Fundamental Finance Theorem

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Fundamental Theorem of Finance

The Fundamental Theorem of Finance

The Fundamental Theorem of Finance (FTAP), in its most crucial incarnation, consists of two key parts that intertwine risk-neutral valuation with the absence of arbitrage opportunities in a financial market. Arbitrage, at its core, refers to the possibility of generating risk-free profits without any initial investment.

First Fundamental Theorem: Absence of Arbitrage Implies the Existence of a Risk-Neutral Probability Measure. This part of the theorem states that if a market is free from arbitrage, then there exists at least one risk-neutral probability measure. A risk-neutral probability measure is a probability distribution that allows us to value assets as if investors were indifferent to risk. In this risk-neutral world, all assets are expected to earn the risk-free rate of return. This doesn’t mean that investors *actually* are risk-neutral, but it provides a powerful mathematical tool for pricing assets.

To understand this better, consider a simple example. Suppose a stock currently trades at $100. In a year, it can either go up to $120 or down to $90. A risk-free bond that pays $105 in a year is also available. If no arbitrage exists, we can find risk-neutral probabilities that, when applied to the stock’s future values, discount back to the current stock price at the risk-free rate. These probabilities would reflect the relative likelihood of the stock going up or down in a risk-neutral setting.

Second Fundamental Theorem: Completeness of the Market Implies the Uniqueness of the Risk-Neutral Probability Measure. This part goes a step further. It states that if the market is *complete*, meaning any contingent claim (a financial instrument whose payoff depends on a future event) can be replicated through a combination of existing assets, then the risk-neutral probability measure is unique. In simpler terms, if we can perfectly hedge any risk using available assets, there’s only one way to consistently price assets without allowing for arbitrage opportunities.

Market completeness is a strong assumption that rarely holds perfectly in the real world. However, it provides a benchmark against which we can assess the efficiency and effectiveness of financial markets. The closer a market is to being complete, the more reliable and precise our risk-neutral valuation becomes.

The FTAP is fundamental because it links the absence of arbitrage (a practical observation) to the existence (and potentially uniqueness) of a risk-neutral probability measure (a powerful theoretical tool). This connection underpins a vast array of financial models, including option pricing models like the Black-Scholes model and other derivatives valuation methods. By identifying and eliminating arbitrage opportunities using risk-neutral probabilities, market participants contribute to a more efficient and stable financial system. The theorem provides a rigorous foundation for understanding how assets are priced and how risks are managed in modern finance.

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