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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 ℚ^{N} or **numeraire measure** is given by

ℚ^{N} (A)=E^{ ℚ}[1_{A}(N(T)B(0))/(B(T)N(0))], for A∈ℱ_{T},

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)σ_{N}(t,N(t))dW^{ℚ}(t) ,

for some function σ_{N}(t,N(t)).
This gives that L(t) has the ℚ-dynamics

dL(t)=L(t)σ_{N}(t,N(t))dW^{ℚ}(t),

so that the Girsanov kernel g changing from ℚ to ℚ^{N}
is given by g(t)=-σ_{N}(t,N(t))^{*} so that

dW^{ℚ}(t)=dW^{ℚN}(t)+σ_{N}(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 ℚ^{N} is

μ^{ℚN}(t,X(t))=μ(t,X(t))+σ(t,X(t))σ_{N}(t,N(t))^{*}

while σ, as usual, is unchanged.

Suppose N(t) is a numeraire on the market then the corresponding martingale measure ℚ

ℚ

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)σ

for some function σ

dL(t)=L(t)σ

so that the Girsanov kernel g changing from ℚ to ℚ

dW

so if an asset X(t) has drift μ(t,X(t)) and diffusion term σ(t,X(t)) under ℚ the drift under ℚ

μ

while σ, as usual, is unchanged.

Questions: Magnus Wiktorsson

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