In the link attached https://www.desmos.com/calculator/dzppcqv2tk?lang=it I made an animated function with the time value graph, it is clickable in the parameters s (the volatility) and k (the strike). You can see that with low volatility (till 20%), the time value is perfectly symmetrical for ITM and OTM, but when volatility increases the time value goes up much much more in ITM than OTM. Could you elaborate on the rationale for this behavior? Thank you very much p.s. this is with option value and extrinsic value integrated https://www.desmos.com/calculator/k7o9w4vd2y?lang=it
It is asymmetrical in linear scale, but should be symmetrical in log scale. You can try to change the x-axis to log scale and see if they are more symmetrical. My understanding is it is due to the lognormal return distribution.