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We observe Euro-call option prices at discrete strikes, say, $C_1>C_2>C_3...$ at several strikes $K_1<K_2<K_3,...$. (The maturity is the same)

It is well-known that the option prices are convex with repsect to $K$. Hence, it is easy to see that, we can just link these points to have a piecewise linear prices line w.r.t. $K$ as an upper bound the prices between $K_i$ and $K_{i+1}$.

However, what about the lower bounds? It there any simple lower bound for the prices between $K_i$ and $K_{i+1}$?

Due to the convexity, it seems that setting the price between $K_i$ and $K_{i+1}$ equal to $C_i$ could violate the convexity.

Jingjings
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  • Are you assuming Black-Scholes? Constant volatility/drift? This is not so clear. – parsiad Jul 28 '16 at 13:44
  • This question is surely more suitable for the quant stack exchange. However, I would not be surprised to find an answer here. –  Jul 28 '16 at 13:45
  • You want a bound independent of the distribution? But if the underlying price is bounded by $S$ then $K>S\implies C(K)=0$, so the price differences are all $0$ beyond some point. Of course you can approximate a bounded distribution as well as you like by letting volatility go to $0$. – lulu Jul 28 '16 at 13:46
  • All I know is simply some prices at several strikes. I don't assume any distribution. The bound should give the same prices at the known strikes – Jingjings Jul 28 '16 at 23:26

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