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Provide option prices for call options stuck at 110,120,130, and 140 for an asset at 100 so that there is no arbitrage. I was not sure how to construct the market quickly.

This is an interview question. I understand that we want to prevent the person from buying and selling a call spread and that the prices would be decreasing. But how do I calculate prices numerically for this question?

Thanks

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2 Answers 2

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You basically need to maintain the convexity with respect to the strike. For example, let's assume that the price of the option struck at 110 is 5 and for the option struck at 140 is 1. Then the price $x$ for the option struck at 120 and the price $y$ for the option struck at 130 satisfy the following inequalities \begin{align*} x &\le \frac{5+y}{2},\\ y &\le \frac{x+1}{2},\\ x &\le \frac{2}{3} \times 5 + \frac{1}{3} \times 1 = \frac{11}{3},\\ y &\le \frac{1}{3} \times 5 + \frac{2}{3} \times 1 = \frac{7}{3}. \end{align*} We can set $x=\frac{11}{3}$ and $y=\frac{7}{3}$.

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You can (also) solve the problem by the repeated application of two rules:

Rule 1. For any two calls (with same maturity) the higher strike call is cheaper than the lower strike call.

Rule 2. For three calls, the middle strike one is cheaper than the linear interpolation of the prices of the other two calls

(Of course, it is the same thing as the convexity property, just a different way to remember it).

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