\documentclass[a4paper,12pt]{article}

\begin{document}

\parindent=0pt

QUESTION

 A European put 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}
Hence verify that the values of
the call and put satisfy the put-call parity formula.


ANSWER

\underline{Europut}: same prices U, D, $r,\ \sigma,\ e^{r\delta
t}$

\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}

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}

$\left.\begin{array}{c}v_2^+=P_2^+=0\\v_2^0+P_2^0=9.79949\end{array}\right\}$
Therefore solve

\begin{eqnarray*}
0&=&61.44549\phi_1^++1.02532\psi_1^+\\
9.79949&=&40.20051\phi_1^++1.02532\psi_1^+
\end{eqnarray*}

$\Rightarrow\left\{\begin{array}{c}\phi_1^+=-0.46126\\\psi_1^+=27.64253\end{array}\right.$

\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}

$\left\{\begin{array}{c}v_2^0=P_2^0=9.79949\\v_2=P_2^-=23.69895\end{array}\right.$
Therefore solve

\begin{eqnarray*}
9.79949&=&40.20051\phi_1^-+1.02532\psi_1^-\\
23.69895&=&26.30105\phi_1^-+1.02532\psi_1^-
\end{eqnarray*}

$\Rightarrow\left\{\begin{array}{c}\phi_1^-=-1.0000\\\psi_1^-=48.76526\end{array}\right.$

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

$P_1^+=v_1^+=49.5764\times(-0.46126)+27.64253=4.77492$


$P_1^-=v_1^-=32.4352\times9-1.0000)+48.76526=16.33006$

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}

$v_1^+=P_1^+=4.77492$

$v_1^-=P_1^-=16.33006$

Therefore solve

\begin{eqnarray*}
4.77492&=&49.5764\phi_0+1.02532\psi_0\\
16.33006&=&32.4352\phi_0+1.02532\psi_0
\end{eqnarray*}

$\Rightarrow\left\{\begin{array}{c}\phi_0=-0.67411\\\psi_0=37.25190\end{array}\right.$

Therefore $v_0=40\times-0.67411+37.25190=10.28750=P_0$

\underline{Summary}

\setlength{\unitlength}{.5in}

\begin{picture}(10,10)

\put(0,5.5){$P_0=10.28750$} \put(0,5){$\phi_0=-0.67411$}
\put(0,4.5){$\psi_0=37.25190$}

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

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

\put(5,3.5){$P_1^-=16.33006$} \put(5,3){$\phi_1^-=-1.0000$}
\put(5,2.5){$\psi_1^-=48.76526$}

\put(7,4){\line(1,1){1}}

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

\put(5,7.5){$P_1^+=4.77492$} \put(5,7){$\phi_1^+=-0.46126$}
\put(5,6.5){$\psi_1^+=27.64253$}

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

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

\put(8,1){$P_2^-=23.69895$}

\put(8,5){$P_2^0=9.79949$}

\put(8,9.5){$P_2^+=0$}

\end{picture}

\begin{tabular}{crcr}
$t=0$&sell short (don't own but trade on account) 0.67411\\&
shares and receive $0.67411\times 40$&=&\$26.96440\\ &invest an
amount&=&$-\$37.25190$\\&Need an extra &&$-\$10.28750$\\ &get from
owner of put option$\Rightarrow$\\& self financing portfolio.\\
$t=\frac{1}{2}$& If up then short 0.46126 shares, i.e. buy\\&
(0.67411-0.46126) shares at 49.5764 at a cost of\\&
$0.21285\times49.5764$&=&$\$10.55234$\\ &Financed by investment
which has risen to\\& $37.25190\times1.02552=\$38.19512$. Pay
$\$10.55234$ for\\& shares to leave
$=38.19512-10.55234$&=&$\$27.64278$ \\&If down then short 1.000
shares i.e. sell on\\&account $(1-0.67411)=0.32589$ and receive
\\&$0.32589\times32.4352$&=&\$10.57031\\&Thus total investment is
now \\& $37.25190\times1.02532+10.57031$&=&\$48.76543
\\ $t=1$&3 possible outcomes, but to save the trees, just\\& consider
the down state, $P_2^-=23.69895$\\& Issuer has a liability of 1
short share,\\& but has saved
$1.02532\times48.76526$&=&\$50.00\\&Must accept 1 share from
owner\\& of put (who exercises) at strike price of\\& \$50. Cost
to issuer $=1\times\$50$&=&$-\$50.00$\\&break even
&$\rightarrow$&0.00
\end{tabular}

What happens to the 1 share the issuer receives? He has to
complete his shorting contract at the end of the account period,
so uses it for that: liability 1 share, owns 1 share, net
result$\rightarrow$ break even. (NB receives no cash for
completing this contract as he's already received money for it at
$t=0,\ t=\frac{1}{2}$).

Call/put parity formula:

$$C(t)-P(t)=S_t-ke^{-r{T-t}}$$

Use $C,\ S,\ k,\ r$ to find $P=C-S+ke^{-r(T-t)}$

\underline{T=1} (Just do up states)

\begin{tabular}{cccc}
&$t=0$&$t=\frac{1}{2}$&$t=1$\\ $C$&2.72610&5.58605&11.44549\\
$S$&40&49.5764&61.44549\\ $k$&50&50&50\\ $T-t$&1&$\frac{1}{2}$&0\\
$C-S+ke^{-r(T-t)}$&$10.28757=P_0$&$4.77515=P_1^+$&$0=P_2^+$
\end{tabular}

The last row is the put prices (up to rounding errors.)



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