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?

(1) The graph assumes constant volatility. So there is no inverse relationship between time value and implied volatility. On the contrary, there is a direct relationship between the two. (2) The option's gamma (delta's slope) is highest ATM because at this point the delta is most sensitive to changes in the price of the underlying. (4) As in (1) the graph assumes constant volatility (i.e. no volatility skew).

By definition (2) is true. But I am looking for explanation behind rate of change of delta (the gamma). Why does it vary and why does it become steeper as we get close to the money? Mathematically, delta computed as a derivative of option price to stock price does depend on other variables such as implied volatility and time to maturity. So if on the next day, the stock price remains the same but implied volatility jumps up by 10%, then there would be a change in delta as well as gamma. In such cases, how do I build a hedged option position if I want to trade volatility and hedge my delta. Because it seems changes in volatility will affect my delta as well. Any insights on this? If you now assume volatility skew, then how do the graphs change?

A correction - Assuming constant volatility, the extrinsic value will be highest at ATM. I am looking for explanation behind rate of change of delta (the gamma). Why does it vary and why does it become steeper as we get close to the money? Mathematically, delta computed as a derivative of option price to stock price does depend on other variables such as implied volatility and time to maturity. So if on the next day, the stock price remains the same but implied volatility jumps up by 10%, then there would be a change in delta as well as gamma. In such cases, how do I build a hedged option position if I want to trade volatility and hedge my delta. Because it seems changes in volatility will affect my delta as well. Any insights on this? If you now assume volatility skew, then how do the graphs change?

There is a graphical interpretation of delta/option price. Once you know it, things become easy. The models in people's mind are visual. Most likely they would provide the output of models in their minds, and are less likely to share the mental models.

I figured this one out. Bigger time value will result in bigger deltas as we get closer to the money.. if we take the percentage change in delta divided by time value, that would be close across different prices. But I still have confusion on how to efficiently delta hedge a portfolio when the delta depends on other factors such as volatility and time to maturity.

Why not bigger deltas will result in bigger time value? I am guessing that you meant by "will result" is "will correspond".

Delta hedging is not a static strategy (i.e. not set and forget). You need to rebalance the hedge as the market moves around and your deltas change. The volatility skew doesn't change the graphs, however as your option moves along the skew the volatility will change and hence the greeks will do as well.