\documentclass[a4paper,12pt]{article}

\begin{document}

\parindent=0pt

QUESTION

A European call option with a strike price \$50 matures in one
year. Divide the one year period into two six-month intervals. The
continuously compounded risk-free rate of return is 5\% and the
volatility is 30\% per annum.

\begin{description}

\item[(a)]
Calculate the up and down factors of the binomial tree model for
the asset price.

\item[(b)]
Find the martingale probability, $q$, of an upstate occurring.

\item[(c)]
If the current stock price is \$40, determine the binomial value
of the option.

\item[(d)]
Hence provide the trading strategy.

\end{description}


ANSWER


\underline{Eurocall}

%this is diagram 1
\setlength{\unitlength}{.5in}

\begin{picture}(10,10)

\put(.5,5){40}

\put(1,5){\line(3,2){3}}

\put(1,5){\line(3,-2){3}}

\put(4,3){32.4352}

\put(5,3){\line(1,1){2}}

\put(5,3){\line(1,-1){2}}

\put(4,7){49.5764}

\put(5,7){\line(1,1){2}}

\put(5,7){\line(1,-1){2}}

\put(7,1){26.30105}

\put(7,5){40.20052}

\put(7,9){61.44549}

\end{picture}

$k=\$50,\ r=0.05,\ \sigma=0.3,\ e^{r\delta t}=1.02532$ in each 6
month period.

\begin{description}

\item[(a)]
Over $\delta t=\frac{1}{2}$ year
$\left\{\begin{array}{c}U=e^{\left[\left(0.05-\frac{0.3^2}{2}\right)
\frac{1}{2}+0.3\sqrt{\frac{1}{2}}\right]}=1.23941\\
D=e^{\left[\left(0.05-\frac{0.3^2}{2}\right)\frac{1}{2}-
0.3\sqrt{\frac{1}{2}}\right]}=0.81088\end{array}\right.$

\item[(b)]
$q=\frac{e^{r\delta t}-D}{U-D}=
\frac{e^{0.05\times\frac{1}{2}}-0.81088}{1.23941-0.81088}=0.50040$

\item[(c)]\

%this is diagram 2
\setlength{\unitlength}{.5in}

\begin{picture}(10,10)

\put(.5,5){$C_0$}

\put(1,5){\line(3,2){3}}

\put(1,5){\line(3,-2){3}}

\put(4,3){$C_1^-$}

\put(5,3){\line(1,1){2}}

\put(5,3){\line(1,-1){2}}

\put(4,7){$C_1^+$}

\put(5,7){\line(1,1){2}}

\put(5,7){\line(1,-1){2}}

\put(7,1){$C_2^-=\max\{61,44549-50,0\}$} \put(7.5,.5){$=11.44549$}

\put(7,5){$C_2^0=\max\{40.20051-50,0\}$} \put(7.5,4.5){$=0$}

\put(7,9){$C_2^+=\max\{26.30105-50,0\}$} \put(7.5,8.5){$=0$}

\end{picture}

Replicating portfolio at $t=\frac{1}{2}$:

\underline{Up state}

\begin{picture}(10,4)

\put(0,2){$v_1^+=49.5764\phi_1^++\psi_1^+$}

\put(1.5,1.2){shares}

\put(2,1.5){\vector(0,1){.4}}

\put(2.5,1.2){cash}

\put(2.8,1.5){\vector(0,1){.4}}

\put(3.5,2){\vector(1,1){1}}

\put(4,2.3){up at $t=1$}

\put(4.6,3){$v_2^+=61.44549\phi_1^++1.025324\psi_1^+$}

\put(3.5,2){\vector(1,-1){1}}

\put(4,1.5){down at $t=1$}

\put(4.6,1){$v_2^0=40.20051\phi_1^++1.02532\psi_1^+$}

\end{picture}

$\left.\begin{array}{c}v_2^+=c_2^+=11.44549\\v_2^0+c_2^0=0\end{array}\right\}$
Therefore solve

\begin{eqnarray*}
11.44549&=&61.44549\phi_1^++1.02532\psi_1^+\\
0&=&40.20051\phi_1^++1.02532\psi_1^+
\end{eqnarray*}

$\Rightarrow\left\{\begin{array}{c}\phi_1^+=0.53874\\\psi_1^+=-21.12274\end{array}\right.$

\underline{Down state}

\begin{picture}(10,4)

\put(0,2){$v_1^-=32.4352\phi_1^-+\psi_1^-$}

\put(3.5,2){\vector(1,1){1}}

\put(4,2.3){up at $t=1$}

\put(4.6,3){$v_2^0=40.20051\phi_1^-+1.02532\psi_1^-$}

\put(3.5,2){\vector(1,-1){1}}

\put(4,1.5){down at $t=1$}

\put(4.6,1){$v_2^-=26.30105\phi_1^-+1.02532\psi_1^-$}

\end{picture}

$v_2^0=v_2^-=0$ Therefore solve

\begin{eqnarray*}
0&=&40.20051\phi_1^-+1.02532\psi_1^-\\
0&=&26.30105\phi_1^-+1.02532\psi_1^-
\end{eqnarray*}

$\Rightarrow\left\{\begin{array}{c}\phi_1^-=0\\\psi_1^-=0\end{array}\right.$

So at $t=\frac{1}{2}$

$C_1^+=v_1^+=49.5764\times0.53874+(-21.12274)=5.58605$


$C_1^-=v_1^-=32.4352\times0+0=0$

At $t=0$

\begin{picture}(10,4)

\put(0,2){$v_0=40\phi_0+\psi_0$}

\put(2.5,2){\vector(1,1){1}}

\put(3,2.3){up at $t=\frac{1}{2}$}

\put(3.6,3){$v_1^+=49.5764\phi_0+1.02532\psi_0$}

\put(2.5,2){\vector(1,-1){1}}

\put(3,1.5){down at $t=\frac{1}{2}$}

\put(3.6,1){$v_2^-=32.4352\phi_0+1.02532\psi_0$}

\end{picture}

$v_1^+=C_1^+=5.58605$

$v_1^-=C_1^-=0$

Therefore solve

\begin{eqnarray*}
5.58605&=&49.5764\phi_0+1.02532\psi_0\\
0&=&32.4352\phi_0+1.02532\psi_0
\end{eqnarray*}

$\Rightarrow\left\{\begin{array}{c}\phi_0=0.32588\\\psi_0=-10.30910\end{array}\right.$

Therefore $v_0=40\times0.32588-10.30910=2.72610=C_0$

\underline{Summary}


%this is diagram 3
\setlength{\unitlength}{.5in}

\begin{picture}(10,10)

\put(0,5.5){$C_0=2.72610$} \put(0,5){$\phi_0=0.32588$}
\put(0,4.5){$\psi_0=-10.30910$}

\put(2,5){\line(3,2){3}}

\put(2,5){\line(3,-2){3}}

\put(5,3.5){$C_1^-=0$} \put(5,3){$\phi_1^-=0$}
\put(5,2.5){$\psi_1^-=0$}

\put(6,3){\line(1,1){2}}

\put(6,3){\line(1,-1){2}}

\put(5,7.5){$C_1^+=5.58605$} \put(5,7){$\phi_1^+=0.53874$}
\put(5,6.5){$\psi_1^+=-21.12274$}

\put(7,8){\line(1,1){1}}

\put(7,6){\line(1,-1){1}}

\put(8,1){$C_2^-=0$}

\put(8,5){$C_2^0=0$}

\put(8,9.5){$C_2^+=11.44549$} \put(8,9){$\phi_2^+=0$}
\put(8,8.5){$\psi_2^+=0$}

\end{picture}

\item[(d)]
\underline{Trading strategy:}

\begin{tabular}{crcr}
$t=0$&buy 0.32588 shares at \$40, cost&=&\$13.03520\\ &finance it
by borrowing&=&$-10.30910$\\&by charging owner of
call&=&$-2.72610$\\&self financing&$\rightarrow$&0.00\\
$t=\frac{1}{2}$&if up then buy an extra $(0.53874-0.32588)$\\
&shares at a cost of $0.21286\times49.5764$&=&\$10.55283\\
&finance by extra borrowing of
$-(21.12274-$\\&$\underbrace{1.02532}_{\textrm{interest on
borrowing}}\times\underbrace{10.30910}_{\textrm{ origional
borrowing}})$&=&$-10.55261$\\ &self financing&$\rightarrow$&0.00\\
&if down then sell all: recieve
$32.4352\times0.32588$&=&\$10.56998\\ &pay back borrowing of
$-1.02532\times10.30910$&=&$-10.57013$\\ &Break
even&$\rightarrow$&0.00
\end{tabular}

Self financing!

The only other interesting one is in upstate at $t=1$. The owner
will exercise and demand 1 share:

\begin{tabular}{rcc}
Issuer must buy $(1-0.53874)$ shares at\\
$-0.46126\times61.44549$&=&$-\$28.34235$\\sells to owner at
$k=\$50$ so receives &=&\$50.00\\&&\$21.65765\\Needs to repay
$21.12274\times1.02532$&=&$-\$21.65757$\\potential loss covered,
break even &$\rightarrow$&0.00
\end{tabular}

\end{description}



\end{document}