The sale of such a security is a swap of conditional future goods for present goods. What decides the cost of the state contingent security? An arbitrage is thus a money machine or, as in this case, an apple machine. Many of the fundamental concepts of Financial Economics are based on the assumption that markets exist in which claims can be traded efficiently at low cost. Arrow-Debreu Model is built upon two different assumptions. The price of this security is the state price of this particular state of the world. The only commodity is the apple and there is no money per se. By using this particular model, one can easily understand the activities like pricing and hedging that are also related to the derivative analysis.
This diversion completed, we return to our world of non-profit dealers. Later we consider more realistic assumptions about the world. A set of swap terms that does not permit arbitrage is arbitrage-free. How to construct a risk-free asset? This is a perfectly general result. The Arrow-Debreu security is a distinguished concept that is very helpful for understanding the analysis of the derivatives.
Add a final trade in which 0. 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. The price of this security is q W The prices q P and q W are the state prices. With this in mind, we return to Party A and Party B. Too good to be true? Assume that Party A comes to you with the proposal that you sign the agreement as Party B.
To be explicit, we refer to swaps of this latter kind as zero-investment strategies. Well and good, but what if one really wanted apples next year if the weather is good. According to the first assumption, there is enough possibility of an equilibrium of competitive nature but the condition is that everyone in the economy should have at least some amount of every kind of available goods in their holdings. Basis Instrument Contracts is playing a major role in popularizing the Arrow-Debreu security. Thus one invests present apples in the hope of obtaining more apples in the future. At present, Basis Instrument Contracts, that is a kind of derivative contract, is becoming very popular. Now consider a security with state-dependent payouts e.
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. He has purchased a riskless bond. If our agent were , he would pay more to insure against the down state than his net gain from the up state would be. The state price is the vector of state prices for all states. In the field of financial economics, Arrow Debreu represents a certain kind of securities product. To insure an even division, it is wise to agree that one party should cut it, and the other should choose his or her piece.
The boxes on the right represent alternative states of the world. In addition to Dealer G, we assume that another, Dealer B, is willing to trade swap : 0. More generally, the price of any security is the amount of the relevant numeraire paid immediately for which the security can be traded. How might one accomplish this? It incorporates: Utility function Indifference curve Maximization under constraint — a decreasing return investment function only; i. State price Пs can be decomposed into the probability of the state, ps, and the price of an expected dollar contingent on state s occurring or say the state price per unit of probability of associated contingent claims , πs.
You are willing to give up 3 apples if the weather is bad in order to increase your consumption if the weather is good. The numbers inside the boxes indicate the payoffs. In effect, the apple is the unit of currency. Consider two people sharing a pizza. In the real world, of course, dealers charge more to sell a security than they are willing to pay to buy it. In particular, she is willing to trade swap : 0.
In the case of the first description, no state is given, since present values are not conditional on future states of the world. We start with the simplest possible example that involves both time and uncertainty. The model derived its name from two eminent economists Kenneth Arrow from America and Gerard Debreu from France. 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. 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. But what about serving as Party A? 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.
Denote the random variable which represents the state as ω; denote tomorrow's random variable as ω 1. Suppose the agent has a standard function which depends on the state of the world. When an opportunity of this type arises, traders will rush to exploit it, causing others to adjust their terms of trade until swap terms involve no arbitrage. The purchase of an atomic security is a swap of present goods for conditional future goods. In principle, any investment vehicle can be considered to be composed of such atomic claims. Note that each such trade has the characteristic of an investment -- today's goods are traded for the prospect of goods in the future.