Numeraire Invariance and application to Option Pricing and Hedging

F. Jamshidian

Numeraire Invariance and application to Option Pricing and Hedging

F. Jamshidian

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic

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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 language	English
Title of host publication	Actuarial and Financial Mathematics Conference
Subtitle of host publication	Interplay between Finance and Insurance
Editors	Michèle Vanmaele, Griselda Deelstra, Ann De Schepper, Jan Dhaene, Huguette Reynaerts, Wim Schoutens, Paul Van Goethem
Publisher	Contact Forum
Pages	3-16
Number of pages	14
Publication status	Published - 14 Feb 2008
Event	Actuarial and Financial Mathematics Conference: Interplay between Finance and Insurance - Academy Palace, Brussels, Belgium Duration: 8 Feb 2008 → 9 Feb 2008

Conference

Conference	Actuarial and Financial Mathematics Conference
Country/Territory	Belgium
City	Brussels
Period	8/02/08 → 9/02/08

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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Actuarial and Financial Mathematical Conference 2008Final published version, 6.57 MB

Cite this

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title = "Numeraire Invariance and application to Option Pricing and Hedging",

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.",

keywords = "predictable representation, IR-59852, self-financing trading strategy, geometric Brownian motion, unique pricing, Itˆo{\textquoteright}s formula, PDE, martingale, exponential Poisson process, arbitrage-free, SDE, Numeraire invariance, Hedging, Markovian, homogeneous payoff",

author = "F. Jamshidian",

year = "2008",

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day = "14",

language = "English",

pages = "3--16",

editor = "Mich{\`e}le Vanmaele and Griselda Deelstra and \{De Schepper\}, Ann and Jan Dhaene and Huguette Reynaerts and Wim Schoutens and \{Van Goethem\}, Paul",

booktitle = "Actuarial and Financial Mathematics Conference",

publisher = "Contact Forum",

address = "Finland",

note = "Actuarial and Financial Mathematics Conference : Interplay between Finance and Insurance ; Conference date: 08-02-2008 Through 09-02-2008",

}

Jamshidian, F 2008, Numeraire Invariance and application to Option Pricing and Hedging. in M Vanmaele, G Deelstra, A De Schepper, J Dhaene, H Reynaerts, W Schoutens & P Van Goethem (eds), Actuarial and Financial Mathematics Conference: Interplay between Finance and Insurance. Contact Forum, pp. 3-16, Actuarial and Financial Mathematics Conference, Brussels, Belgium, 8/02/08.

Numeraire Invariance and application to Option Pricing and Hedging. / Jamshidian, F.
Actuarial and Financial Mathematics Conference: Interplay between Finance and Insurance. ed. / Michèle Vanmaele; Griselda Deelstra; Ann De Schepper; Jan Dhaene; Huguette Reynaerts; Wim Schoutens; Paul Van Goethem. Contact Forum, 2008. p. 3-16.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic

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N2 - 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.

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KW - martingale

KW - exponential Poisson process

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A2 - Van Goethem, Paul

PB - Contact Forum

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Y2 - 8 February 2008 through 9 February 2008

ER -

Numeraire Invariance and application to Option Pricing and Hedging

Abstract

Conference

Keywords

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