What is a local martingale?
In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Local martingales are essential in stochastic analysis (see Itō calculus, semimartingale, and Girsanov theorem).
Is Brownian motion local martingale?
In particular, a (continuous) local martingale is a deterministic time change of Brownian motion if and only if its quadratic variation is a deterministic function (and absolutely continuous with respect to Lebesgue measure).
What is the theorem that is used to change the measure and induce a shift in the drift of the above process?
In probability theory, the Girsanov theorem (named after Igor Vladimirovich Girsanov) describes how the dynamics of stochastic processes change when the original measure is changed to an equivalent probability measure.
What is a change of measure?
Roughly speaking, a change of measure can change the drift of a diffusion process but not the noise. The Girsanov formula is the formula for the L that does the change. If they can be, the two probability 2 Page 3 measures are absolutely continuous with respect to each other, or equivalent.
What is equivalent martingale measure?
In mathematical finance, a risk-neutral measure (also called an equilibrium measure, or equivalent martingale measure) is a probability measure such that each share price is exactly equal to the discounted expectation of the share price under this measure.
Why do we need change of measure?
Changes of probability measure are important in mathematical finance because they allow you to express derivative prices in “risk-neutral” form as an expected discounted sum of dividends. This in turn allows many derivative prices (e.g. options) to be computed in closed-form.
Is Black Scholes risk neutral?
Economists Fischer Black and Myron Scholes demonstrated in 1968 that a dynamic revision of a portfolio removes the expected return of the security, thus inventing the risk neutral argument.
What is P measure?
Also known as the real-world measure, the P-measure is a way of measuring probability based on historical data rather than based on assumptions of the existence of a risk-free rate and absence of arbitrage in the market. The P-measure is typically used for the purpose of risk measurement.