Law of One Price in Arbitrage Free Markets

SUMMARY

The Law of One Price (LOP) states that identical assets or securities must trade at the same price across all markets when adjusted for costs and currency differences. This fundamental principle underpins modern financial theory and forms the basis for arbitrage-free pricing models. When price discrepancies occur, arbitrage opportunities emerge that, when exploited, help restore price consistency.

Mathematical foundation

The Law of One Price can be expressed mathematically for two identical assets $A$ and $B$ trading in different markets:

$P_A = P_B \cdot (1 + c)$

Where:

$P_A$ is the price of asset A
$P_B$ is the price of asset B
$c$ represents transaction costs, including currency conversion if applicable

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Arbitrage mechanism and price convergence

When the Law of One Price is violated, arbitrageurs can profit by:

Buying the asset in the cheaper market
Simultaneously selling in the more expensive market
Capturing the price differential minus costs

This arbitrage activity creates:

Buying pressure in the lower-priced market, pushing prices up
Selling pressure in the higher-priced market, pushing prices down

The process continues until prices converge to equilibrium, minus transaction costs.

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Applications in modern markets

Cross-exchange trading

The Law of One Price is particularly relevant for high-frequency trading strategies that exploit brief price discrepancies across exchanges. These strategies often employ ultra-low latency data feeds to identify and capture arbitrage opportunities.

Derivatives pricing

The principle is fundamental to derivatives pricing, where:

Options on the same underlying with identical strikes and expiries should have consistent prices
Futures contracts should align with spot prices plus carrying costs
Put-call parity relationships must hold to prevent arbitrage

Market efficiency indicators

Persistent violations of the Law of One Price may indicate:

Market inefficiencies
Structural barriers to arbitrage
Liquidity constraints
Regulatory restrictions

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Limitations and real-world considerations

Transaction costs

Real-world implementation must account for:

Trading fees
Bid-ask spreads
Clearing costs
Market impact costs

Market frictions

Several factors can prevent perfect arbitrage:

Latency between markets
Capital constraints
Short-selling restrictions
Different trading hours across markets
Regulatory barriers

Risk considerations

Arbitrage strategies based on the Law of One Price face:

Execution risk
Counterparty risk
Technology risk
Regulatory risk

Market structure implications

The Law of One Price influences:

Market fragmentation and consolidation
Development of smart order routing systems
Cross-border trading infrastructure
Market making strategies

Modern electronic markets have increased the efficiency of arbitrage mechanisms, making prices converge faster and reducing the magnitude of discrepancies. This has led to more sophisticated arbitrage strategies and enhanced market efficiency.

Role in financial theory

The Law of One Price is central to:

Asset pricing models
Portfolio theory
Risk-neutral pricing frameworks
Market efficiency concepts

It provides a theoretical foundation for understanding how prices should behave in efficient markets and helps identify potential market inefficiencies when violations occur.