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Consider three European call options with strikes $K_1<K_2<K_3$ all at the same expiration time T. If we assume the absence of arbitrage at all earlier times t, there is a derived equation from the properties of option $$C(K_2)<\frac{K_3-K_2}{K_3-K_1}C(K_1)+\frac{K_2-K_1}{K_3-K_1}C(K_3)$$ Could someone explain how to think about this equation?

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  • $\begingroup$ The relation above is not true. Could you please check? $\endgroup$
    – MainCom
    Mar 24, 2021 at 7:03
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    $\begingroup$ What do you want to know? This relation doesn't hold. It'll be different if the right hand is a sum instead of an inequality. Check for that. $\endgroup$
    – simsalabim
    Mar 24, 2021 at 8:13
  • $\begingroup$ Sorry about the typo. The right hand should be a sum. Could you explain how to show this relationship? $\endgroup$ Mar 24, 2021 at 16:17

1 Answer 1

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$C(K)$ is a convex function of strike, therefore it holds that:

$$C\left( t K_1 + (1-t) K_2 \right) \leq t C\left( K_1 \right) + (1-t) C\left( K_2 \right) $$ with $t = \frac{K_3-K_2}{K_3-K_1}$.

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