three pricing and hedging approaches: 1 The method based on the idea of replication of a contingent claim. It can only be applied to attainable contingent claim in a complete or incomplete model and it yields the hedging strategy and arbitrage price process. 2 The method relying on the concept of a martingale measure, which can be used in either a complete or an incomplete model. It furnishes the unique arbitrage price process for any attainable claim and a possible arbitrage price process for a non-attainable contingent claim. In the latter case, the price process depends on the choice of a risk-neutral probability. 3 The backward induction approach in which a multi-period market model is decomposed into a family of single-period models. Pricing is performed in a recursive way starting from the date T 1 and moving step-by-step towards the initial date 0. Hedging strategy can also be computed provided that the claim is attainable. M. Rutkowski (USydney) 6: Multi-Period Market Models 55 / 55