Equivalent martingale measure

Definition

An equivalent martingale measure is a probability measure under which the discounted price process of financial assets becomes a martingale. This concept is crucial in pricing derivatives and ensuring that no arbitrage opportunities exist in a financial market. The existence of an equivalent martingale measure indicates that the market can be properly modeled to reflect risk-neutral valuation, leading to consistent pricing models like the Black-Scholes model.

5 Must Know Facts For Your Next Test

1. The existence of an equivalent martingale measure is essential for ensuring that asset prices can be modeled without arbitrage opportunities.
2. In the context of the Black-Scholes model, the equivalent martingale measure allows for the simplification of option pricing by transforming real-world probabilities into risk-neutral probabilities.
3. The Radon-Nikodym theorem provides a mathematical framework for establishing the relationship between different probability measures, which is fundamental in defining equivalent martingale measures.
4. The Girsanov theorem explains how to change the probability measure so that a given stochastic process becomes a martingale, which is key to establishing equivalent martingale measures.
5. Using an equivalent martingale measure allows for consistent pricing across different financial instruments and helps in hedging strategies.

Review Questions

• How does the concept of an equivalent martingale measure relate to the pricing of derivatives?
  • An equivalent martingale measure allows for the pricing of derivatives by transforming the real-world probability distribution into a risk-neutral framework. This means that under this measure, the expected discounted future cash flows of derivatives can be calculated using the risk-free rate, which simplifies pricing models. Without this measure, pricing would be inconsistent and potentially expose traders to arbitrage opportunities.
• Discuss the role of the Radon-Nikodym theorem in establishing an equivalent martingale measure within financial modeling.
  • The Radon-Nikodym theorem plays a pivotal role in financial modeling as it provides the necessary mathematical foundation to change from one probability measure to another. In establishing an equivalent martingale measure, it allows us to express changes in probability densities, thus facilitating the transition from real-world dynamics to risk-neutral dynamics. This transformation is crucial for ensuring consistent valuation and pricing within derivatives markets.
• Evaluate the implications of not having an equivalent martingale measure in a financial market and how this affects trading strategies.
  • If a financial market lacks an equivalent martingale measure, it implies that arbitrage opportunities may exist, leading to inconsistencies in asset pricing. Traders could exploit these opportunities for risk-free profits, causing market inefficiencies. This would undermine confidence in trading strategies and models like Black-Scholes, which rely on such measures to ensure fair pricing and proper risk assessment. Ultimately, without this framework, markets could become unstable and unpredictable.

Related terms

Risk-neutral valuation: A method of pricing derivatives by assuming that investors are indifferent to risk, allowing expected returns to be calculated using the risk-free rate.

No-arbitrage principle: The concept that there should be no opportunity to make a profit without risk, ensuring that similar assets are priced equally in an efficient market.

Martingale:

A stochastic process where the conditional expectation of future values, given present information, is equal to the current value, indicating fair game behavior.