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QUESTION

Prove the following simple bounds on European call options on an
asset that pays no dividends:

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\item[(a)]
$C\leq S$

\item[(b)]
$C\geq S-k\exp(-r[T-t])$

\item[(c)]
If two otherwise identical calls have exercise prices $K_1$ and
$K_2$ with $K_1<K_2$, then

$$0\leq C(S,t,K_1)-C(S,t,k_2)\leq k_2-k_1.$$

\item[(d)]
If two otherwise identical call options have expiry times $T_1$
and $T_2$ and $T_1<T_2$ then

$$C(S,t,T_1)\leq C(S,t,T_2).$$

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ANSWER

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\item[(a)]
$C=\max(S-k,0)$ so obviously with $k>0,\ o<C<S.$

\item[(b)]
Consider $S-c=S-\max(S-k,o)$

Now RHS has spread from $S-(S-k)=k (S>k)$ to $S-0=S\ (S<k)$

Therefore $\begin{array}{ll}S-c=k,&S>k\\S-c=S,&S<k\end{array}$

Therefore $S-c>k$.

\item[(c)]
$C(S,t,k_1)-C(S,t,k_2)=\max(S-k_1,0)-\max(S-k_2,0)$

Now if

\begin{eqnarray}
S<k_1<k_2,RHS&=&0\\ k_1<S<k_2,RHS&=&S-k_1\\
k_1<k_2<S,RHS&=&S-k_1-S+k_2=k_2-k_1
\end{eqnarray}

Now consider (4) versus (5). In (4), $S<k_2$ so (5) is $>$(4).
Also $S-k_1>0$ since $S>k_1$ therefore (4)$>$(3)

Therefore (5)$>$(4)$>$(3)

Therefore $)\leq C(S,t,k_1)-C(S,t,k_2)\leq k_2-k_1$

\item[(d)]
$T_1<T_2$

Prove: $C(S,y,T_1)\leq C(S,t,T_2)$

If result is not true (i.e. $C(S,t,T_1)>C(S,t,T_2)$),buy
longer-dated call and write the other.

This violates the arbitrage concept since you receive $C(S,t,T_1)$
and payout $C(S,t,T_2)$ making profit of $C(S,t,T_1)-C(S,t,T_2)>0$
therefore must have $C(S,t,T_1)\leq C(S,t,T_2).$

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