# Option proofing: Analytical solution for option math

How do I prove the following equation:

P(X=100)≤(P(X=110)-P(X=90))/2

I am not sure how to start and whether it involves using the Black-Sholes formula or not (something like this: https://www.youtube.com/watch?v=LM6iMfHbQDs&t=939s). Also, please note that this is an option price of a put, not a probability.

Thank you!

• I think what you want is actually $P(X=100)≤\frac{P(X=110) + P(X=90))}{2}$.That is, the convexity of the option price. – Gordon Aug 18 '20 at 18:31
• I think yes, there is a mistake in the sign. – S_Star Aug 18 '20 at 19:53

• It's definitely true that you can compare BS fomulae for different options and strikes. But if all you want to show is that the K=90 and the K=110 puts must sum to give more than twice the price of the K=100 put, then the above is sufficient. At expiry, there is a positive probability of the portfolio being worth more and NO chance it is worth less. Since today's price is the discounted future price in the RN measure (ie. $C(0) = \delta(0,t){\mathbb E}[C(t)]$), this means the portfolio MUST, ALWAYS be more expensive. – StackG Aug 18 '20 at 7:48