Stock Numeraire Change and Log-Price Dynamics

Summary

The document derives the dynamics of the logarithm of a geometric Brownian motion after changing to the stock numeraire measure. The change of measure is motivated by rewriting an expectation involving the terminal stock price times its logarithm as an expectation under the stock measure. The answer uses the requirement that the money market account divided by the stock is a martingale under that measure to determine the stock’s drift, then applies Itô’s formula to obtain the log-price drift.

The result is contrasted with the familiar log-price drift under the bond numeraire. The derivation also flags a notation ambiguity: the question calls the stock growth rate the risk-free rate, while the answer distinguishes the physical-measure growth rate from the risk-free rate. The formulas therefore depend on keeping measure and drift conventions explicit. The text gives a compact theoretical derivation but no empirical evidence or trading strategy.

Key ideas

• Changing to the stock numeraire rewrites a stock-weighted expectation as an expectation under a new measure.
• Under the stock measure, the money market account divided by the stock must be a martingale.
• That martingale condition determines the stock drift under the new measure.
• Itô’s formula then gives the log-price dynamics, with a drift different from the bond-numeraire case.
• Care is needed to distinguish the physical stock growth rate from the risk-free rate.

Tags

# Change-of-measure: Dynamics of $\log(S_t)$ with $S_t$ as numeraire


# Change-of-measure: Dynamics of $\log(S_t)$ with $S_t$ as numeraire












Let $S$ be a GBM with dynamics $dS_t/S_t=rdt+\sigma dW_t$. We want to compute the following expected value: \begin{align*} \mathbb{E}(S_T\log(S_T)). \end{align*} Using a change of measure we can write \begin{align*} \mathbb{E}(S_T\log(S_T)) =S_0\mathbb{\widehat E}(\log(S_T)) \end{align*} where $\mathbb{\widehat P}$ is the measure with $S$ as numeraire. How do we finish this problem? What are the dynamics of $(\log(S_t))_{t\geq 0}$ under $\mathbb{\widehat P}$?

## Answer by spaceisdarkgreen (score 7, accepted)

https://quant.stackexchange.com/a/35829

Under the stock numeraire measure, $\frac{B_t}{S_t}$ is a Martingale. We can compute $$d\frac{B_t}{S_t}= \frac{1}{S_t}dB_t -\frac{1}{S_t^2}B_tdS_t+\frac{1}{S_t^3}B_t\sigma^2S_t^2dt\\=\frac{B_t}{S_t}\left(rdt -\mu dt -\sigma dW_t +\sigma^2dt\right)$$ so the growth rate $\mu$ that makes this a Martingale is $$ \mu = r+\sigma^2.$$

So the growth rate of the stock under the stock numeraire measure is $r+\sigma^2$.

Then, applying Ito as usual, you can find that $\log(S_t)$ follows Brownian motion with drift $r+\frac{1}{2}\sigma^2.$ (This is in contrast to $r-\frac{1}{2}\sigma^2$ in the usual case with the bond as numeraire.)

EDIT

Looking back, I see that I missed the fine print that you called the growth rate of the stock $r$. I hope it's clear that I started in the physical measure, called the stock growth rate $\mu$ and used $r$ to refer to the risk free rate.

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