Numeraire Invariance and application to Option Pricing and Hedging

F. Jamshidian

    Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademic

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    Abstract

    Numeraire invariance is a well-known technique in option pricing and hedging theory. It takes a convenient asset as the numeraire, as if it were the medium of exchange, and expresses all other asset and option prices in units of this numeraire. Since the price of the numeraire relative to itself is identically 1 at all times, this reduces pricing and hedging to a market with zero-interest rates. A somewhat controversial implication is that the modelling focus should be more on the asset price ratios rather than on the asset price processes themselves. The idea of numeraire invariance is already implicit in Merton (1973), and since then many authors have contributed to its development. After a brief survey of its origins, we state and prove the numeraire invariance principle for general semimartingale price processes, following essentially Duffie [3]. We then present its application to unique pricing in arbitrage-free models and discuss nondegeneracy and unique hedging.
    Original languageEnglish
    Title of host publicationActuarial and Financial Mathematics Conference
    Subtitle of host publicationInterplay between Finance and Insurance
    EditorsMichèle Vanmaele, Griselda Deelstra, Ann De Schepper, Jan Dhaene, Huguette Reynaerts, Wim Schoutens, Paul Van Goethem
    PublisherContact Forum
    Pages3-16
    Number of pages14
    Publication statusPublished - 14 Feb 2008
    EventActuarial and Financial Mathematics Conference: Interplay between Finance and Insurance - Academy Palace, Brussels, Belgium
    Duration: 8 Feb 20089 Feb 2008

    Conference

    ConferenceActuarial and Financial Mathematics Conference
    CountryBelgium
    CityBrussels
    Period8/02/089/02/08

    Keywords

    • predictable representation
    • IR-59852
    • self-financing trading strategy
    • geometric Brownian motion
    • unique pricing
    • Itˆo’s formula
    • PDE
    • martingale
    • exponential Poisson process
    • arbitrage-free
    • SDE
    • Numeraire invariance
    • Hedging
    • Markovian
    • homogeneous payoff

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