Hi, I have some confusions about some options graphs. I've attached the graph of option price vs stock price, delta vs stock price, gamma vs stock price and vega vs stock price. (1) Option value vs Stock Price: As can be seen from the graph the time value of the option increases as the stock price moves towards the strike (S < K) and then decreases as the stock price increases further (S > K). My question is how does this relate to implied volatility? In most cases, the option prices exhibit a volatility skew/smile where OTM/ITM options have higher implied vola than ATM options. So there seems to be an inverse relationship between time value of option and implied volatility. This is not clear to me. (2) Option Delta vs Stock Price: Delta ranges from 0 to 1 as we move OTM to ITM with respect to the strike price. My question is in the variations in the rate of increase of delta at different points in time (S << K, S < K, S = K, S > K, S >> K). At the money, the slope is much steeper compared to other points. Why is this and is this related to the time value discussed in (1)? (3) Option Gamma vs Stock Price: This graph can be explained from (2) as gamma is just the rate of change of delta with respect to the stock price. (4) Option vega vs Stock Price: This graph is completely contrary to my thinking. I thought due to the vola skew, the OTM/ITM options should have higher implied vols hence vega than ATM options but this graph shows something completely different. Why?