Theta and Vega, example is inside

Theta is a measure of the rate of decline of option’s time-value resulting from the passage of time (time decay).
Theta provides an estimate of the dollar amount that an option price will lose each day due to the passage of time and there is no move in either the stock price or volatility.


Example:
The price of ABC May 50 Call with 25 days to expiration is $3. Its theta is -0.10. The price of ABC Jul 50 Call with 85 days to expiration is $4.8, and the theta is -0.03. When one day passes and there is no change in ABC stock price as well as the implied volatility of either options, the value of ABC May 50 Call will decrease by $0.10 to $2.9, and the value of ABC Jul 50 Call will drop by $0.03 to $4.77.


Theta of ATM, ITM & OTM Option

Theta is typically highest for ATM options, and is progressively lower as options are ITM and OTM.
This makes sense because ATM options have the highest time value component, so they have more time value to lose over time than an ITM or OTM option.


For ATM option, Theta increases as an option get closer to the expiration date.
In contrast, for ITM & OTM options, Theta decreases as an option is approaching expiration. The above effects are particularly observed in the last few weeks (about 30 days) before expiration.

The Impact of Implied Volatility (IV) and Time Remaining to Expiration on Theta

Theta (time decay) would increase sharply in the last few weeks before expiration and can severely undermine a long option holder's position, particularly if Implied Volatility (IV) is also decreasing at the same time. This is because theta is higher when either volatility is lower or there are fewer days to expiration.

Vega measures the sensitivity of an option’s price to changes in Implied Volatility (IV). Vega estimates how much an option price would change when volatility changes 1%.


Example:
The current price of ABC May 50 Call is $3, with Vega 0.20 and the volatility of ABC stock is 35%. If the volatility of ABC increases to 36%, the ABC May 50 Call’s price will rise to $3.20. If the volatility of ABC drops to 34%, the ABC May 50 Call’s value will drop to $2.80.


Vega of ATM, ITM & OTM Option
The impact of volatility changes is greater for ATM options than for the ITM & OTM options.
Vega is highest for ATM options, and is gradually lower as options are ITM and OTM.
This means that the when there is a change in volatility, the value of ATM options will change the most. This makes sense because ATM options have the highest time value component, and changes in Implied Volatility would only affect the time value portion of an option’s price.
Comparing between ITM & OTM options, the impact of volatility changes is greater for OTM options than it is for ITM options


The Impact of Time Remaining to Expiration on Vega
Assuming all other things unchanged, Vega falls when volatility drops or the option gets closer to expiration.
Vega is higher when there is more time remaining to expiration. This makes sense because options with more time remaining to expiration have larger portion of time value, and it is the time value that is affected by changes in volatility.