Call Option Price Convex Function of Strike Price

[nag_san]

Hi

P1 Book3 Chapter 14 Page 176 says "Under any model, the price of a call option is always a convex function of the Strike Price" and then it shows Figure 14.6 where as the Strike price increases (X Axis), the call price increases (Y Axis). It would make sense if It's a convex function of the Stock Price but can someone provide an example to visualize Call Option being a Convex Function of the Strike Price. Does it imply that if Strike price increases so does the call price. If Strike price increases relative to the Stock Price, then Call option price decreases per BSM (tried few examples) -ie out of the Money Call- so the statement that Call price being a Convex function of the Strike price isn't very intuitive without examples. Any explanation or examples would be helpful.

 

[David Harper CFA FRM]

Hi @nag_san That's a mistake (good catch!). It's also unusual to plot option value against strike price (however, please note it is typical to plot implied volatility against K or K/S).

It is a true statement that regular (aka, vanilla) option value is always convex. Please see below. Consistent with your observation, almost all illustrations of this sort show the plots in the left column; i.e., against stock price. If we remove the X-axis label on GARP's 14.6 (alternatively if we replace "Call" with "Put"), then its shape refers (you can see) to either the call-versus-stock (upper left) or the put-versus-strike (lower right). Here is the XLS I used (a minor variation of my learning BSM/greeks XLS) https://www.dropbox.com/s/i6y23e514rc18z2/071720-option-convex.xlsx?dl=0 Thank you!

 

[nag_san]

Thanks @David Harper CFA FRM that's a relief and makes a lot of sense now.