General Arbitrage Relations

Submitted by loner on 7 January, 2008 - 12:49pm

Derivatives
Finance

In the market for standard options, if there are no riskless arbitrage opportunities, options must satisfy the four general arbitrage relations given in the attached picture.

Hedge Relation:

$<br /> S\ge C \ge \max \left ( S - K, Sd^{-t} - Kr^{-t}\right )$
Bull Spread Relation:
for $K_1 < K_2$ ,

$<br /> C\left ( K_1\right ) > C\left ( K_2\right ) \wedge C\left ( K_1\right ) - C\left ( K_2\right ) < K_2 - K_1$
Butterfly Spread Relation:
for $K_1 < K_2 < K_3$ equally spaced,

$<br /> C\left ( K_2\right ) < 0.5 \times \left [ C\left (K_1\right ) + C\left (K_3\right )\right]$
Time Spread Relation:
for $t_1 < t_2$ ,

$<br /> C\left (t_2\right ) \ge C\left (t_1\right )$