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22 2. Single Period Market Models 2.2 The Formal Set-Up Let us define formally the simple case of a single-period model consisting only of a risk-free security, called the bond, and a risky security, called the stock. We then illustrate the economic principles that will also be considered later on when dealing with more complex models. The model is specified by the following elements: • Two trading dates, the initial date t = 0 and the final date T = 1. • A finite sample space ⌦ = {!1 , !2 , . . . , !n } with n < 1. Each ele- mentary event !i , i = 1, . . . , n, represents a ‘state of the world’, also called scenario; the investors at time t = 0 have no information on the outcome, while at time T = 1 they know exactly which scenario has materialized. • A probability measure P on (⌦, P(⌦)), where P(⌦) is the powerset of the sample space, such that P(!i ) := P({!i }) > 04 for all i = 1, . . . , n. • A bank account, called the ‘bond’, with initial value B0 = 1 and deter- ministic final value B1 = 1 + r, where r is the risk-free market interest rate, fixed at time t = 0. • A risky security, called the ‘stock’, whose initial value S0 is known (quoted on the market) and whose final value S1 is a random variable on (⌦, P(⌦)). Definition 2.1. In a single-period market model, a trading strategy is a triple of real constants (V0 , ↵, ), where V0 denotes the value of the initial investment, while ↵ and denote the number of shares of the stock and the units of the bond, respectively, to be held in the portfolio over the time 4 By choosing a non-degenerate probability measure we avoid ambiguities when dealing with equalities of random variables: if X, Y are two random variables on (⌦, P(⌦), P), usually X = Y means equality P-almost surely, while in our setting it means X(!) = Y (!) for all ! 2 ⌦. 2.2 The Formal Set-Up 23 interval (0, 1]. The value of the trading strategy, also called portfolio value, at time T = 1 is V1 = ↵S1 + B1 . (2.2) Note that the investor decides the trading strategy at the initial date, then they apply it over the next time period and wait for the outcome at the final date. We will restrict our consideration to the strategies involving niether cash inflow nor outflow aside the initial investment, therefore the following relation must be satisfied: V0 = ↵S0 + B0 = ↵S0 + . (2.3) This means that the whole and only amount of money invested in the bond and stock over the time interval (0, 1] is exactly given by the initial investment of the strategy. Such trading strategies are called self-financing. Since selffinancing trading strategies are univocally specified by the stock and bond investments ↵, , or equivalently by the initial investment V0 and either ↵ or , henceforth we will shorter the notation and write just (↵, ) for a self-financing trading strategy. Definition 2.2. The gain of a trading strategy (V0 , ↵, ) over the time period (0, 1] is a random variable on (⌦, P(⌦)), defined by G := r + ↵(S1 S0 ). (2.4) By simple rearrangement of the equations (2.2),(2.4), we get the following self-financing condition, equivalent to (2.3), that relates the final value to the gain of a trading strategy: V1 = V0 + G. (2.5) In order to compare prices at di↵erent times, it is convenient to normalize the market values with respect to the bond. Thus, we define the discounted stock price S ⇤ as St⇤ := St , Bt t = 0, 1, 24 2. Single Period Market Models the discounted value V ⇤ of a trading strategy (↵, ) as Vt⇤ := + ↵St⇤ = Vt , Bt t = 0, 1, and the discounted gain G⇤ of a trading strategy (↵, ) as G⇤ := ↵(S1⇤ S0⇤ ). The self-financing condition for the discounted market is the same as for the original market: V1⇤ = V0 + G⇤ . 2.3 (2.6) Economic Considerations and the No-Arbitrage condition As anticipated in Lecture 1, it is desirable that the mathematical model of a financial market satisfies some reasonable economic principles. In the following, we see three related criteria, the strongest of which will be taken as assumption for the models we study. Definition 2.3. A trading strategy with value V is said to be dominant if there exists another trading strategy, with value Ṽ , such that V0 = Ṽ0 and V1 > Ṽ1 . Note that by equalities and inequalities of random variables, we mean that they hold in every scenario, for instance V1 (!i ) > Ṽ1 (!i ) for all i = 1, . . . , n in Definition 2.3. The existence of dominant strategies in the market has a few equivalent conditions. Lemma 2.4. There exists a dominant strategy if and only if there exists a trading strategy, with value V , satisfying V0 = 0 and V1 > 0. Proof. If a strategy as in the statement of Lemma 2.4 exists, it is also dominant, since it dominates the strategy that starts with zero money and does not invest in anything. Conversely, if a strategy (↵, ) dominates a strategy 2.3 Economic Considerations and the NA condition (˜ ↵, ˜), then by defining (¯ ↵, ¯) = (↵, ) 25 (˜ ↵, ˜), we get a strategy as in the statement of Lemma 2.4. Lemma 2.4 describes the situation where an investor starts with zero money and is able to end up with a sure strictly positive gain. Obviously this situation is neither realistic nor reasonable and must be avoided. An analogous reasoning is valid for a strategy as in Lemma 2.5. Lemma 2.5. There exists a dominant strategy if and only if there exists a trading strategy, with value V , satisfying V0 < 0 and V1 0. Proof. Prove it as an exercise. If we interpret the final value of an investment strategy as a contingent claim, the initial value of such investment would be interpreted as the price of such claim. This implies that the existence of a dominating strategy in the market also leads to illogical pricing, because two claims with a surely di↵erent value would have the same price at time t = 0. Instead, the pricing of contingent claims would be logically consistent if the price of any claim was unique and a claim that pays more than another one in every state of the world had a higher price at time t = 0. There is a mathematical formalization of these properties. Definition 2.6. A linear pricing measure is a non-negative vector ⇡ = (⇡1 , . . . , ⇡n ) 2 Rn+ such that the value V of any self-financing trading strategy satisfies V0 = n X ⇡i V1⇤ (!i ). i=1 If there is a linear pricing measure, then the inconsistency disappears. An equivalent way to characterize linear pricing measures is shown in the following lemma and it also gives an explanation for the appellation “measure”. Lemma 2.7. ⇡ is a linear pricing measure if and only if it is a probability measure on (⌦, P(⌦)) and satisfies S0⇤ = n X i=1 ⇡i S1⇤ (!i ). (2.7) 26 2. Single Period Market Models Proof. If ⇡ is a linear pricing measure, by Definition 2.6 we have + ↵S0⇤ = n X ⇡i ( + ↵S1⇤ (!i )) . (2.8) i=1 By taking ↵ = 0, we get Pn i=1 ⇡i = 1, which makes ⇡ a probability mea- sure; then by taking ↵ = 1 we get the equation (2.7). Conversely, if ⇡ is a probability measure satisfying (2.7), then (2.8) holds by substitution. Note that, if a linear pricing measure exists, by (2.7) and Definition 2.6, both the initial stock price and the initial value of any strategy are equal to the expectation under such probability measure of the final discounted stock price and of the final discounted value of the strategy, respectively. This reminds us of the risk-neutral probability measure seen in the introductory examples of Lecture 2. However, the two objects do not coincide, but we will see that the risk-neutral probability is just a special kind of linear pricing measure. Lemma 2.8 shows that models which exclude dominant strategies can be economically reasonable, in that they allow for a logically consistent pricing of claims. Lemma 2.8. There exists a linear pricing measure if and only if there are no dominant strategies. The proof pivots on the linear programming duality theory and we do not show it. The assumption of non-existence of dominant strategies is stronger than the assumption mentioned in Lecture 1 and formalized as follows: Definition 2.9. The law of one price holds if there do not exist two trading strategies, with value V and Ṽ respectively, such that V1 = Ṽ1 but V0 > Ṽ0 . It is clear how this law is crucial to avoid ambiguity about the initial price of any claim. Lemma 2.10. If there are no dominant strategies, then the law of one price holds. 2.3 Economic Considerations and the NA condition 27 Proof. If the law of one price does not hold, then by definition there exist two trading strategies (¯ ↵, ¯) and (˜ ↵, ˜) such that V̄1 = Ṽ1 but V̄0 > Ṽ0 . This implies that G̃ > Ḡ. By defining a new strategy ↵ = ↵ ˜ ↵ ¯, = ↵S0 , this has value V0 = 0, V1 > 0. Thus a dominant strategy exists. Remark 2.11. The converse of Lemma 2.10 is not true. Indeed, you can find a dominant strategy even in a model that satisfies the law of one price. Example 2.12 presents a situation of this kind. Example 2.12. Consider a single-period market model specified by the following data: the sample space contains only two elementary events, ⌦ = {!1 , !2 }, the interest rate is r = 1, the stock price at time 0 is S0 = 10 and takes the following values at time T = 1, depending of the ‘state of the world’: S1 (!1 ) = 12, S1 (!2 ) = 8. In this model, the law of one price is satisfied: for all X = (X1 , X2 ) 2 R2 , there exists a unique trading strategy (↵, ) such that V1 (!1 ) = ↵S1 (!1 ) + (1 + r) = X1 , V1 (!2 ) = ↵S1 (!2 ) + (1 + r) = X2 . However, we can define a dominant strategy by taking ↵ = 1 and = 10, whose value at time 0 is V0 = 0 and at time T = 1 is V1 (!1 ) = 8 > 0, V1 (!2 ) = 12 > 0. Once we agreed upon restricting our attention to market models that do not allow for the existence of dominant trading strategies, a further restriction seems necessary: would you allow for trading strategies that starts with a null initial investment, do not lose any money and end up with a strictly positive gain in at least one (but not all) scenario? If you care for the equilibrium of the market model, certainly not. This situation corresponds to the concept of arbitrage opportunity mentioned in Lecture 1 and formalized below. Definition 2.13. An arbitrage opportunity is a (self-financing) trading strategy with value V such that: (i) V0 = 0, 28 2. Single Period Market Models (ii) V1 0, (iii) EP [V1 ] > 0, i.e. P(V1 > 0) > 0. Lemma 2.14. If there exists a dominant strategy, then there exists an arbitrage opportunity. The statement is trivially true, by definition. Remark 2.15. The converse of Lemma 2.14 is not true. Indeed, you may find an arbitrage opportunity in a model excluding dominant strategies. Example 2.16 presents a situation of this kind. Example 2.16. Consider a single-period market model specified by the following data: the sample space contains only two elementary events, ⌦ = {!1 , !2 }, the interest rate is r = 0, the stock price at time 0 is S0 = 10 and takes the following values at time T = 1, depending of the ‘state of the world’: S1 (!1 ) = 12, S1 (!2 ) = 10. The strategy defined by ↵ = 1, = 10 is an arbitrage opportunity, because: V0 = 0 and V1 (!1 ) = ↵S1 (!1 )+ (1+r) = 12 10 = 2, V1 (!2 ) = ↵S1 (!2 )+ (1+r) = 10 10 = 0. Nevertheless, we can prove that there are no dominant strategies in the market. Indeed: ⇡ = (0, 1) is a linear pricing measure, by Lemma 2.7, then Lemma 2.8 concludes the proof. The set of conditions in Definition 2.13 can be equivalently rewritten in terms of the gain of the trading strategy. Lemma 2.17. (↵, ) is an arbitrage opportunity if and only if it satisfies (i) and its gain G satisfies: (a) G 0, (b) EP [G] > 0. Proof. Suppose that (↵, ) is an arbitrage opportunity. Then, by definition, G = V1 V0 0 and EP [G] = EP [V1 ] V0 = EP [V1 ] > 0. Conversely, suppose 2.3 Economic Considerations and the NA condition that (i), (a) and (b) are satisfied by some trading strategy (↵, ). Then V1 = V0 + G = G 0 and EP [V1 ] = EP [G] > 0. We can resume the relation between the models satisfying the conditions explained in this section as: NA ⇢ No dominant strategies ⇢ law of one price, where we denote by NA the no-arbitrage condition, i.e. the non-existence of arbitrage opportunities. This means that market models can be divided in four categories: there are no arbitrage opportunities, there are arbitrage opportunities but no dominant strategies, there are dominant strategies but the law of one price holds, and the law of one price does not hold. An example of model that does not even satisfy the law of one price is given in Example 2.18. Example 2.18. Consider the market model specified in Example 2.12 but change the specification of the stock price at time T = 1 in the state !2 as S1 (!2 ) = S1 (!1 ) = 12. Then, for any given real constant X 2 R, there exist infinitely many trading strategies whose final value is V1 = X. That is because we have to solve one equation in two variables: V1 = ↵S1 + (1 + r) = X. We want to work with models of the first category, that is satisfying the NA condition. This is of course a theoretical assumption, because in real markets much more frictions are involved and the markets are far from being ideal. Nevertheless, in very liquid markets any arbitrage possibility that arises is quickly eliminated (by all the investors taking advantage of it) and the market equilibrium is restored. So our assumption is reasonable. The NA condition is not easy to be checked, but we will see in future chapters a fundamental theorem that provides us with an equivalent condition involving a special probability measure. 29 30 2. Single Period Market Models 2.4 The Binomial Model on a Single Period A special case of this simple one period model is given by the so called Binomial or Cox-Ross-Rubinstein (CRR) model, from the names of the authors who introduced it in 1979. We will consider here the single-period case of the model, which is in general a multi-period model. We will study in detail the general setting of the binomial model in the next chapter. Assumption 2.4.1 (Single-period binomial model). The initial stock price S0 is strictly positive and the stock price at the final date T = 1 is proportional to the initial stock price, in the following way: S1 = S0 ⇠, where ⇠ is a Bernoulli-like random variable, taking only two values u, d, u > d > 0, with probabilities P(⇠ = u) = p, P(⇠ = d) = 1 p. The sample space is the path space, that is the set of possible trajectories for the stock price: ⌦ = {(S0 , S0 u), (S0 , S0 d)}. Definition 2.19. A European contingent claim (ECC) is represented by a random variable X on (⌦, P(⌦)) and it is said to be attainable if there exists a self-financing trading strategy with final value V1 = X. Such trading strategy, if it exists, is called a replicating, or hedging, strategy for X. Now, given a European claim X, consider the problem of hedging, that is to find the hedging strategy for X, if it exists. Since the claim is a random variable on the path space, it can take only two values, which we denote by X u in the scenario (S0 , S0 u) and X d in the scenario (S0 , S0 d). By imposing the replicating condition, we get: ( ↵S0 u + (1 + r) = X u ↵S0 d + (1 + r) = X d 2.4 The Binomial Model on a Single Period Solving for ↵, 31 we find Xu Xd , (u d)S0 uX d dX u = (1 + r) 1 u d Thus the initial investment needed to replicate the derivative is ↵ = (2.9) (2.10) V0 = ↵S0 + = 1 (1 + r)(u = (1 + r) 1 d) (1 + r qX u + (1 d)X u + (u (1 + r))X d q)X d where 1+r d . (2.11) u d Note that q 2 (0, 1) if and only if d < 1 + r < u. In this case, we can define q= a probability measure Q by Q(⇠ = u) = q, Q(⇠ = d) = 1 q, (2.12) and the initial value of the hedging strategy can be rewritten as V0 = EQ [X ⇤ ] , (2.13) where X ⇤ = (1 + r) 1 X is the discounted claim. Furthermore, the initial stock price can also be rewritten as the expectation under Q of the discounted final stock value: S0 = EQ [S1⇤ ]. Thus, Q is the unique risk-neutral probability of the model. The appellation “risk-neutral” refers to the fact that, under Q, a risk-neutral investor, i.e. an investor who has no preference between the expected value of a payo↵ and the random payo↵ itself, is indi↵erent to investing either on shares of the stock or on bonds, since they have the same expected payo↵. Note that the risk-neutral probability is only a theoretical object and it is unrelated to the market (or real-world, or objective) probability P that specifies the dynamics of the stock price in the model. We remark that the only property shared by P and Q is that they both assign a strictly positive value to every possible event. 32 2. Single Period Market Models Theorem 2.20. The single-period binomial model satisfies the NA condition if and only if d < 1 + r < u. Proof. We leave the proof of the implication ) of the statement for homework. (() If d < 1+r < u, we have already proved that we can define a probability measure Q by (2.11) and that under this measure the initial stock price S0 is equal to the expected value of the discounted stock price at time T = 1. Suppose that an arbitrage opportunity (↵, ) exists, by Definition 2.13, it must satisfy the conditions (i)-(iii). Now, (i) translates to V0 = ↵S0 + =0 and (ii) to V1 = ↵S1 + (1 + r) 0. By taking the expectation under Q, we get EQ [V1 ] = ↵EQ [S1 ] + (1 + r) = ↵(1 + r)S0 + (1 + r) = (1 + r)V0 = 0. Thus, condition (iii) cannot be satisfied: we found a contradiction. Henceforth we assume that the model satisfies the NA condition: Assumption 2.4.2. We consider a single-period binomial model with d < 1 + r < u. We have also seen that for any given European claim X there exists a hedging strategy, defined in (2.9)-(2.10). By the lemmas proved in Section 2.3, the law of one price holds, hence the time 0 value of the hedging strategy determines the price of the derivative. At this point, another concern arises: by introducing the derivative security X in the market, with price determined by (2.13), are we introducing a possibility for arbitrage opportunities? We would like that the price of the derivative is such that also the secondary 2.4 The Binomial Model on a Single Period market composed of the bond, the stock, and the derivative itself, is free of arbitrage opportunities. In this case, the price of the derivative is called the arbitrage-free price. Fortunately, this is true for the initial value of the hedging strategy: Theorem 2.21 (Thm 2.2.1 in ?). V0 = EQ [X ⇤ ] is the unique arbitrage-free price for the European contingent claim X. We defer the proof to the multi period case. 33