# Girsanov's Theorem for Multiple Risky Assets

Girsanov's theorem provides the measure transformation from probability measure P to Q such that-

$dW_t^Q=dW_t^P+\lambda dt\implies \xi_tW_t^Q$ is a martingale under the P measure where $\xi_t=e^{-\lambda W_t^P-\frac{\lambda^2}{2}t}$ and $W_t^X$ is the Weiner's Process under probability measure $X$.

For single assets this can be utilised to convert their dynamics from objective probability measure P to risk-neutral measure Q.

$\frac{dS}{S}=\mu dt+\sigma dW^P=(\mu-\lambda\sigma)dt+\sigma dW^Q$.

Setting $\lambda=\frac{\mu-r_f}{\sigma}=$ Sharpe Ratio of asset S, we get-

$\frac{dS}{S}=r_fdt+\sigma dW^Q$.

Suppose we have two tradeable assets in a market with the same source of risk $W$.

$\frac{dS_1}{S_1}=\mu_1 dt+\sigma_1 dW^P$

$\frac{dS_2}{S_2}=\mu_2 dt+\sigma_2 dW^P$

What $\lambda$ should be chosen in this case for making the transformation from P to Q? If we choose the Sharpe Ratio of the 1st asset, then we'll end up getting-

$\frac{dS_1}{S_1}=r_f dt+\sigma_1 dW^Q$

$\frac{dS_2}{S_2}=(\mu_2-\sigma_2\frac{\mu_1-r_f}{\sigma_1}) dt+\sigma_2 dW^Q$

Similarly for choosing the other Sharpe Ratio.

It seems intuitive to choose the greater of the two ratios for making the transformation, but that'll result in the other asset having an expected return lesser than $r_f$ in the risk-neutral measure. This would imply that participants simply choose to never trade that asset. Does this reasoning seem correct?

If two assets have the same source of risk $W^P$ then the no arbitrage opportunity condition implies that their Sharpe ratios are the same, i.e. $\frac{\mu_1-r_f}{\sigma_1} = \frac{\mu_2-r_f}{\sigma_2}$. This is a textbook exercise and is easily proven by building a risk free portfolio long one share of the first asset and short $\frac{S_1 \sigma_1}{S_2 \sigma_2}$ shares of the second asset.