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QUESTION
\begin{description}
\item[(a)]
An investor wishes to trade in options on an asset whose current
price one year from the maturity date of an option is \$25, the
exercise price of the option is \$20, the risk-free interest rate
is 5\% per annum and the asset volatility is 20\% per annum.
Calculate by what amount the asset price has to change for the
purchaser of a European call option to break even giving your
answer to 4 decimal places?
\item[(b)]
Write down the call-put parity formula for European options. Hence
repeat part (a) but for a European put.
\item[(c)]
Sketch the qualitative behaviour of the European call and put
values over the lifetime of the option as a function of the
underlying asset price.
\item[(d)]
Calculate the initial price of the call option in part (a) if the
asset pays a continuous dividend of $DS$ where $S$ is the asset
price and $D=0.01$.
\end{description}
You may assume that the solution of the Black-Scholes equation for
a European call option, paying no dividends, is given by,
$$c(S,t)=SN(d_1)-K\exp(-r(T-t))N(d_2),$$
$$d_1=\frac{\log\left(\frac{S}{K}\right)+
\left(r+\frac{\sigma^2}{2}\right)(T-t)}{\sigma\sqrt{T-t}},$$
$$d_2=\frac{\log\left(\frac{S}{K}\right)+
\left(r-\frac{\sigma^2}{2}\right)(T-t)}{\sigma\sqrt{T-t}}.$$
ANSWER
\begin{description}
\item[(a)]
$T=1,\ S_0=25,\ K=20,\ r=0.05,\ \sigma=0.2$
For the holder of a Eurocall, the asset price must rise by the
following to break even:
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\put(4,1.5){\line(1,1){2}}
\put(.5,1.6){$C_0$}
\put(3.9,2.1){$K$}
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\put(6,2){$S$}
\put(0,3.6){$C(S,t)$}
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Payoff at $t=T$.
Therefore the price must rise to $K+C$ for the holder to break
even. If the initial asset price is $S_0$, requires final asset
price is $K+C_0$ so the rise must be $K+C_0-S_0$ Therefore we need
to know the initial premium at $S_0$.
Use the formula given at $t=0$.
\begin{eqnarray*}
C(S_0,0)&=&S_0N(d_1(0))-Ke^{-rT}N(d_2(0))\\
d_1(0)&=&\frac{\left(\log\left(\frac{S_0}{K}\right)+
\left(r+\frac{\sigma^2}{2}T\right)\right)}{\sigma\sqrt{T}}\\
d_2(0)&=&\frac{\left(\log\left(\frac{S_0}{K}\right)+
\left(r-\frac{\sigma^2}{2}T\right)\right)}{\sigma\sqrt{T}}
\end{eqnarray*}
Feed in the above data to get
$\left.\begin{array}{c}d_1=1.47\\d_2=1.27\end{array}\right\}$ to 2
d.p.
We need to find $N(1.47)$ and $N(1.27)$. From the tables,
$N(1.47)=0.9297,\ N(1.27)=0.8980$
$$C(S,0)=25\times0.9292-20e^{-0.05}\times0.8980=6.1459$$
Therefore to break even they need a new price of
$K+C-0=20+6.1459=26.1459$
Therefore the current price needs to rise by $K+C_0-S=1.1459$.
\item[(b)]
The call-put parity formula is $$C(S,t)-P(S