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\textbf{Question}

In the question YOU MAY ASSUME
\begin{description}
\item{(i)}
that all Calls and Puts are of European type,

\item{(ii)}
that (whether bought long or short) all Calls may be purchased for a
constant value $C$ and all Puts may be purchased for a constant value
$P$,

\item{(iii)}
that all calculations are to be made from the point of view of the
holder of the option (rather than that of the writer).
\end{description}

\begin{description}
%Question 2a
\item{(a)}
A STRADDLE is an option strategy that consists of a position where one
is a long one Call and long one Put, both with same strike $E$ and
expiry $T$. At expiry the underlying has a value $S(T)$. What
conditions must $S(T)$ satisfy in order for a straddle to be
profitable?

Draw a profit diagram for a straddle, plotting the profit at expiry
against $S(T)$. If an investor buys a straddle, what view is she or he
taking of the likely behaviour of the underlying?

%Question 2b
\item{(b)}
A BUTTERFLY SPREAD is an option strategy that consists of a position
where one is long one Call with a strike $E-K$, long one call with a
strike $E+K$ and short two calls, both with strike $E$, where the
constant $K$ satisfies $4C<K<E$. All Calls are assumed to have the
same expiry $T$. At expiry the underlying has a value $S(T)$. What
conditions must $S(T)$ satisfy in order for a butterfly spread to be
profitable? 

Draw a profit diagram for a butterfly spread, plotting the profit at
expiry against $S(T)$. If an investor buys a butterfly spread, what
view is she or he taking of the likely behaviour of the underlying?
\end{description}

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\textbf{Answer}

\begin{description}
%Question 2a
\item{(a)}
The total payoff from a straddle is that due to one call and one put,
i.e.
$$\rm{max}(S-E,0)+\rm{max}(E-S,0).$$
In order for the straddle to be profitable to the holder we therefore
require
$$\rm{max}(S-E,0)+\rm{max}(E-S,0)-C-P>)$$

There are two cases:
\begin{description}
\item{(i)} $S>E$ in which case profit$=S_E+0-C-P$ and so for
profit$>0$ we need $S>E+C+P$

\item{(ii)} $S<E$ in which case profit$=0+E-S-C-P$ and so for
profit$>0$ we need $S<E-C-P$.
\end{description}
So for profit we need
$$\rm{EITHER}\ S>E+C+P \ \ \ \rm{OR} \ \ \ S<E-C-P$$

Profit diagram:-
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(Diagram must have all these labels for full marks)

-Investor buys a straddle if she or he believes that the underlying
 will move a long way in price, but doesn't know which way.

\item{(b)}
 Payoff from Butterfly Spread=
$$\rm{max}(S-E+K,0)+\rm{max}(S-E-K,0)-2\rm{max}(S-E,0)$$
$\Rightarrow \rm{Profit}=\rm{max}(S-E+K,0)+\rm{max}(S-E-K,0)-
2\rm{max}(S-E,0)- 4C$

There are now 4 cases to consider:-

$\begin{array}{cl} (i) & S<E-K \\ &\rm{Profit} =0+0-2(0)-4C\\
& =-4C\\
(ii) & E-K <S<E \\ &\rm{Profit}=S-E+K+0-2(0)-4C\\
& = S-E+K-4C\\
(iii) & E<S<E+K \ \ \\ &\rm{Profit}=S-E+K+0-2(S-E)-4C\\
& = -S+E+K-4C\\
(iv) & E+K<S \\ &\rm{Profit}=S-E+K+S-E-K-2S+2E-4C\\
& =-4C
\end{array}$

Thus to be profitable we need
\begin{eqnarray*}
S-E+K-4C>0 & \Rightarrow & S>4C+E-K \ \ \ \rm{or}\\
-S+E+K-4C>0 & \Rightarrow & S< -4C+E+K\\
\rm{i.e.} E-K+4C < & S & < E + K -4C
\end{eqnarray*}

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(Again, need all labels for full marks)

An investor buys a butterfly spread if she or he considers that
changes in the price of the underlying will be small.

\end{description}

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