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- 1. ECO568 - Uncertainty and Financial Markets Arthur Charpentier and Alfred Galichon Ecole polytechnique, January 6 2009 1. No-Arbitrage and the Law of One Price
- 2. General Introduction (1) Features of Financial Economics Abundance of …nancial data, empirical testability. Interactions, collective behavior. Equilibrium. Partial vs. General: how do you trust your model versus the market?
- 3. General Introduction (2) Rationality Preferences + beliefs => choices "Rational" preferences. - “more” is always preferred to “less” - really? - Risk-aversion: people buy insurance - but they also purchase lottery tickets Rational beliefs: agents’subjective probabilities = objective probabilities (stationarity; no learning, no insider information, no overcon…dence)
- 4. General Introduction (3) S. Ross’"three stylized facts" about …nancial markets: E¢ cient Markets. Asset returns are unpredictable. Then what is this course useful for? Risk Premium. Risky assets have higher expected returns (ie. Lower prices) than safe ones. By how much? Correlation. Asset returns are cross-correlated. How to use it? We shall come up with a consistent theory accounting for these facts.
- 5. Course outline Lecture 1 (Jan 6): No Arbitrage and the Law of One Price Lecture 2 (Jan 13): The Demand for Risk Lecture 3 (Jan 20): Investment Theory (Static case): MVA, CAPM Lecture 4 (Jan 21): Informational aspects Lecture 5 (Feb 10) Investment Theory (Dynamic case, discrete time) Lecture 6 (Feb 17): Investment Theory (Dynamic case, continuous time) Lecture 7 (March 3): Term Structure Models Lecture 8 (March 10): Empirical Puzzles in Asset Pricing Lecture 9 (March 17): Risk management and Financial bubbles Exam: March 24 Course webpage: http://groups.google.com/group/x-eco568.
- 6. Course organization The class is organized as 1h45 lecture, then 1h45 section ("petite classe"). The lecture slides are distributed in class, and posted online after each lecture. The "Polycopié de cours" by G. Demange and G. Laroque is avaibable from the Scolarité. Students are advised to look at it, but the lecture slides are self-contained. Oral participation is strongly encouraged!
- 7. Today’s lecture outline General introduction No-Arbitrage and the Law of One Price Section: Arbitrage Pricing Theory, Cox-Ross-Rubinstein
- 8. The setting contingent states at time t = 1: ! 2 f1; :::; g. Examples. ! 2 f0; 1g (no accident/accident); ! value the stock market at time t = 1. K assets available at time t = 0: k 2 f1; :::; Kg. the tableau a! k is the value of asset k in contingent state !, pk price of asset k at time t = 0. A market is the data a! k ; pk . Portfolio: combination of assets, zk quantity of asset k. Portfolio price (t = 0): P k pkzk. Portfolio value (t = 1): P k a! k zk.
- 9. Example. Consider the following tableau, where lines correspond to assets, and columns to states a = 0 1 2 1 1 0 ! ; p = 3 1 what is the value of portfolio of 5 assets k = 1 and 2 asset k = 2 in the state ! = 2?
- 10. Arbitrage opportunity De…nition. An arbitrage portfolio is a portfolio zk such that P k a! k zk 0 for all ! and P k pkzk 0, at least one of these E + 1 inequalities being strict. A market is arbitrage free if there is no arbitrage portfolio. Proposition. A market a! k ; pk without arbitrage opportunity satis…es the Law of One Price: if for two assets k and l, a! k = a! l for every state !, then the assets have the same price: pk = pl. Example. Consider a = 0 B @ 2 0 1 1 2 4 3 2 5 1 C A ; and p = 0 B @ 2 1 2 1 C A. The portfolio z = ( 1; 1; 1) is an arbitrage portfolio, as P k a! k zk = 0 for every !, and P k pkzk = 1.
- 11. State-price vectors and the No-Arbitrage theorem The No-Arbitrage Theorem. A market a! k ; pk is arbitrage-free if and only if there is a vector q! such that: - q! > 0 for every state ! - pk = P ! a! k q! for every asset k q! is then called a state-price vector. Remark. 1. Although q! is in general not unique, this does not contradict the law of one price, as every state-price vector q! yields the same set of asset prices. 2. This notion does not involve the "statistical" probability per se (apart from its support).
- 12. Example. Consider a = 0 B @ 2 0 1 1 2 4 3 2 5 1 C A ; and p = 0 B @ 1 1 2 1 C A. The market is arbitrage- free, and q = (3=7; 0; 1=7) is a state-price vector. Sketch of the proof of the No-Arbitrage theorem. ( is easy: P k pkzk = P k;! q!a! k zk = P k;! q! a! k zk , thus P k a! k zk 0 implies P k pkzk 0. For the converse, we note that the investor is looking for the portfolio with minimal cost which gets him a positive revenue with certainty in every state of the world. The investor solves V = infz P k zkpk s.t. P k zka! k 0, and V is to be interpreted as the sure gain of the investor. By positive homogeneity, V = 0 or V = 1, and there is No arbitrage opportunity if and only if V = 0. Now, write the Lagrangian V = infz supq 0 P k zkpk P !;k zka! k q!, where
- 13. q! are the Lagrange multipliers associated to the constraints. But by duality, one can invert the inf and the sup, and V = supq 0 infz P k zk pk P ! a! k q! . Therefore there is No arbitrage opportunity if and only if there is a vector q such that inf z X k zk pk X ! a! k q! ! = 0 hence P ! q!a! k = pk for all k. It remains to explain why one can choose q! > 0. Remember, we have introduced the q! as Lagrange multipliers in the investor’s constrained optimization problem. The null portfolio z = 0 is a solution to the constrained optimization problem which has all the constraints saturated. By a well-known result on linear programming, all the corresponding Lagrange multipliers q! can be chosen strictly positive. Remark: Game-theoretic interpretation. The proof has a game-theoretic interpretation as a game between Investor and the Market, which sets the state
- 14. prices in order to minimize the Investor’s gain. The Market’s strategy is to choose a state-price vector q!, while the Investor’s strategy is to pick up his investment portfolio zk. The value of this game (for the Market) is then: V = inf z sup q 0 X k zk pk X ! a! k q! ! V is interpreted as the worse outcome for the Market facing a rational Investor. This is a zero-sum game: the value of the game for the Investor is V , the opposite og the value of the game for the Market. The duality principle is equivalent to a min-max theorem, which precisely says that the value of this game will be the same regardless whom (of the Investor or the Market) plays …rst.
- 15. Consequence: Arrow-Debreu prices De…nition. An Arrow-Debreu asset is an asset yielding 1 in state !0, and 0 otherwise. Denote a! !0 = 1f!=!0g. Proposition. If q is a state-price vector, then the price of a! !0 is q!0. Moreover, if there exists a portfolio yielding 1 in state !0 and 0 otherwise, then its price equals q!0. Example: Digital options, call options. Suppose a stock price a! can take up to a …nite number of values v1; :::; vN. Then the digital option of strike vk is the option d! vk = 1fa!=vkg. This is precisely an Arrow-Debreu price.
- 16. Risk neutral probability Denote k = 0 the riskless asset, ie. a! 0 = 1 for every state !, and call p0 the price of that asset. Introduce r the riskfree return r such that 1 + r = 1 p0 . By the law of one price, p0 = P ! q!, thus 1 = P ! q! (1 + r). Denoting ! = q! (1 + r), one has ! 0 and P ! ! = 1, thus ! can be interpreted as a probability: this is the risk neutral probability. One has p = 1 1+r P !;k !a! k zk = 1 1+rE! hP k a! k zk i , where P k a! k zk is the portfolio contingent value at t = 1. This probability has no reason to coincide with the "statistical" probability, the di¤erence comes from the agents risk aversion. More on this soon! One de…nes r! k = a! k pk the return of asset k in state !. One has for each k, P ! !r! k = 1+r, thus every return have the same risk-neutral probability, which is the riskless return.
- 17. Example: Call option. The riskfree asset is a! 0 = 1 + r with unit price, and consider the stock a! 1 with initial price S, where we suppose that the world has only two states, ! = h in which case ah 1 = S (1 + h), and ! = h in which case al 1 = S (1 + l). Suppose (1 + h) S > K > (1 + l) S, and consider the call option c! on a1 with strike K. One has c! = a! 1 K + , so c! = (1 + h) S K if ! = h, and c! = 0 if ! = l. One looks for the risk-neutral probability !. One has l (1 + l)+ h (1 + h) = 1+r, thus h = r l h l. Thus the call price is C = h 1+r ((1 + h) S K) = r l (1+r)(h l) ((1 + h) S K). Alternatively, one could also have looked for the value of a replicating portfolio to hedge the value of the call. Call (z0; z1) such a portfolio; one has z0 (1 + r) + z1 (1 + h) S = (1 + h) S K z0 (1 + r) + z1 (1 + l) S = 0
- 18. which solves into z0 = (1+l)[(1+h)S K] (1+r)(h l) and z1 = (1+h)S K (h l)S . By the law of one price, the call value equals the initial value of the replicating portfolio, thus C = r l (1+r)(h l) ((1 + h) S K).
- 19. Complete markets De…nition. An asset class a! k is complete if for every contingent payo¤ c!, there exists a portfolio zk such that c! = P k a! k zk. Proposition. An asset class a! k is complete if and only if the rank of the matrix a! k is equal to the number of states . In that case, the state-price vector q! is unique, and a contingent payo¤ c! has price p = P ! q!c!. Important consequences: 1. When the markets are complete, there are at least as many assets than there are states: K. 2. One can therefore eliminate E redundant assets which are linear com- binations of independent assets, and suppose in practice that = K.
- 20. Example 1. Consider a = 0 B @ 2 0 1 1 2 4 3 2 5 1 C A ; and p = 0 B @ 1 1 2 1 C A. The market is arbitrage-free, but not complete as the …rst two columns sum up to the third one. Example 2: options render the market complete. Consider a single asset a = 5 2 3 with price 2. Consider the market made of a!, and call options of strike 2 and 3, of payo¤ respectively (a! 2)+ and (a! 3)+ . The market tablean can be written 0 B @ 5 2 3 3 0 1 2 0 0 1 C A, and the asset prices are 0 B @ 1 p p0 1 C A. Provided p0 < p < 1, the new market is complete and arbitrage free.
- 21. Arbitrage bounds* Proposition. Given an arbitrage-free market M = a! k ; pk , de…ne PM the set of probabilities wich are risk-neutral probabilities for this market. (i) PM is a convex set. (ii) PM is empty if and only if M o¤ers arbitrage opportunities. (iii) PM is reduced to a point if and only if the market M is complete. (iv) If and 0 2 PM, then 0 ? M, that is P ! 0 ! a! k = 0 for all k.
- 22. We introduce the notion of replicable claim. De…nition. Given a contingent claim c!, one says that c! is replicable in the market M if there exists a portfolio z such that c! = X k zka! k for all !. We use the following result to introduce the notion of arbitrage price bounds: De…nition. For a general contingent claim c!, not necessarily replicable in the market M, de…ne the lower and upper arbitrage price bounds of the claim as v# (c) = 1 1 + r # (c) and v" (c) = 1 1 + r " (c) , where # (c) = inf 2PM fE ! [c!]g and " = 1 1 + r sup 2PM fE ! [c!]g :
- 23. We conclude with a result relating the notion of arbitrage price bounds and the notion of replicability. Theorem. The claim c! is replicable in the market M if and only if # (c) = " (c). The set of No-Arbitrage prices for c! is given by h v# (c) ; v" (c) i . Remark. If the claim c! is not replicable, then adding it to the market with a price within arbitrage bounds will reduce the set PM: its dimension will decrease by one.
- 24. Dynamic arbitrage In the dynamic case, one observes the evolution of the assets over time, and one can rebalance one’s portfolio over time. We take the time steps to be discrete. Call a!jt the value of asset a at time t and in state !, where !jt = !0!1:::!t (each time step brings on a new piece of randomness). One writes !j (t + 1) = (!jt; !t), and we shall suppose there is no uncertainty at date t = 0, namely !0 has only one value. A strategy is the description of a portfolio in the dynamic case, allowing re- balancing over time: formally a strategy zk (!jt) is the composition of the investor’s portfolio at time t in the state of the world !. Note that a strategy can depend on !0, !1,..., !t, but not on !t+k: investors can have memory,
- 25. but cannot predict the future. The portfolio (liquidation) value at time t is P k zk (!jt) a !jt k . An arbitrage opportunity is a strategy z such that P k zk (!0) a !0 k 0 and P k zk (!jt 1) a !jt k P k zk (!jt) a !jt k . The interpretation is that such strategy has a negative price and allows positive consumption c !jt z = X k zk (!jt 1) a !jt k zk (!jt) a !jt k 0 at each time step. De…nition. Markets are said to be dynamically complete if for every contingent claim c!jt, there exists a portfolio z such that c !jt z = c!jt.
- 26. Theorem. The three following conditions are equivalent: (i) There is no arbitrage (ii) There is a vector q!jt > 0 such that P k zk (!0) a !0 k = P !jt q!jtc !jt z , where c !jt z is de…ned as above (iii) For every !jt, there exists a vector q!t+1 (!jt) > 0 such that for all k, a !jt k = P !t+1 a !jt;!t+1 k q!t+1 (!jt). Remark. One sees that q!jt and q!t+1 (!jt) are related by q!jt = q!t+1 (!jt) q!t (!jt 1) :::q!1 (!j0), and q!t+1 (!jt) = q!jt+1 q!jt .
- 27. Reference for the course Campbell, J., Viceira, L. Strategic Assets Allocation, Oxford. Demange, G., Laroque, G., Finance et Economie de l’Incertain, Economica. Ingersoll, J., Theory of Financial Decision Making, Rowman & Little…eld. Mas-Colell, Whinston, Green, Microeconomic Theory, Oxford. Ross, S., Neoclassical Finance, Princeton. Thank you!

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