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What is Girsanov's Theorem?
Girsanov's Theorem is a fundamental result in stochastic calculus and mathematical finance. It provides a mathematical framework for transforming a stochastic process under one probability measure into an equivalent process under a different probability measure, particularly in the context of continuous-time financial models.
Girsanov's Theorem is particularly relevant in the context of financial modeling, where it is used to derive the risk-neutral measure. The risk-neutral measure is an artificial probability measure under which the expected returns of risky assets are adjusted to be risk-free. It simplifies the pricing and valuation of derivatives by assuming a risk-free investment opportunity.
The Theorem is closely related to the Martingale Representation Theorem which provides a way to express a stochastic process as a stochastic integral with respect to a certain martingale under the changed probability measure. This representation is essential for pricing derivatives and constructing optimal trading strategies. The Theorem also involves the construction of a Radon-Nikodym derivative, also known as the Girsanov density or Girsanov kernel. This derivative specifies the transformation between the original and the new probability measure. It quantifies the adjustments needed to account for the change in probability measure and the associated drift term.
Girsanov's Theorem is widely used in option pricing and risk management. It enables the derivation of risk-neutral pricing formulas, such as the Black-Scholes-Merton formula, by changing the measure to the risk-neutral measure and integrating the derivative payoff under the transformed measure. However, it assumes certain conditions, such as the absence of arbitrage opportunities and the completeness of the financial market. It may not hold in the presence of transaction costs, market frictions, or other complexities. Careful considerations of market assumptions and limitations are necessary when applying Girsanov's Theorem in practice.
Girsanov's Theorem plays a crucial role in quantitative finance, allowing for the modeling and pricing of derivative securities under the risk-neutral measure. It facilitates the analysis of financial market dynamics and the construction of optimal trading strategies in continuous-time models.
Girsanov's Theorem is covered in more detail in module 3 of the CQF program.