\documentclass[a4paper,12pt]{article}

\begin{document}

\parindent=0pt

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. Compare the binomial and continuous answers.


ANSWER

$T=1,\ k=\$50,\ r=5\%=0.05,\ \sigma=30\%=0.3,\ S_0=\$40$ No
dividend $\Rightarrow D=0$.

\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*}

Plug in numbers: at $t=0$ for initial premium (NB $\log=\log_e$)

\begin{eqnarray*}
d_1&=&\frac{\left[\log\left(\frac{40}{50}\right)+
\left(0.05+\frac{0.3^2}{2}\right)\right]}{0.3}=-0.4271\\
d_2&=&\frac{\left[\log\left(\frac{40}{50}\right)+
\left(0.05-\frac{0.3^2}{2}\right)\right]}{0.3}=-0.7271
\end{eqnarray*}

Look up in table: Table only works to 2 d.p. so look up
$N(-0.43)=0.3336$ and $N(-0.73)=0.2327$

Therefore $C(S_0,0)=40\times0.3336-50\times
e^{-0.5}\times0.2327=2.2764$

Binomial value is $2.7261$.

\underline{Put}

$P(S,t)=-se^{-D(T-t)}N(-d_1)+ke^{-r(T-t)}N(-d_2)$

$d_1$ and $d_2$ are the same as above.

\begin{eqnarray*}
P(S_0,0)&=&-40\times N(+0.43)+50e^{-0.05}N(0.73)\\
&=&-40\times0.6664+50e^{-0.05}\times0.7673\\ &=&9.8379
\end{eqnarray*}

Binomial value is 10.28750.



\end{document}
