Numeraire Invariance and application to Option Pricing and Hedging

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Jamshidian, Farshid (2008) Numeraire Invariance and application to Option Pricing and Hedging.

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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.
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