Complete market

What Is Complete Market?

A complete market, a core concept in Financial Economics, is a theoretical market setting where every possible future economic state or contingency can be insured against or replicated through existing financial instruments. In essence, it is a market where there are enough unique financial assets to allow investors to construct portfolios that can generate any desired payoff in any future state of the world²¹, ²². This means that for every conceivable future economic scenario, there exists a security or a combination of securities whose payoff perfectly matches that scenario²⁰.

The concept of a complete market implies that all risks, both aggregate and idiosyncratic, can be fully diversified away or transferred among market participants. This theoretical ideal serves as a benchmark for understanding how financial markets function and how derivative securities play a role in facilitating risk transfer and price discovery. In such a market, there are no "missing" markets for any specific risk, allowing for optimal allocation of resources and efficient pricing of assets.

History and Origin

The foundational concept of complete markets is deeply intertwined with the development of general equilibrium theory in economics. Pioneering work by economists Kenneth Arrow and Gerard Debreu in the mid-20th century laid the theoretical groundwork. They introduced the idea of "Arrow-Debreu securities," which are hypothetical financial instruments that pay out one unit of value in a specific future "state of the world" and zero in all other states¹⁹. If a complete set of these Arrow-Debreu securities existed for every possible future state, the market would be considered complete¹⁸.

Arrow and Debreu's contributions, for which they were awarded the Nobel Memorial Prize in Economic Sciences, demonstrated that under certain conditions, a competitive economy with complete markets could achieve a Pareto-optimal allocation of resources, meaning no individual could be made better off without making someone else worse off¹⁷. While these theoretical securities do not exist in practice, the concept provided a powerful framework for understanding market efficiency and the role of financial innovation. The work of Fisher Black and Myron Scholes on option pricing further contributed to the understanding of how financial instruments can "complete" a market by allowing for the replication of various payoffs. Their model, while based on specific assumptions, showed how a portfolio of an underlying asset and a risk-free bond could replicate the payoff of an option, suggesting a pathway towards market completeness in certain contexts¹⁶.

Key Takeaways

• A complete market is a theoretical financial market where all possible future economic outcomes can be hedged or replicated using available financial instruments.¹⁵
• In a complete market, there are no uninsurable risks, and all information is fully reflected in asset prices.
• The concept is foundational to asset pricing theory and the valuation of complex financial products.
• Complete markets facilitate efficient risk sharing and resource allocation among market participants.
• Real-world financial markets are considered incomplete markets due to various frictions and the absence of instruments for every conceivable contingency.

Formula and Calculation

While there isn't a single universal formula to calculate market completeness, the concept is often demonstrated through the ability to replicate any arbitrary payoff using a set of existing assets. This is related to the idea of a "spanning condition."

In a simplified discrete-state model, if there are (S) possible future states of the world and (N) available securities, for a market to be complete, the number of linearly independent securities (N) must be at least equal to the number of states (S), i.e., (N \ge S).

The pricing of a contingent claim in a complete market relies on the concept of risk-neutral probability. If a market is complete and free of arbitrage, then the price of any contingent claim can be determined by its expected payoff under risk-neutral probabilities, discounted at the risk-free rate.

The value (V_0) of a contingent claim at time 0 can be expressed as:

Where:

• (V_0) = Present value of the contingent claim
• (r) = Risk-free interest rate
• (T) = Time to maturity
• (S) = Total number of possible future states
• (Q(s)) = Risk-neutral probability of state (s) occurring
• (V_T(s)) = Payoff of the contingent claim in state (s) at time (T)

This formula underscores how, in a complete market, the valuation of any financial instrument becomes a matter of calculating the expected value of its payoffs under a specific probability measure (risk-neutral) and discounting it back to the present. The absence of no-arbitrage condition is crucial for this valuation framework.

Interpreting the Complete Market

Interpreting the complete market concept involves understanding its implications for financial theory and practice. In a theoretically complete market, every investor can perfectly tailor their portfolio to match their unique risk preferences and consumption needs across all future states of the world. This means that if an investor desires a specific payoff if, for example, a particular stock price reaches a certain level, they can either find an existing instrument that provides that exact payoff or construct a portfolio of existing instruments that perfectly replicates it.

The completeness of a market simplifies financial modeling and analysis significantly. It implies that there is a unique set of prices, known as state prices, for each future state of the world¹⁴. These state prices represent the present value of receiving one unit of currency if and only if a specific state occurs. The existence of these state prices means that any financial contract, regardless of its complexity, can be priced by simply summing the product of its payoff in each state and the corresponding state price.

In essence, a complete market allows for perfect hedging of all risks and ensures that all market participants face the same marginal rates of substitution across states, leading to an optimal allocation of resources.

Hypothetical Example

Imagine a simplified economy with only two possible future states next year: "Boom" or "Recession."
Suppose there are two basic assets available:

1. Stock A: Currently trades at $100. If "Boom," it will be worth $120. If "Recession," it will be worth $80.
2. Risk-Free Bond: Currently trades at $95. If "Boom," it will be worth $100. If "Recession," it will also be worth $100.

In this simple economy, the market is complete because any payoff in the "Boom" or "Recession" state can be replicated by a combination of Stock A and the Risk-Free Bond.

Let's say an investor wants to create a portfolio that pays $100 in "Boom" and $90 in "Recession." Can they achieve this in this complete market?

Let (x_A) be the number of units of Stock A and (x_B) be the number of units of the Risk-Free Bond. We set up a system of equations for the payoffs:

• Boom state: (120x_A + 100x_B = 100)
• Recession state: (80x_A + 100x_B = 90)

Subtracting the second equation from the first:
(40x_A = 10 \implies x_A = 0.25)

Substitute (x_A) back into the first equation:
(120(0.25) + 100x_B = 100)
(30 + 100x_B = 100)
(100x_B = 70 \implies x_B = 0.70)

So, the investor needs to buy 0.25 units of Stock A and 0.70 units of the Risk-Free Bond. The cost of this portfolio today would be:
Cost = (0.25 \times $100 + 0.70 \times $95 = $25 + $66.50 = $91.50)

This example demonstrates the concept of replication within a complete market. Because we could find a combination of existing securities to perfectly match the desired payoff in both states, the market is considered complete for this specific two-state, two-asset scenario.

Practical Applications

While a perfectly complete market is a theoretical ideal, the concept has significant practical applications in finance, particularly in the realm of derivatives and risk management. Financial institutions and market participants strive to "complete" markets by creating and trading instruments that allow for more precise hedging and transfer of risk.

• Derivatives Trading: The proliferation of complex derivative securities such as options, futures, and swaps can be seen as an attempt to introduce instruments that span a wider range of future contingencies. These instruments enable investors to manage specific risks (e.g., interest rate risk, currency risk, commodity price risk) that might not be directly insurable through traditional assets. For example, the Federal Reserve Bank of Boston highlights how derivatives are essential tools for hedging various exposures¹³.
• Arbitrage Pricing Theory: The absence of arbitrage opportunities is a cornerstone of complete markets. In practice, financial professionals constantly seek to identify and exploit arbitrage opportunities, which, when they exist, indicate market incompleteness or inefficiencies. The rapid trading activity driven by arbitrageurs helps push markets towards a state where prices reflect all available information, aligning with the principles of market efficiency.
• Regulatory Frameworks: Regulators consider market completeness when designing policies related to financial stability and consumer protection. Understanding where markets are incomplete helps identify areas where specific risks remain unhedged, potentially leading to systemic vulnerabilities. While aiming for perfect completeness is impractical, regulations may encourage the development of certain financial products or market structures that enhance risk transfer. The Federal Reserve Board, for instance, focuses on derivatives and trading activities in their regulatory oversight, acknowledging their role in risk management¹².
• Quantitative Finance: The complete market framework is fundamental to quantitative finance, particularly in the development of sophisticated pricing models for financial instruments. Models like the Black-Scholes model for options rely on the assumption of a dynamically complete market, even if it is an idealized simplification¹⁰, ¹¹. This theoretical underpinning allows quants to derive fair values for complex instruments.

Limitations and Criticisms

Despite its theoretical elegance and utility as a benchmark, the concept of a complete market faces several significant limitations and criticisms in the context of real-world financial systems.

Firstly, genuine complete markets do not exist in practice. The number of conceivable future states of the world is effectively infinite, making it impossible to create a unique financial instrument for every single one⁹. Real markets are always, to some extent, incomplete markets. This incompleteness arises from various factors, including transaction costs, information asymmetries, and the inherent difficulty in foreseeing and contracting for all future contingencies.⁸

Secondly, even if theoretically possible, the cost of creating and trading such an exhaustive set of instruments would be prohibitive. Transaction costs, including brokerage fees, bid-ask spreads, and taxes, prevent perfect arbitrage and replication strategies from being perfectly frictionless⁷.

Thirdly, the assumption of continuous trading, often implicit in complete market models like the Black-Scholes-Merton model, is not realistic⁶. Markets trade in discrete time, and there can be significant price gaps between trading opportunities. This discontinuity means that perfect dynamic hedging, which is crucial for achieving completeness in many models, is not fully achievable. The CFA Institute acknowledges that models like Black-Scholes rely on assumptions that may not hold true in real-world situations, such as constant volatility and no transaction costs⁵.

Furthermore, market imperfections, such as liquidity constraints, regulatory restrictions, and behavioral biases of investors, further contribute to market incompleteness. These factors limit the ability of market participants to fully transfer or hedge all desired risks. The presence of these frictions means that real-world supply and demand dynamics can lead to prices that deviate from theoretical complete market valuations.

Complete Market vs. Incomplete Market

The distinction between a complete market and an incomplete market is fundamental in financial economics, highlighting the gap between theoretical ideals and real-world complexity.

The primary area of confusion arises because both concepts relate to market efficiency and the ability to manage risk. However, a complete market represents a perfect, frictionless environment where all contingencies are covered, while an incomplete market acknowledges the practical limitations and imperfections inherent in real-world financial systems.

FAQs

What does "complete market" mean in simple terms?

In simple terms, a complete market is a perfect financial market where you can create a financial instrument or strategy to get a specific payoff for any possible future event. It means there's a way to insure against or bet on every single outcome, no matter how specific.¹, ²

Why is the concept of a complete market important?

The concept of a complete market is crucial because it provides a theoretical benchmark for understanding how financial markets should function under ideal conditions. It helps economists and financial professionals analyze market efficiency, design new financial products, and develop asset pricing models, even though real markets are not perfectly complete.

Are real-world markets complete?

No, real-world markets are not perfectly complete. They are generally considered incomplete markets due to various factors like transaction costs, regulations, information asymmetries, and the sheer number of unpredictable future events. While financial innovations, particularly derivative securities, help increase market completeness, they cannot achieve it perfectly.

How do derivatives relate to complete markets?

Derivatives, such as options and futures, play a significant role in making real-world markets more "complete." They allow investors to create synthetic positions that replicate payoffs for specific future scenarios, thereby providing ways to manage or transfer risks that might otherwise be uninsurable. For example, by using options, investors can construct portfolios that pay off only if a stock price moves in a particular way, effectively creating tailored contingent claim payoffs.

What are some implications of market incompleteness?

Market incompleteness implies that some risks cannot be perfectly hedged or transferred, leading to potential welfare losses and inefficiencies in the allocation of capital. It also means that there might not be a unique, universally accepted way to price certain complex financial instruments, and opportunities for arbitrage might temporarily exist.