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2015 | OriginalPaper | Buchkapitel

6. Ito’s Lemma and Its Applications

verfasst von : Carl Chiarella, Xue-Zhong He, Christina Sklibosios Nikitopoulos

Erschienen in: Derivative Security Pricing

Verlag: Springer Berlin Heidelberg

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Abstract

This chapter introduces Ito’s lemma, which is one of the most important tools of stochastic analysis in finance. It relates the change in the price of the derivative security to the change in the price of the underlying asset. Applications of Ito’s lemma to geometric Brownian motion asset price process, the Ornstein–Uhlenbeck process, and Brownian bridge process are discussed in detail. Extension and applications of Ito’s lemma in several variables are also included.

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Vorheriges Kapitel Manipulating Stochastic Differential Equations and Stochastic Integrals

Nächstes Kapitel The Continuous Hedging Argument

We do not give a formal mathematical definition of what is meant by the term “a solution of a stochastic differential equation”. But essentially it means what we see in Eq. (6.16). The value of the process for x up to time t is a function of the initial condition, time t and the underlying driving stochastic process (here z(t) − z(0)).

Note that the operations \(\mathbb{E}_{0}\) and ∫ ₀ ^t commute (i.e. their order may be interchanged) as may be easily shown by appealing to the definition of the integral.

Note that m(t) is a deterministic quantity so we can use ordinary calculus here.

The result at the second equality uses results demonstrated in Sect. 5.2.

Note that the σ of the discussion of this section is equivalent to \(\sqrt{D}\) of the discussion in Sect. 4.3.2.

In the current notation the values are σ = 1, k = 1, x(0) = 1.

In this subsection the lim_t → 1 y(t) should be interpreted as in the mean square sense.

Fubini’s theorem is discussed in Sect. 22.4

Abramowitz, M., & Stegun, I. A. (1970). The impact of jump risks on nomminal interest rates and foreign exchange rates. Review of Quantitative Finance and Accounting, 2, 17–31.

Cox, J. C., Ingersoll, J. E., & Ross, S. A. (1985b). An intertemporal general equilibrium model of asset prices. Econometrica, 53(2), 363–384.CrossRef

Heath, D., Jarrow, R., & Morton, A. (1992a). Bond pricing and the term structure of interest rates: A new methodology for continent claim valuations. Econometrica, 60(1), 77–105.CrossRef

Oksendal, B. (2003). Stochastic differential equations (6th ed.). New York: Springer.CrossRef

Titel: Ito’s Lemma and Its Applications
verfasst von: Carl Chiarella
Xue-Zhong He
Christina Sklibosios Nikitopoulos
Verlag: Springer Berlin Heidelberg
Buch: Derivative Security Pricing
Print ISBN: 978-3-662-45905-8

Electronic ISBN: 978-3-662-45906-5

Copyright-Jahr: 2015
DOI: https://doi.org/10.1007/978-3-662-45906-5_6