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\begin{document}

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QUESTION

Draw the expiry payoff diagrams for each of the following
portfolios (ignore premium costs):

\begin{description}

\item[(a)]
Short one share, long two calls with exercise price $K$ (this one
is called a straddle);

\item[(b)]
Long one call and one put, both with exercise price $K$ (this is
also a straddle);

\item[(c)]
Long one call, and two puts, all with exercise price $K$ (a
strap);

\item[(d)]
Long one put and two calls, all with exercise price $K$ (a strip);

\item[(e)]
Long one call with exercise price $K_1$ and one put with exercise
price $K_2$; compare the three cases $K_1>k_2$ (also a strangle),
$k_1=k_2$ and $k_1<k_2$;

\item[(f)]
As in (e), but also short one call and one put with exercise price
$K$, where now $k_1<K<k_2$ (a butterfly spread).

\end{description}


ANSWER

In what follows payoff is drawn for the owner of the portfolio and
ignores premium.

\begin{description}

\item[(a)]
\lq\lq Short'' means you have a liability to but a share rather
than being \lq\lq long'' which means you own it. Thus the payoff
from a short share is

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\put(1.1,3.8){Payoff of short share}

\put(3.8,.2){$-S$}

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Payoff of a call to the owner is $=\max(s-k,0)$

Thus payoff of 2 calls is $=2\times\max(s-k,0)$

Thus the total payoff of portfolio is $=-s+2\max{s-k,0}$

Graphically

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\put(1,0){\vector(0,1){5}}

\put(0,4.8){payoff}

\put(3,2.1){$k$}

\put(5,2.1){$2k$}

\put(5,2){\circle*{.1}}

\put(0.8,1.7){0}

\put(1,2){\circle*{.1}}

\put(.8,0){$-k$}

\put(3,0){\circle*{.1}}

\put(6,2){$S$}

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

\put(3,0){\line(1,1){4}}

\put(2.,1){$-s+0$}

\put(5.5,4){$-s+2(s-k)=s-2k$}

\put(6,1){\lq\lq STRADDLE''}

\end{picture}

\item[(b)]
Long call payoff$=\max(s-k,0)$

Long put payoff$=\max(k-s,o)$

Total
payoff$=\max(s-k,0)+\max(k-s,0)=\left\{\begin{array}{cc}k-s,&k\geq
s\\s-k,&k\leq s\end{array}\right.$ Similar shape to other \lq\lq
straddle'' hence name.

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\put(4,1){\circle*{.1}}

\put(0.8,0.7){0}

\put(.8,4){$k$}

\put(1,4){\circle*{.1}}

\put(6,1){$S$}

\put(5,.3){\lq\lq STRADDLE''}

\put(1,4){\line(1,-1){3}}

\put(4,1){\line(1,1){3}}

\put(2.4,3){$k-s$}

\put(5.2,3){$s-k$}

\end{picture}

\item[(c)]
Long one call $=\max(s-k,0)$

Long two puts$=2\max(k-s,0)$

Total
payoff$=\max(s-k,0)+2\max(k-s,0)=\left\{\begin{array}{cc}s-k,&s\geq
k\\2(k-s),&s\leq k\end{array}\right.$

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\put(1,0){\vector(0,1){5}}

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\put(4,1.1){$k$}

\put(4,1){\circle*{.1}}

\put(0.8,0.7){0}

\put(.6,4){$2k$}

\put(1,4){\circle*{.1}}

\put(6,1){$S$}

\put(5,.3){\lq\lq STRAP''}

\put(1,4){\line(1,-1){3}}

\put(4,1){\line(2,1){3}}

\put(2.4,3){$2k-2s$}

\put(5.2,2){$s-k$}

\end{picture}


\item[(d)]
Long one put$=\max(k-s,0)$

Long two calls$=2\max(s-k,0)$

Total
payoff$=\max(k-s,0)+2\max(s-k,0)=\left\{\begin{array}{cc}2(s-k),&s\geq
k\\k-s,&s\leq k\end{array}\right.$

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\put(4,1.1){$k$}

\put(4,1){\circle*{.1}}

\put(0.8,0.7){0}

\put(.8,2.5){$k$}

\put(1,2.5){\circle*{.1}}

\put(6,1){$S$}

\put(5,.3){\lq\lq STRIP''}

\put(1,2.5){\line(2,-1){3}}

\put(4,1){\line(1,1){3}}

\put(2.4,2){$k-s$}

\put(5.2,3){$2s-2k$}

\end{picture}

\item[(e)]
Long one call, strike $k_1=\max(s-k_1,0)$

Long one put, strike $k_2=\max(k_2-s,0)$

$\underline{k_1>k_2}$

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\put(.6,3){$k_2$}

\put(1,3){\circle*{.1}}

\put(3,.7){$k_2$}

\put(3,1){\makebox(0,0){x}}

\put(5,.7){$k_1$}

\put(5,1){\makebox(0,0){x}}

\put(0.8,.7){0}

\put(1,3){\line(1,-1){2}}

\put(3,1.02){\line(1,0){2}}

\put(5,1){\line(1,1){2}}

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$\underline{k_1=k_2}$

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$\underline{k_1<k_2}$

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\put(.6,3){$k_2$}

\put(1,3){\circle*{.1}}

\put(0,2){$k_2-k_1$}

\put(1,2){\circle*{.1}}

\put(3,.7){$k_1$}

\put(3,1){\circle*{.1}}

\put(5,.7){$k_2$}

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\put(3,2){\line(1,0){2}}

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\item[(f)]
Short one call$=-\max(s-k,0)$

Short one put$=-\max(k-s,0),\ k_1<k<k_2$

Long one call$=\max(s-k,0)$

Long one put$=\max(k_2-s,0)$

Total payoff$=\max(k_2-s,0)+\max(s-k_1,0)-\max(k-s,0)-\max(s-k,0)$

Split into several ranges

\begin{tabular}{ccc}
$0<s<k_1(<k<k_2)\Rightarrow$&$-(k-s)+k_2-s$&$=k_2-k$\\
$0<k_1<s<k<k_2\Rightarrow$&$-(k-s)+s-k_1+k_2-s$&$=k_2-k-k_1+s$\\
$0<k_1<k<s<k_2\Rightarrow$&$-(s-k)+s-k_1+k_2-s$&$=k_2+k-k_1-s$\\
$0<k_1<k<k_2<s\Rightarrow$&$-(s-k)+s-k_1$&$=k-k_1$
\end{tabular}

Graphically:

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\put(6,1){$S$}

\put(2,.7){$k_1$}

\put(3,.7){$k$}

\put(4,.7){$k_2$}

\put(0,2){$k-k_1$}

\put(0,2.5){$k_2-k$}

\put(0,4){$k_2-k_1$}

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\put(2,2.5){\line(2,3){1}}

\put(3,4){\line(1,-2){1}}

\put(4,2){\line(1,0){2}}

\end{picture}

\end{description}




\end{document}
