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\title{The No-Arbitrage Property Under A Change of Num\'eraire}
\author{Mingxin Xu} 
\date{Computational Finance Seminar: April 8}
\maketitle
\begin{abstract}
The fundamental idea in risk-neutral pricing is that asset
prices discounted by money market accounts are martingales under a
risk-neutral measure.  However, under certain circumstances, other choices
of num\'eraires are more useful.  For example, to price bond options, we
change from risk-neutral measure to forward measure and use bond prices as
num\'eraires.  To find a simple one-dimensional integro-differential
equation for pricing arithmetic Asian options, the stock price is the
proper num\'eraire to use. Of course, to build models of exchange rates,
we will face two different currencies and therefore two different
discounting factors as num\'eraires.  A desirable property for the
definition of no-arbitrage would be the stability under a change of
num\'eraire.  Unfortunately, this is not the case in the definition of No
Free Lunch with Vanishing Risk (NFLVR) which is a version of no-arbitrage
used by Delbaen and Schachermayer to prove the equivalence of no-arbitrage
and existence of risk-neutral measure in semimartingale models.  We
will show an example when there is no-arbitrage under one currency and
arbitrage under another.  Then we will give a condition under which the
no-arbitrage property is preserved under the change of num\'eraire.  This
talk is mainly based on the paper `The No-Arbitrage Property Under A
Change of Num\'eraire', by Delbaen and Schachermayer.
\end{abstract}
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