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QUESTION

Calculate the initial premium and the trading strategy for the
asset/bond replicating portfolio for a European put option on the
following data:

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Strike \$50, Maturity 1 year, two intervals;

Continuously compounded annual risk-free rate 5.60\%;

Volatility 20\%; current price \$50

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ANSWER

$k=50,\ r=0.056\Rightarrow e^{r\delta
t}=e^{0.056\times0.5}=1.02840,\ \sigma=0.2,\ s_0=50\\
U=e^{\left[\left(0.056-\frac{0.2^2}{2}\right)\times\frac{1}{2}
+0.2\sqrt{\frac{1}{2}}\right]}=1.17283\\
D=e^{\left[\left(0.056-\frac{0.2^2}{2}\right)\times\frac{1}{2}
-0.2\sqrt{\frac{1}{2}}\right]}=0.88389$

\underline{Europut} Summary:

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\begin{picture}(10,10)

\put(0,5.5){$P_0=2.583869$} \put(0,5){$\phi_0=-0.36796$}
\put(0,4.5){$\psi_0=20.98206$} \put(0,4){$S_0=50$}

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

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

\put(5,3.5){$P_1^-=5.315974$} \put(5,3){$\phi_1^-=-0.85648$}
\put(5,2.5){$\psi_1^-=43.16786$} \put(5,2){$S_1^-=44.1945$}

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

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

\put(5,7.5){$P_1^+=0$} \put(5,7){$\phi_1^+=0$}
\put(5,6.5){$\psi_1^+=0$} \put(5,6){$S_1^+=58.6415$}

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

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

\put(8,1){$P_2^-=10.93692$} \put(8,.5){$S_2^-=39.06308$}

\put(8,5){$P_2^0=0$} \put(8,4.5){$S_2^0=51.83264$}

\put(8,9.5){$P_2^+=0$} \put(8,9){$S_2^+=68.77651$}

\end{picture}



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