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Questions and Answers
Q1. Prove that the price of a call option is a convex function of the strike price.
Answer:
To prove the call option is a convex function of the strike price we consider the price as a function of strike, and prove that a line joining any 2 points on the graph of this function lies on or above the graph.
This is a necessary and sufficient property of convex functions.
For K1 < K2: 
for all 
.
With this in mind, consider a ptf long 
call options struck at K1, long (1-) struck at K2 and short one option struck at K1 + (1-)K2.
Since the final payoff is convex in strike, the above equation holds at the expiry of the options. Therefore our ptf has non-negative value at expiry.
By no-arbitrage considerations, this implies that the ptf must be of non-negative value at all times before expiry, and hence that the equation above hold at these times and we have proved the convexity of the call option price.