The Framework of the Arrow Debreu Model Finance Essay

The Arrow-Debreu Model, also referred to as the Arrow-Debreu-McKenzie model is often used as a general reference for other microeconomic models. It was named after Kenneth Arrow, Gerard Debreu and Lionel W. McKenzie. When compared to prior models, the Arrow-Debreu Model radically generalized the notion of a commodity, differentiating them by time and place of delivery. Hence, for instance, 'rice in India in July" and 'rice in Bangladesh in September" are regarded as different commodities. The Arrow-Debreu Model usually applies to economies with complete markets, in which there exists a market for every time period and forward prices for all commodity at all times and in all places. The Arrow-Debreu Model states that aggregate supplies will equal aggregate demand for every commodity in the economy, should the following assumptions be made: Convexity is the measure of the curvature in the relationship between price and interest. It is often used as a risk management tool to help measure and manage the amount of market risk to which a portfolio may be exposed to. The relationship between the price and the yield is generally inversely correlated, meaning that when the price of the product is high, it yields lower interest rates. Perfect Competition In economic theory, perfect competition is achieved when there are no participants large enough to have the market power to set the price of a homogeneous product; this is as defined by Roberts, J. (1987). We understand that every participant in the market is a price taker and there are no partial influences on the price of the product, the price is determined solely on the demand and supply of the product. This demand and supply is determined by the forces of the market. Although there are very few perfectly competitive markets, if any, out there, some buyers and sellers in some auction type markets satisfy the concept. Demand Independence This means that the demand for the product is independent of any conditional factors in the market. For instance, the demand is not conditional to the supply of another product or even any seasonal changes. The Arrow-Debreu Model is a central model in the General Equilibrium Theory. This theory studies the supply and demand fundamentals in an economy, with the objective of proving that all prices are at equilibrium. Developed by economist Leon Walras, this theory studies a theoretical economic system in which all consumers are utility maximizers and firms are perfectly competitive. The model further goes on to prove that such a stable, unique equilibrium can exist under these conditions. (https://www.economyprofessor.com/economictheories/general-equilibrium-theory.php) Mainly, the ADM is built on two distinct assumptions: Equilibrium of competitive nature is possible on the condition that everyone in the economy should have at least some amount of every kind of available goods in their holdings. Secondly, there is a huge amount of labour resources available in the market that can be utilized for producing the required commodities and services. In the field of financial economics, Arrow Debreu represents a certain kind of securities product. The Arrow-Debreu security is a distinguished concept that is very helpful for understanding the analysis of the derivatives. By using this particular model, one can easily understand the activities like pricing and hedging that are also related to the derivative analysis. On the other hand, the Arrow-Debreu Model is also used in areas like financial engineering and many more. Although the theory was criticized by various eminent economists, but the truth is that the Arrow-Debreu Model is very important for the derivative industry and helps the industry to grow at a rapid pace.A At present, Basis Instrument Contracts, a kind of derivative contract, is becoming very popular. This concept is tractable enough and is providing the Black Scholes' analysis a new dimension. At the same time, the Basis Instrument Contracts is also applying the analysis of Black Scholes' in different markets. Basis Instrument Contracts is playing a major role in popularizing the Arrow-Debreu security.

FRAMEWORK OF THE ARROW-DEBREU MODEL

In order to understand fully the mathematical structure of the Arrow-Debreu Model, we explore in more details the Arrow-Debreu securities and the Basic Instrument Contract in order to understand how the model applies in reality.

Arrow-Debreu Securities

Arrow-Debreu securities are assets that payoff in a specific state of nature at a certain time (LeBaron, 2005). They are useful building blocks to think about other assets even though they do not exist in reality. In order to understand these securities, we look into the following example: Let us assume that S is the set of all states that the world can be in tomorrow. Hence, for each s in S, there is a corresponding Arrow-Debreu security that pays off 1 if s happens, or otherwise 0 is paid. For instance, if s could be; Sunny tomorrow and No rain and S1 is at $4 and S2 at $5 and then if s happens then you will receive $4. Although this is not practical for a large number of securities, it is possible to apply this to a limited number. Suppose Adam wants to receive: State 1- $6, State 2- $8 and in State 3- $25. All Adam needs to do is buy 6 Arrow-Debreu securities for state 1, 8 for state 3 and 25 for state 3. With Arrow-Debreu securities we can replicate this stock to build a portfolio. Furthermore, with arbitrage principles we know that the price of any two things paying the same amount in every state should be the same (LeBaron, 2005). This would lead the price of the stock to be the same as the price of the Arrow-Debreu.

Basic Instrument Contract (BICs)

The basic instrument contract is an important contract when analysing financial derivatives. It is a form of representative derivative contract that helps the derivatives to be replicated in a static manner (https://finance.mapsofworld.com/finance-theory/concepts/basis-instrument-contract.html). This can be done in a market where there are many periods of trading. In order to make the definition more comprehensible, we need to first stress on the characteristics of BICs (https://knol.google.com/k/introduction-to-basis-instruments-contracts-bics-for-mathematics-finance-and#Defining_BICs): A BIC involves and identifies two parties; one or more buyers (named B) and one or more sellers (named A) The definition of each contract comprises of three dates; firstly, the contract agreement date- the date at which the binding rights and obligations are agreed upon. Secondly, the premium payment date [t (i)] - the date at which B complies with his part of the agreement by paying seller A an amount of units (known as premium of contract). Thirdly, the contract expiry date- the payout payment date [t(j)] at which the seller A pays the buyer an amount in units of basis currency The agreements on the hedging BICs would have been contracted before the lapse of liquidity; this would ensure that there is no risk that the position cannot dynamically hedge. Furthermore, a BICs market would help structurally mitigate market volatility in market and reduce transaction costs because hedging activity is substantially reduced to the settlement of agreements contracted beforehand. Hence we can see here that BICs generally uphold the concept of the Arrow-Debreu model with a greater degree of realism. Trading BICs will allow agents to be involved with the Arrow-Debreu model. This is because both of them are involved with only two possible outcomes, you either receive money or you don't and also both of them do not allow for riskless arbitrage modelling.

SHORTCOMINGS OF THE MODEL

It has become rather pass© to criticize the Arrow-Debreu model on empirical grounds, i.e. by showing how the results are not robust under certain small relaxations of idealized assumptions. At every turn, the ADM assumes some unrealistic assumption in order to realize the logical structure of the model.

Non-Convexities

Radner (1968) extended the Arrow-Debreu model to include agents with differing information about the economy. He found that when information to the environment, the Arrow-Debreu contingent claims equilibrium can achieve an optimum. This is relative to a given structure of information. However, as Radner points out, if the agent receives information about the trading behaviour of other market participants, than externalities arise. These externalities often distort preferences or may proceed to diminish the optimality of the competitive equilibrium. It is particularly important to note that the set-up cost of gathering information (may be independent of the scale of production) can introduce non-convexity into the production possibility set. As we know that non-convexities violate the basic assumption of Arrow-Debreu model.

Uncertainty about Market Prices

Another issue arising from the Arrow-Debreu model is the concern of whether ex ante optimally or ex post optimality was the appropriate measure of efficiency. When regarding practicality, Arrow's optimum is a normative dead end. Given Arrows claim on optimal distribution and supposing the occurrence of some event, Guth (2007) asked whether in the even to f the distribution of real goods resulting from the given distribution of contingent claims is a Pareto optimal distribution of real goods. Starr (1973) claims that for the pure exchange economy, the Arrow-Debreu equilibrium will be ex Post Pareto optimal. A Pareto optimal outcome is such that no-one can be better without making someone else worse off (https://moneyterms.co.uk/pareto-optimal/). This is only possible if all the market participants assign the same probability value to the given state (s) occurring. In this context, Starr refers to this property as universally similar beliefs. Hence, this means that the uncertainty and information about the environment and the behaviour of other market participants, proves that the Arrow- Debreu model has some shortcomings as one of its main assumptions is that the price is certain.

CONCLUSION AND FUTURE IMPLICATIONS

This paper attempts to explore the theoretical framework of the Arrow-Debreu model. The model imposes the limitations of (1) assuming that locational choices are made in the context of a capitalist economy and (2) excluding governmental or more generally, collective action. Within the framework of the model, it has been procured that general equilibrium among market participants and factors can exists, when the implications are put into practice. The Arrow-Debreu model is one that is based on a set of assumptions that is rather difficult to find in today's world. Formulated into a purely mathematical form, the Arrow-Debreu model can be easily modifies into spatial models with proper definitions of the commodities based on the commodity's location or time of delivery. Furthermore, when commodities are specified to be conditional on various states of the world, the model can be easily incorporated with expectations and uncertainties. This model has been subject to criticism that many of the assumptions it makes does not fit in the real economy. However it is important to note that the criticism is not unique to the Arrow-Debreu model but rather it applies to all general equilibrium models. There are many ways of improving the already proven flaw of the Arrow Debreu Model. One such way is provided by Radner. Radner (1985) in his work explains that the deficiency of Arrow-Debreu theory as inadequate treatment of money, the stock market, and active markets at every date can be improved for better results. In order to correct these deficiencies, he explains that for future extensions of the Arrow-Debreu model should include; Uncertainty about future prices as well as uncertainties about the environment A method for producers to compare net revenues at different dats and across states of the world Consumers facing a sequence of budget constraints over time, rather than the single present net worth budget constraint of the Arrow-Debreu model. Speculation in future markets by storage, hedging, etc. Agents attempt to forecast future prices based on information about the environment and other market participants behaviour up until that point in time. Through these changes, we will be able to see a more real-life implication of the ADM, thus giving us more insight to the model. The model is one that, although correct from where it stands, is not able to be emulated to fit the practices of the real economy as of today. The assumptions, on which the Arrow-Debreu model is based, are very rare to find, for instance, a perfectly competitive market is rather impractical in today's reality. Hence, we are in need for more economists to work on the Arrow-Debreu model to modify it to suit reality as it is today.

Risk Neutral Valuation/Pricing

In finance we can estimate a real option's value by using either Risk neutral valuation approach or Binomial/decision trees. Risk Neutral Valuation is a straight forward and essential in option pricing theory. The principle states that we can with complete impunity assume the world is risk neutral when pricing option (Hull, 2006). The resulting prices are correct not just in a risk neutral world, but in other worlds as well. For example: let us illustrate that risk neutral valuation gives the same answer as no arbitrage arguments. Suppose the stock price is currently $30 and will move either up to $32 or down to $28 at the end of 3 months. The option considered is a European call option with a strike price of $31 and an expiration date in 3 months. The risk-free interest rate is 12% per annum. Let us define p as the probability of an upward movement in the stock price in a risk-neutral world. We can argue that the expected return on the stock in a risk neutral world must be the risk free rate of 12%. This means that p must satisfy 32p+ 28(1-p) = 30e0.12*3/12 or 4p= 30e0.12*3/12- 28. That is, p must be 0.6523. At the end of the 3 months, the call option has a 0.6523 probability of being worth 1 and a 0.3477 probability of being worth zero. Its expected value is therefore 0.6523*1 + 0.3477*0= 0.6523. In a risk neutral world this should be discounted at the risk free rate. The value of the option today is therefore 0.6523e-0.12*3/12 or $0.633. This is the same as the value obtained earlier, demonstrating that no arbitrage arguments and risk neutral valuation give the same answer. If we put it in a two step tree diagram: 34.2 32 30 29.8 28 26.2 It should be emphasized that p is the probability of an up movement in a risk neutral world. In general this is not the same as the probability of an up movement in the real world. In our example p= 0.6523. When the probability of an up movement is 0.6523, the expected return on both the stock and the option is the risk free rate of 12%. Suppose that, in the real world, the expected return on the stock is 16% and p* is the probability of an up movement. It follows that 32p*+ 28(1-p*) = 30e0.16*3/12 so that p* = 0.7041. The expected payoff from the option in the real world is then given by p* x 1 + (1-p*) x 0. This is 0.7041. Unfortunately it is not easy to know the correct discount rate to apply to the expected payoff in the real world. A position in a call option is riskier than a position in the stock. As a result the discount rate to be applied to the payoff from a call option is greater than 16%. Without knowing the option's value, we do not know how much greater than 16% it should be. Using risk neutral valuation is convenient because we know that in a risk neutral world the expected return on all assets (and therefore the discount rate to use for all expected payoffs) is the risk free rate. Besides having a connection with the binomial model, risk neutral valuation also arises from one key property of the Black Scholes Merton differential equation. This property is that the equation does not involve any variables that are affected by the risk preferences of investors. The variables that do appear in the equation are the current stock price, time, stock price volatility and the risk free rate of interest. All are independent of risk preferences. The Black Scholes Merton differential equation would not be independent of risk preferences if it involved the expected return, Aµ, on the stock. This is because the value of Aµ does depend on risk preferences. The higher the level of risk aversion by investor, the higher Aµ will be for any given stock. It is fortunate that Aµ happens to drop out in the derivation of the differential equation. Because the Black Scholes Merton differential equation is independent of risk preferences, an in In some case there is an existing model for a financial option that corresponds to the real option in question. Sometimes, however there is not such a model and financial engineering techniques must be used. Many financial engineering methods are very complex and are more suitable to be explained in an advanced finance course. On the other hand, Risk Neutral valuation is one of the methods which are reasonably easy to implement with simulation analysis. It is also similar to the certainty equivalent method in that a risky variable is replaced with one that can be discounted at the risk free rate. For example: Cordova Software, which produced antivirus software, is considering a project with uncertain future cash flows. Discounting these cash flows at a 14% cost of capital gives a present value of $51.08 million. The cost of the project is $50 million, so it has an expected NPV of $1.08 million. Given the uncertain market demand for the software, the resulting NPV could be much higher or much lower. However Cordova has certain software licenses that allow it to defer the project for a year. If it waits, it will learn more about the demand for the software and will implement the project only if the value of those future cash flows is greater than the cost of $50 million. If in case, the present value of the project's future cash flow is $44.80 million, excluding the $50 million cost of implementing the project. We expect this value to grow at a rate of 14%, which is the cost of capital for this type of project. However we know that the rate of growth is very uncertain and could either be much higher or lower than 14%. Let's assume that the variance of the growth rate is 20%. Given a starting value ($44.80), a growth rate (14%) and a variance of the growth rate (20%), option pricing techniques assume that the resulting value at a future date comes from a lognormal distribution. Because we know the distribution of future values, we could use simulation to repeatedly draw a random variable from that lognormal distribution. For example, suppose we simulate a future value at Year 1 for the project and it is $75 million. Since this is above $50 million cost, we would implement the project in this random draw of the simulation. The payoff is $25 million, and we could find the present value of the payoff if we knew the appropriate discount rate. We could then draw a new random variable and simulate a new value at Year 1. Suppose the new value is $44 million. In this draw of the simulation, we would not implement the project, and the payoff is $0. We could repeat this process many thousands of times and then take the average of all the resulting present values, which is our estimate of the value of the option to implement the project in one year. Unfortunately we do not know the appropriate discount rate. Therefore, we turn to risk neutral valuation. Instead of assuming that the project's value grows at the risk free rate of 6%. Growing at 6% rate instead of a 14% rate would reduce the resulting project value at Year 1, the time we must exercise the option. For example, suppose our first simulation run produces a $55 project value, based on the $44.8 starting value, a 20% variance of the growth rate, and a 6% growth rate instead of the true 14% growth rate. The payoff is only $5 ($55-$50). However, we now discount the $5 payoff at the risk free rate to find its present value. Note that this procedure is analogous to the certainty equivalent approach in which we reduce the value of the risky future cash flow but then discount at the risk free rate. We can repeat the simulation many times, finding the present value of the payoff when discounted at the risk free rate. The average present value of all the outcomes from the simulation is the estimate of the simulation is the estimate of the real option's value. We used the risk neutral approach to simulate the value was $7.19 million. With 200,000 simulations, the average value was $6.97 million. Besides using the simulation analysis method, we can also use the probabilistic methods in estimates real option's value. However, the main difficulty in intuitively grasping the risk neutral valuation concept arises from the fact that the probabilistic methods and tools used were not primarily developed from a financial pricing perspective (Cox, 1985). This results in the use of two different languages, one from economics and the other from mathematics, which can easily be confusing. From the economist's view, risk-neutral probabilities are state prices compounded with the risk free rate. One can always use this translation in order to make economic sense when using the tools from probability theory. It should be mentioned that the risk neutral probability concept is only useful for arbitrage-free pricing. An arbitrage-free price is not necessarily a fair price, or the correct price; it is only a market consistent price. Moreover there are two general conclusions, first, if a market participant was buying (selling) a redundant asset above (below) its arbitrage-free price, then we can say that there would be a more efficient way for this market participant to express his view, namely via the replication strategy; secondly, if a market participant was buying (selling) the underlying asset of a redundant security, where the redundant security trades below (above) its arbitrage-free price, then there would be a more efficient way for this market participant to express his view, again via the replication strategy.

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