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QUESTION

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

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

Continuously compounded annual risk-free rate 5\%;

Volatility 30\%; Current price \$50.

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



ANSWER

$k=50,\ r=0.05,\ \sigma=0.3$ (higher than question 3), $S_0=50$
(different from question 1),

$U=$ same as question 1$=1.23941,\\ D=$ same as question
1$=0.81088$

\underline{Eurocall} Summary:

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\begin{picture}(10,10)

\put(0,5.5){$C_0=6.504431$} \put(0,5){$\phi_0=0.610595$}
\put(0,4.5){$\psi_0=-24.0253$} \put(0,4){$S_0=50$}

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

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

\put(5,3.5){$C_1^-=0.122325$} \put(5,3){$\phi_1^-=0.014426$}
\put(5,2.5){$\psi_1^-=-0.46256$} \put(5,2){$S_1^-=40.544$}

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

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

\put(5,7.5){$C_1^+=13.20524$} \put(5,7){$\phi_1^+=1$}
\put(5,6.5){$\psi_1^+=-48.7653$} \put(5,6){$S_1^+=61.9705$}

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

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

\put(8,1){$C_2^-=0$} \put(8,.5){$S_2^-=32.87632$}

\put(8,5){$C_2^0=0.250639$} \put(8,4.5){$S_2^0=50.25064$}

\put(8,9.5){$C_2^+=26.80686$} \put(8,9){$S_2^+=76.80686$}

\end{picture}

$C_0$ increases with $\sigma$ increasing. Logic: more
volatility=more risk=more insurance needed=higher premiums.




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