What Is Completeness?
Completeness, in the context of financial markets, refers to a theoretical state where it is possible for investors to construct a portfolio of existing financial instruments that can perfectly replicate any desired future payoff, regardless of the future state of the world. This concept is central to financial markets theory and asset pricing. A complete market implies that there are enough unique contingent claims available to effectively transfer wealth across all possible future economic scenarios, thereby allowing for ideal risk sharing and consumption smoothing. In such a market, all future uncertainties can be fully hedged or speculated upon, and there is no uninsurable risk.9
History and Origin
The foundational concept of complete markets emerged from mid-20th-century economic theory, notably through the work of economists Kenneth Arrow and Gérard Debreu. Their contributions, for which they were awarded Nobel Memorial Prizes, laid the groundwork for general equilibrium theory under uncertainty, where the existence of a complete set of Arrow-Debreu securities allows for efficient allocation of risk. A significant development in the application of complete market theory to finance came with the work of J. Michael Harrison and David M. Kreps. Their seminal 1979 paper, "Martingales and Arbitrage in a Multiperiod Securities Markets," formalized the theory of risk-neutral valuation and demonstrated how, under certain conditions, continuous trading in a limited number of assets could effectively create a complete market for pricing complex derivative securities. 7, 8This mathematical framework became fundamental to modern option pricing models, such as the Black-Scholes-Merton model.
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Key Takeaways
- Completeness in financial markets implies the ability to perfectly hedge or replicate any future payoff.
- It is a theoretical construct often assumed in advanced asset pricing models.
- Complete markets facilitate optimal risk sharing and economic equilibrium.
- The concept is closely tied to the absence of arbitrage opportunities.
- Real-world markets are generally considered incomplete due to various frictions.
Interpreting Market Completeness
In a complete market, the price of any financial asset can be determined precisely based on the prices of a smaller set of fundamental financial instruments. This means that if markets are complete, any future uncertain payoff can be synthetically created by trading in the available securities. For market participants, completeness would imply that they can achieve their desired level of risk management and allocation of resources across all potential future economic states. This ideal scenario simplifies the analysis of market efficiency and the valuation of complex financial products, as all risks are theoretically priced and tradable.
Hypothetical Example
Imagine a simplified economy with only two possible future states next year: "boom" or "recession."
- State 1 (Boom): A specific company's stock, "GrowthCo," is expected to trade at $120.
- State 2 (Recession): GrowthCo stock is expected to trade at $60.
In a complete market, investors could combine GrowthCo stock with a risk-free asset (like a bond) in such a way that they could guarantee any desired payoff in either state. For example, if an investor wanted a guaranteed $100 regardless of the state, they could construct a portfolio. If they wished to fully hedge against a recession, they could create a synthetic security that pays off more in the recession state, offsetting losses from GrowthCo stock. This is possible because there are enough independent assets to "span" (cover) all possible future outcomes. The ability to create such precise payoffs for every state of the world is the essence of market completeness.
Practical Applications
While perfectly complete markets are a theoretical ideal, the concept of completeness has significant practical implications in finance, particularly in the pricing and hedging of derivative securities. The Black-Scholes-Merton model, for instance, assumes a dynamically complete market where options can be perfectly replicated by continuously trading the underlying asset and a risk-free bond. 5This assumption allows for the precise calculation of option prices.
Beyond derivatives, the degree of market completeness can influence phenomena like international trade imbalances. Research by the Federal Reserve Bank of Boston suggests that incomplete markets can lead to trade imbalances because countries may not be able to fully hedge against national income risks through international financial assets. 4Similarly, in financial reporting, the concept of completeness is crucial. The Securities and Exchange Commission (SEC) emphasizes the completeness of financial statements, requiring companies to provide all material information necessary for investors to make informed decisions.
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Limitations and Criticisms
Despite its theoretical elegance, the assumption of complete markets faces significant limitations and criticisms when applied to the real world. Actual financial markets are widely considered to be incomplete markets. Several factors contribute to this incompleteness:
- Information asymmetry: Not all market participants have access to the same information, making it impossible to price all contingent claims accurately.
- Transaction costs: Real-world trading involves costs such as commissions, bid-ask spreads, and taxes, which prevent perfect replication strategies.
- Limited tradability: Many risks, such as individual labor income risk or certain types of catastrophic events, cannot be directly hedged through publicly traded financial instruments.
- Behavioral factors: Investor behavior, often influenced by cognitive biases, can lead to market outcomes that deviate from the rational expectations assumed in complete market models.
- Market frictions: Regulations, liquidity constraints, and borrowing constraints further restrict the ability to create all possible contingent payoffs.
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Critics argue that models assuming complete markets can lead to counterfactual predictions and may not fully capture the complexities of actual economic phenomena. For example, the "equity premium puzzle," which describes the historically high excess returns of stocks over risk-free assets, is often cited as a phenomenon that standard complete market models struggle to explain. Economists like Thomas J. Sargent have highlighted the need to analyze welfare and market outcomes in settings that acknowledge the inherent incompleteness of markets.
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Completeness vs. Incomplete Markets
The distinction between complete and incomplete markets lies at the heart of modern financial economics.
| Feature | Complete Markets | Incomplete Markets |
|---|---|---|
| Definition | All possible future states of the world can be perfectly hedged or replicated through existing financial instruments. | Not all future states can be perfectly hedged or replicated. |
| Risk Exposure | No uninsurable risk. | Some risk remains unhedged or uninsurable. |
| Risk Sharing | Optimal and efficient risk sharing among participants. | Suboptimal risk sharing; agents may face precautionary saving. |
| Arbitrage | Absence of arbitrage opportunities is guaranteed. | Arbitrage opportunities may exist if new securities are introduced. |
| Realism | A theoretical ideal used in models. | Reflects real-world market conditions. |
| Implications | Simplifies asset pricing and allows for precise valuation. | Introduces complexities like liquidity risk and potential inefficiencies. |
In a complete market, any contingent claim can be replicated, meaning its price is determined by arbitrage. In incomplete markets, this is not always true, leading to a range of possible prices for certain claims and making portfolio management more complex.
FAQs
What is an Arrow-Debreu security in the context of market completeness?
An Arrow-Debreu security is a hypothetical financial contract that pays out one unit of a currency or good in exactly one specific future state of the world and zero in all other states. In a theoretical complete market, a full set of such securities, one for each possible future state, would exist, allowing investors to perfectly customize their payoffs.
Why are real-world financial markets considered incomplete?
Real-world markets are incomplete due to various factors, including transaction costs, information asymmetry, limited availability of financial instruments to cover every possible contingency, and legal or regulatory restrictions. It's practically impossible to create and trade a security for every conceivable future event.
How does market completeness relate to derivative pricing?
The assumption of market completeness is crucial for many derivative securities pricing models, such as the Black-Scholes-Merton model. These models often assume that a derivative's payoff can be perfectly replicated by continuously trading the underlying asset and a risk-free asset. This replicating portfolio allows for a precise, arbitrage-free price calculation for the derivative.
What are the consequences of incomplete markets for investors?
Incomplete markets mean that investors cannot perfectly hedge against all forms of risk. This leads to residual risk exposure and may necessitate strategies like precautionary saving to mitigate uninsurable risks. It also means that risk sharing may not be perfectly efficient, and there might be a broader range of possible valuations for certain assets.