Brownian Motion Under a Numeraire Forward Measure

Summary

The document addresses the Brownian motion adjustment used when changing from a risk-neutral measure to a forward measure associated with a numeraire. The response says that both formulas in the question are incorrect: the Radon–Nikodym density should include the numeraire ratio and the bank-account discount factor, expressed through the accumulated short rate.

It then gives the Brownian drift adjustment in terms of the numeraire’s quadratic variation, and specializes to a zero-coupon bond numeraire, where the adjustment uses the bond’s volatility coefficient. Levy’s characterization theorem is identified as the route for proving that the adjusted process is Brownian under the new measure. The excerpt states the corrected relationships and proof tool but does not carry out the proof. Applying the result requires consistent definitions of numeraire volatility and the relevant measure-change assumptions.

Key ideas

• A measure change to a numeraire measure uses a density involving the numeraire and the bank account.
• The Brownian motion under the new measure includes a drift adjustment linked to numeraire volatility.
• For a zero-coupon bond numeraire, the adjustment is expressed using that bond’s volatility coefficient.
• Levy’s characterization theorem can establish that the adjusted process is Brownian under the new measure.
• The excerpt gives the correction and proof approach but leaves the proof steps unstated.

Tags

# Prove that $d\hat{W}_t = dW_t - \frac{1}{N_t} \cdot dN_t\cdot dW_t$ gives a Brownian motion under forward measure


# Prove that $d\hat{W}_t = dW_t - \frac{1}{N_t} \cdot dN_t\cdot dW_t$ gives a Brownian motion under forward measure












Let $N_t$ be a numeraire and $(W_t)$ be the standard Brownian motion under the risk-neutral probability measure $P$.

Recall that forward measure $\hat{P}$ is defined as the Radon-Nikodym derivative: $$\frac{d\hat{P}}{d P} = e^{-\int_0^t r_s \,ds}\frac{N_t}{N_0}$$ where $r_s$ is risk-free interest rate.

Whenever I want to change the underlying measure to forward measure (take bond as numeraire), I always uses the equation $$d\hat{W}_t = dW_t - \frac{1}{N_t} \cdot dN_t\cdot dW_t.$$ However, I am not able to prove that equation above implies that $(\hat{W}_t)$ is a Brownian motion under the forward measure $\hat{P}$.

## Answer by AXH (score 2)

https://quant.stackexchange.com/a/53096

Both equations you have provided are incorrect.

The first equation should read:

$ \frac{d \hat{P} } {d P}(t) = \frac{N_t}{N_0} \frac{\beta_0}{\beta_t} $

where $\beta_t = \exp \int_{0}^{t} r_s ds$.

The second equation should read

$ d \hat{W}_t = d W_t - \sqrt{\frac{ d \langle N \rangle_t }{N^2_t}} dt $

Choosing $N_t = P_{tT}$ means that we have

$ d \hat{W}_t = d W_t - \sigma_{tT} dt $

where $\sigma_{tT}$ is the volatility coefficient of the zero coupon bond.

Now one can use Levy's characterization theorem to show that $\hat{W}$ is a $\hat{P}$-Brownian motion.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.