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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 30\%; current price \$50

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What do you deduce about the way the premium behaves with
volatility for a put option?


ANSWER

$k=50,\ r=0.056\Rightarrow e^{r\delta t}=1.02840,\ \sigma=0.3,\
s_0=50\\
U=e^{\left[\left(0.056-\frac{0.3^2}{2}\right)\times\frac{1}{2}
+0.3\sqrt{\frac{1}{2}}\right]}=1.24313\\
D=e^{\left[\left(0.056-\frac{0.3^2}{2}\right)\times\frac{1}{2}
-0.3\sqrt{\frac{1}{2}}\right]}=0.81332$

\underline{Europut} Summary:

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\begin{picture}(10,10)

\put(0,5.5){$P_0=3.994391$} \put(0,5){$\phi_0=-0.3826$}
\put(0,4.5){$\psi_0=-23.12461$} \put(0,4){$S_0=50$}

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

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

\put(5,3.5){$P_1^-=8.222358$} \put(5,3){$\phi_1^-=0.96835$}
\put(5,2.5){$\psi_1^-=-47.60145$} \put(5,2){$S_1^-=40.666$}

\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){$S_1^+=62.1565$}

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

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

\put(8,1){$P_2^-=19.92553$} \put(8,.5){$S_2^-=33.07447$}

\put(8,5){$P_2^0=0$} \put(8,4.5){$S_2^0=50.55312$}

\put(8,9.5){$P_2^+=0$} \put(8,9){$S_2^+=77.26861$}

\end{picture}

Premium $P_0$ increases as $\sigma$ increases. (Same as call,
higher risk$\Rightarrow$ higher premiums.)





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