Bivariate Black-Sholes Model

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!

The error is in the application of Girsanov theorem.

We have multivariate Black-Sholes market, however I apply one-dimensional Girsanov theorem.

I should apply multi-dimensional Girsanov theorem.

Then there would be now such equations, except the case for $\rho=1$.

The alike task is formulated here

The solution is here

http://wwwf.imperial.ac.uk/~mdavis/course_material/SDEIRM/IRM08_SOLUTIONS3.PDF

Thank you guys and sorry for disturbing!

Mikhail.