Moreover, there is no reason to settle for such a small gain. The boxes on the right represent alternative states of the world. Arrow-Debreu Model is capable of analyzing the exact situations of those markets that are very competitive. The names of the states are indicated at the tops of the boxes. We focus initially on a tree that will produce: 63 apples if the weather is good 48 apples if the weather is bad This is shown in the figure below. A particular example is the risk-free asset this replicates the first point of slide 5 It has a payoff of 1 in each state of nature at the end of the period We have the following: L3: Arrow-Debreu Securities Risk Neutral Probability and P Solving exercise 5. Note that the ability to make a trade is central to the definition of a price.
In standard parlance, this piece of paper or its electronic equivalent would be termed a security. The box on the left refers to present value. Denote the random variable which represents the state as ω; denote tomorrow's random variable as ω 1. The Arrow-Debreu security is a distinguished concept that is very helpful for understanding the analysis of the derivatives. Lecture 5: Arrow-Debreu Economy The following topics are covered: Arrow-Debreu securities Pricing Arrow-Debreu securities Optimal portfolios of Arrow Debreu securities How Arrow-Debreu securities differ framework differs from the standard utility maximization? What decides the cost of the state contingent security? Quadruple them and the gain is quadrupled.
Thus a set of atomic security prices is sufficient for accomplishing any desired trade. To formalize: An arbitrage provides a positive net payoff in at least one time and state and no negative net payoff in any time and state. And what would be the terms of trade? Well and good, but what if one really wanted apples next year if the weather is good. The only commodity is the apple and there is no money per se. Note that the latter two descriptions include the item apple , the number of units one , the time at which delivery is to be made 0 or 1 , and the state of the world that must obtain for delivery to be made good or bad weather. Now consider a security with state-dependent payouts e. To answer these questions, consider the following agreement: Party A promises to pay party B: 6 apples if the weather is good Party B promises to pay party A: 3 apples if the weather is bad Neither party pays the other anything today on signing Such an agreement is called a swap in financial parlance.
What would it mean to trade good weather apples for bad weather apples? Hence the often-used name: the time-state paradigm. If one desires an answer based on ethical considerations, other disciplines will have to be invoked. For detailed computations, of course, the specifics of a proposed transaction may need to be taken into account and the relevant price bid or ask used. Time proceeds from left to right. The numbers inside the boxes indicate the payoffs.
Each projector can be addressed as a strategic marker of a game path, and leads to the matrix form of an Arrow-Debreu security, therefore, we call it game Arrow-Debreu projector. But what about serving as Party A? It characterizes promised future payments in terms of both the times at which payments are to be made and the states of the world that must obtain for payments to be made. How about adding an additional asset 0,1,3? Later we consider more realistic assumptions about the world. We begin with the assumption that dealers stand ready to trade atomic claims and to do so without cost. The more likely a move to W is, the higher the price q W gets, since q W insures the agent against the occurrence of state W. The seller of this insurance would demand a higher premium if the economy is efficient.
In addition to Dealer G, we assume that another, Dealer B, is willing to trade swap : 0. But what of the other possible type of trade in this world? This example constitutes an arbitrage -- every trader's dream. A set of swap terms that does not permit arbitrage is arbitrage-free. Too good to be true? Add a final trade in which 0. In , a state-price security, also called an Arrow-Debreu security from its origins in the , a pure security, or a primitive security is a contract that agrees to pay one unit of a a currency or a commodity if a particular state occurs at a particular time in the future and pays zero numeraire in all the other states.
In the real world, of course, dealers charge more to sell a security than they are willing to pay to buy it. The only type of productive investment in this economy is, not surprisingly, an apple tree. Try the sum of state prices 0 and 1. In our examples we assume unrealistically that there is no such spread and hence that there is but one price. Analogously, for a indicating a continuum of possible states, the value is found by over the. With this in mind, we return to Party A and Party B. While, in traditional financial economics, the Arrow-Debreu securities pay one unit of numeraire per state of nature, in the present game setting, they pay one unit of payoff per game path at the beginning of the game, however this analogy may be taken it must be addressed with some care, since these are not securities, rather, they represent, projectively, strategic choice chains in the game, so that the price of a projector P ˆ f n…21 the Arrow-Debreu price is the price of a strategic choice and, therefore, the result of the strategic evaluation of the game by the different players.
Basis Instrument Contracts is playing a major role in popularizing the Arrow-Debreu security. But is 6 apples the best that you can do? The price of this security is q W The prices q P and q W are the state prices. How might one accomplish this? Note that each such trade has the characteristic of an investment -- today's goods are traded for the prospect of goods in the future. In principle, any investment vehicle can be considered to be composed of such atomic claims. At present, Basis Instrument Contracts, that is a kind of derivative contract, is becoming very popular.
According to the second assumption, there is a huge amount of labor resources available in the market that can be utilized for producing the required commodities and services. Is this a fair deal? This can better be understood in standard financial terms. The spread between the ask selling and bid buying price provides compensation for the market-making function. . Suppose the agent has a standard function which depends on the state of the world. One but only one of the states of the world in a vertical position will take place at the time in question. On the other hand, the Arrow-Debreu Model is also used in areas like financial engineering and many more.