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What is Itô's lemma?
Itô's lemma, also known as the Itô-Doeblin Formula, is a fundamental result in stochastic calculus. It provides a rule for differentiating stochastic processes involving Brownian motion. Named after the Japanese mathematician Kiyoshi Itô, this lemma plays a central role in the development of stochastic calculus and its applications in quantitative finance. It is extensively used in mathematical finance for pricing and risk management of derivative securities. It allows for the derivation of stochastic differential equations and facilitates the calculation of various quantities, such as option prices, implied volatilities, and hedging strategies, through the application of stochastic calculus.
Key aspects of Itô's lemma to consider are:
Stochastic Differential Equation (SDE): Itô's lemma is used to differentiate stochastic processes described by stochastic differential equations (SDEs). SDEs incorporate the effects of random fluctuations, such as Brownian motion, into the evolution of a variable over time. The lemma helps compute the differential of a function involving these stochastic processes.
Chain Rule for Stochastic Calculus: Itô's lemma is a stochastic analog of the chain rule in ordinary calculus. It provides a way to compute the derivative of a function that involves a stochastic process by considering the contributions from both the drift (deterministic) and diffusion (stochastic) components.
Expansion of Stochastic Processes: Itô's lemma expands a function of a stochastic process into a Taylor series expansion, considering the impact of the stochastic term (Brownian motion) on the higher-order differentials. The lemma takes into account the quadratic variation of the Brownian motion, which captures its random and non-linear behavior.
Limitations and Assumptions: Itô's lemma assumes that the stochastic process follows a continuous-time, continuous-sample path. It is based on the assumption of Itô diffusion, which represents a specific type of stochastic process. Deviations from these assumptions may require modifications to the application of Itô's lemma.
Itô's lemma is covered in more detail in module 1 of the CQF program.