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Change of numeraire

A numeraire is a positive self-financing portfolio, which serves as the discounting factor on the market. The usual numeraire is the bank account B, which corresponds to the martingale measure ℚ. If we change numeraire we should change the martingale measure accordingly.
Suppose N(t) is a numeraire on the market then the corresponding martingale measure ℚN or numeraire measure is given by
N (A)=E[1A(N(T)B(0))/(B(T)N(0))], for A∈ℱT,
that is the LR-process L(t)=(B(0)N(t))/(B(t)N(0)). Since N(t)/B(t) is a ℚ-martingale we get that N(t) has the ℚ-dynamics
d N(t)=r(t)N(t)dt+N(t)σN(t,N(t))dW(t) ,
for some function σN(t,N(t)). This gives that L(t) has the ℚ-dynamics
dL(t)=L(t)σN(t,N(t))dW(t),
so that the Girsanov kernel g changing from ℚ to ℚN is given by g(t)=-σN(t,N(t))* so that
dW(t)=dWN(t)+σN(t,N(t))*dt.
so if an asset X(t) has drift μ(t,X(t)) and diffusion term σ(t,X(t)) under ℚ the drift under ℚN is
μN(t,X(t))=μ(t,X(t))+σ(t,X(t))σN(t,N(t))*
while σ, as usual, is unchanged.

 

Questions: Magnus Wiktorsson
Last update: 2011 Sep 17 15:56:42. Validate: HTML CSS

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