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QUESTION

Using the table provided, calculate the continuous-time premiums
of the European call and put options of questions 1 and 2 of
exercises 4 if the underlying asset pays a continuoue dividend of
$0.01Sdt$, where $S$ is the asset price at time $t$.

ANSWER

Use formula with $D=0.01$ in:

\underline{call}
\begin{eqnarray*}
C(S,t)&=&se^{-D(T-t)}N(d_1)-ke^{-r(T-t)}N(d_2)\\
d_1&=&\frac{\left[\log\left(\frac{S}{k}\right)+
(r-D+\frac{1}{2}\sigma^2)(T-t)\right]}{\sigma\sqrt{T-t}}\\
d_1&=&\frac{\left[\log\left(\frac{S}{k}\right)+
(r-D-\frac{1}{2}\sigma^2)(T-t)\right]}{\sigma\sqrt{T-t}}
\end{eqnarray*}

Therefore at $t=0$, initial premium given by
\begin{eqnarray*}
d_1&=&\frac{\left[\log\left(\frac{40}{50}\right)+
\left(0.05-0.01+\frac{0.3^2}{2}\right)\right]}{0.3}=-0.4605\\
d_2&=&\frac{\left[\log\left(\frac{40}{50}\right)+
\left(0.05-0.01-\frac{0.3^2}{2}\right)\right]}{0.3}=-0.7605
\end{eqnarray*}

$N(-0.46)=0.3228\\N(-0.76)=0.2236$

Therefore
\begin{eqnarray*}
C(40,0)&=&40\times
e^{-0.01}\times(0.3228)-50e^{-0.05}\times0.2236\\
&=&12.7835-10.6347\\&=&2.1488
\end{eqnarray*}

which is less than the value of the option without a continuous
dividend (since asset price will be falling by $DSdt$).

\underline{Put}
\begin{eqnarray*}
P(S,t)&=&-se^{-D(T-t)}N(-d_1)+ke^{-r(T-t)}N(-d_2)\\ &=&-40\times
e^{-0.01}N(+0.46)+50e^{-0.05}N(+0.76)\\ &=&-40\times
e^{-0.01}\times0.6772+50\times e^{-0.05}\times0.7764\\ &=&10.1083
\end{eqnarray*}



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