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QUESTION

Two shares follow geometric Brownian motions, i.e.,

$$ds_1=S_1(\mu_1 dt+\sigma_1dW_1)$$

$$ds_20S_2(\mu_2dt+\sigma_2dW_2)$$

The price changes are correlated with the function coefficient
$\rho$, i.e. $dW_1dW_2=\rho dt$. Starting from Taylor'S theorem
extended to stochastic variables, find the stochastic differential
equation satisfied by a function $f(S_1,S_2)$.


ANSWER

Taylor for 2 variables, %S_1$ and $S_2$.

$$df=\frac{\partial f}{\partial S_1}ds_1+\frac{\partial
f}{\partial S_2}ds_2+\frac{1}{2}\frac{\partial^2f}{\partial
S_1^2}(ds_1)^2+\frac{\partial^2f}{\partial S_1\partial
S_2}ds_1ds_2+\frac{\partial^2f}{\partial S_2}(ds_2)^2+\ldots$$ (no
explicit t dependence). But $ds_i=S_i(\mu_i dt+\sigma_idw_i)$

\begin{eqnarray*}
\Rightarrow df&=&\frac{\partial f}{\partial
S_1}S_1(\mu_1dt+\sigma_1dw_1)+ \frac{\partial f}{\partial
S_2}S_2(\mu_2dt+\sigma_2dw_2)+\\&&
\frac{1}{2}\frac{\partial^2f}{\partial
S_1^2}S_1^2(\mu_1dt+\sigma_1dw_1)^2+
\frac{1}{2}\frac{\partial^2f}{\partial
S_2}S_2^2(\mu_2dt+\sigma_2dw_2)^2+\\&& \frac{\partial^2f}{\partial
S_1\partial S_2}S_1S_1(\mu_1dt+\sigma_1dw_1)(\mu_2dt+
\sigma_2dw_2)\\ &=&\left(\frac{\partial f}{\partial S_1}S_1\mu_1+
\frac{\partial f}{\partial S_2}S_2\mu_2\right)dt+ \frac{\partial
f}{\partial S_1}S_1\sigma_1dw_1+ \frac{\partial f}{\partial
S_2}S_2\sigma_2dw_2\\&& +\frac{1}{2}\frac{\partial^2f}{\partial
S_1^2}S_1^2\left(\mu_1^2(dt)^2+ 2\mu_1\sigma_1dtdw_1+
\sigma_1^2(dw_1)^2\right)\\&&+
\frac{1}{2}\frac{\partial^2f}{\partial
S_2^2}S_2^2\left(\mu_2^2(dt)^2 +2\mu_2\sigma_2dtdw_2
+\sigma_1^2(dw_2)^2\right)\\&& +\frac{\partial^2f}{\partial
S_1\partial S_2}S_1S_2\left(\mu_1\mu_2dt^2+
\mu_1dtdw_2\sigma_2+\mu_2\sigma_1dtdw_1
+\sigma_1\sigma_2dw_1dw_2\right)
\end{eqnarray*}

Now, eliminate the smaller terms: $(dt)^2=0,\ dtdw_1=0,\
dtdw_2=0,\ (dw_1)^2=dt,\ (dw_2)^2=dt$ (since $w_1,w_2$ are both
$\in N(0,t)$) and $dw_1dw_2=\rho dt$ as per hint.

Collecting like terms

\begin{eqnarray*}
df&=&\left(\frac{\partial f}{\partial S_1}S_1\mu_1+\frac{\partial
f}{\partial
S_2}S_2\mu_2+\frac{1}{2}S_1^2\sigma_1^2\frac{\partial^2f}{\partial
S_1^2}+\frac{\partial^2f}{\partial S_1\partial
S_2}S_1S_2\sigma_1\sigma_2+\frac{1}{2}S_262+\sigma_2^2\frac{\partial^2f}{\partial
S_2^2}\right)dt\\&&+\left(\frac{\partial f}{\partial
S_1}S_1\sigma_1dw_1+\frac{\partial f}{\partial
S_2}S_2\sigma_2dw_2\right)
\end{eqnarray*}




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