Let us propose bivariate Black-Sholes Model. Assume, we have an arbitrage-free complete market.
$r_{f}$ is risk-free rate.
Under real-world measure $P$:
$dS_{1} (t)=S_{1} (t) [\mu_{1}dt+\sigma_{1}dW_{1,t}^{P}]$
$dS_{2} (t)=S_{2} (t) [\mu_{2}dt+\sigma_{2}dW_{2,t}^{P}]$
$corr(W_{1,t},W_{2,t})=\rho$
where $W_{i,t}^{P}$ is standard Brownian Motion under $P$. I have found out that in this model it holds: $\mu_{2}-r_{f}=\sigma_{2}(\rho+\sqrt{1-\rho^{2}})\frac{\mu_{1}-r_{f}}{\sigma_{1}}$
But I am not sure, whether the deirvation is right.
Does it hold?
The derivation is presented here:
From the fact of correlation between two Brownian motions,it holds: $W_{2,t}^{P}=\rho W_{1,t}^{P}+\sqrt{1-\rho^{2}}W_{0,t}^{P}$
where $W_{0,t}^{P}$ is Standard Brownian Motion under $P$, indepndent of $W_{1,t}^{P}$.
Hence, one can rewrite:
$dS_{1} (t)=S_{1} (t) [\mu_{1}dt+\sigma_{1}dW_{1,t}^{P}]$
$dS_{2} (t)=S_{2} (t) [\mu_{2}dt+\sigma_{2}\rho dW_{1,t}^{P}+\sigma_{2}\sqrt{1-\rho^{2}}W_{0,t}^{P}]$
We propose the new measure $Q$, defined by: $\frac{dQ}{dP}=exp\{\frac{r_{f}-\mu_{1}}{\sigma_{1}}W_{1,t}-\frac{1}{2}(\frac{r_{f}-\mu_{1}}{\sigma_{1}})^{2}t\}$
Under this measure
$W_{1,t}^{Q}=W_{1,t}^{P}+\frac{\mu_{1}-r_{f}}{\sigma_{1}}$
$W_{0,t}^{Q}=W_{0,t}^{P}+\frac{\mu_{1}-r_{f}}{\sigma_{1}}$
are Standard Brownian Motions under $Q$.
Let us define $W_{2,t}^{Q}=\rho W_{1,t}^{Q}+\sqrt{1-\rho^{2}}W_{0,t}^{Q}$, which is also Standard Browniam Motion under $Q$ correlated with $W_{1,t}^{Q}$ with coefficient $\rho$
Hence for assets it holds:
$dS_{1} (t)=S_{1} (t) [r_{f}dt+\sigma_{1}\frac{\mu_{1}-r_{f}}{\sigma_{1}}+\sigma_{1}dW_{1,t}^{P}]=S_{1} (t) [r_{f}dt+\sigma_{1}dW_{1,t}^{Q}]$
$dS_{2} (t)=S_{2} (t) [r_{f}dt+\mu_{2}dt-r_{f}dt+\sigma_{2}\rho (dW_{1,t}^{P}+\frac{\mu_{1}-r_{f}}{\sigma_{1}})-\sigma_{2}\rho\frac{\mu_{1}-r_{f}}{\sigma_{1}} +\sigma_{2}\sqrt{1-\rho^{2}}(W_{0,t}^{P}+\frac{\mu_{1}-r_{f}}{\sigma_{1}})-\sigma_{2}\sqrt{1-\rho^{2}}\frac{\mu_{1}-r_{f}}{\sigma_{1}}]=S_{2} (t) [r_{f}dt+\sigma_{2}\rho dW_{1,t}^{Q}+\sigma_{2}\sqrt{1-\rho^{2}}dW_{0,t}^{Q}+{(\mu_{2}-r_{f})-\sigma_{2}(\rho+\sqrt{1-\rho^{2}})\frac{\mu_{1}-r_{f}}{\sigma_{1}}}dt]=S_{2} (t) [r_{f}dt+\sigma_{2}dW_{2,t}^{Q}+{(\mu_{2}-r_{f})-\sigma_{2}(\rho+\sqrt{1-\rho^{2}})\frac{\mu_{1}-r_{f}}{\sigma_{1}}}dt]$
One can see that discounted process of $S_{1}(t)$ is Martingale, hence measure $Q$ is Eqivalent Martingale Measure.
Accodring to Second Theorem of Asset Pricing it is unique.
Hence, $e^{-r_{f}t} S_{2}(t)$ should be as well martingale.
Hence,
$(\mu_{2}-r_{f})-\sigma_{2}(\rho+\sqrt{1-\rho^{2}})\frac{\mu_{1}-r_{f}}{\sigma_{1}}=0$
$\mu_{2}-r_{f}=\frac{\sigma_{2}}{\sigma_{1}}(\rho+\sqrt{1-\rho^{2}})(\mu_{1}-r_{f})$.
Thank you in advance!
