Vega
The change in option value for a 1-point change in implied volatility. Both calls and puts have positive vega - more volatility means more chance of a large favourable move, and the downside is capped by the option structure.
Vega is largest at the money and increases with time to expiry. A 2-year option has far more vega than a 1-week option, because there is much more time for volatility to matter.
Practical consequence: to trade a view on volatility, use longer-dated options. To trade a view on a specific event, use short-dated ones where gamma dominates.
Theta
The value lost per day from time passing, holding everything else constant. Long options have negative theta.
Theta is largest at the money and accelerates as expiry approaches for at-the-money options - the famous decay curve. An at-the-money option loses value slowly at first and then very fast in the final weeks.
Out-of-the-money options decay differently: they lose value more steadily and are nearly worthless well before expiry.
The trade-off
Vega and theta are two sides of one position:
- Long options: long vega, long gamma, short theta. You benefit from volatility rising and from the market moving, and you pay rent every day.
- Short options: short vega, short gamma, long theta. You collect rent and lose if the market moves more than priced.
The whole options business is deciding whether the rent is worth the convexity - which is the same question as whether implied volatility exceeds what will be realised.
The interview question
"You are long a straddle. The stock does not move for a week. What happened to your P&L?"
You lost theta and probably vega too, since implied vol typically falls when nothing happens. Being long a straddle is a bet on movement, and no movement is the losing outcome even though you have no directional exposure.
Practise in finance and derivatives.