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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 $K_1&lt;K_2$, 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 $T_1&lt;T_2$ 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 $k&gt;0,\ o&lt;C&lt;S.$

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

Now RHS has spread from $S-(S-k)=k (S&gt;k)$ to $S-0=S\ (S&lt;k)$

Therefore $\begin{array}{ll}S-c=k,&amp;S&gt;k\\S-c=S,&amp;S&lt;k\end{array}$

Therefore $S-c&gt;k$.

\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&lt;k_1&lt;k_2,RHS&amp;=&amp;0\\ k_1&lt;S&lt;k_2,RHS&amp;=&amp;S-k_1\\
k_1&lt;k_2&lt;S,RHS&amp;=&amp;S-k_1-S+k_2=k_2-k_1
\end{eqnarray}

Now consider (4) versus (5). In (4), $S&lt;k_2$ so (5) is $&gt;$(4).
Also $S-k_1&gt;0$ since $S&gt;k_1$ therefore (4)$&gt;$(3)

Therefore (5)$&gt;$(4)$&gt;$(3)

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

\item[(d)]
$T_1&lt;T_2$

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

If result is not true (i.e. $C(S,t,T_1)&gt;C(S,t,T_2)$),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 $C(S,t,T_1)-C(S,t,T_2)&gt;0$
therefore must have $C(S,t,T_1)\leq C(S,t,T_2).$

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