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QUESTION

Let $S$ behave lognormally such that $ds=S(\mu dt+\sigma dW)$.
Write down It\^{o}'s lemma for a general (deterministic) $f(S)$.
Hence find the stochastic differential equations satisfied by:

\begin{description}

\item[(i)]
$f(S)=As+B$;

\item[(ii)]
$g(S)=S^n$,

\end{description}
where $A,\ B,\ n$ are constants.


ANSWER

$$ds=S(\mu dt+\sigma dw)$$

\begin{eqnarray*}
df&=&\frac{\partial f}{\partial
S}ds+\frac{1}{2}\frac{\partial^2f}{ds^2}(ds)^2\textrm{ no time
dependence}\\ df&=&\frac{\partial f}{\partial S}(S\mu dt+S\sigma
dw)+\frac{1}{2}\frac{\partial^2f}{\partial S^2}(S^2(\mu dt+\sigma
dw)^2)\\ &=&f'S\mu dt+f'S\sigma
dw+\frac{1}{2}f''S^2(\mu(dt)^2+2\mu dt dw \sigma+\sigma^2(dw)^2)
\end{eqnarray*}

But rule of thumb $\Rightarrow (dt)^2=0,\ (dtdw)=)$ (i.e.
negligible with respect to dt,) and$(dw)^2=dt$. Therefore

\begin{eqnarray*}
df&=&f'S\mu dt+f'S\sigma dw+\frac{1}{"}f''S^2\sigma^2dt\\
\textrm{or }df&=&f'S\sigma dw+(f'S\mu+\frac{1}{2}f''S^2\sigma^2)dt
\end{eqnarray*}

\begin{description}

\item[(i)]
\begin{eqnarray*}
f(S)&=&As+B,\ f'=A,\ f''=0,\ As=f-B\\ df&=&As\sigma dw+As\mu dt\\
df&=&(f-B)[\sigma dw+\mu dt]
\end{eqnarray*}

\item[(ii)]
\begin{eqnarray*}
g(S)&=&S^n\Rightarrow g'=ns^{n-1},\ g''=n(n-1)S^{n-2}\\
dg&=&ns^{n-1}.S\sigma
dw+\left[ns^{n-1}S\mu+\frac{n(n-1)S^{n-2}S^2\sigma^2}{2}\right]dt\\
dg&=&g\left[n\sigma
dw+\left(n\mu+\frac{n(n-1)}{2}\sigma^2\right)dt\right]\textrm{ as
}g=S^n\\ \textrm{ or }dg&=&ng\left[\sigma
dw+\left(\mu+\frac{(n-1)}{2}\sigma^2\right)dt\right]
\end{eqnarray*}

\end{description}



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