Which of the following statements are true:I.

Implied volatility is the volatility that is priced in the current market prices. In other words, we know the formula for pricing an option, and we find the value of the volatility for which we get the current market price. It is the volatility that the market is ascribing to that security. Therefore statement I is not correct.

Implied volatility in the market tends to vary across strikes, even with the same underlying. There is a certain level of demand for out of the money puts, for example, from investors trying to set a floor for their losses. At the same time, there is a natural supply of out of the money calls from investors trying to earn some relatively risk free premiums. These factors cause options with different strikes to be priced in a way as to give different estimates of market volatility at each strike *& resulting in the 'volatility smile'). Therefore statement II is not correct.

The shape of the implied volatility curve is called the 'volatility smile', because of the shape it sometimes takes. Statement III is correct.

Theta is the rate of change of an option's price with the passage of time. It causes the value of the option to decrease over time, such that if it is out of the money and close to expiry, the price of the option approaches zero. Theta is therefore negative by definition. Statement IV is not correct. Remember that theta will be positive for short positions.