Time value of option

CFA Level I Derivatives
#1

For American or European options (puts or calls), is there a point where the time value can be zero or even negative?

I remember reading somewhere that any option will sell for atleast its intrinsic value which would indicate time value of option is never negative but I am not sure how dividends etc might play a role on this?

Thanks

#2

For a put option that is far in-the-money, with a (fairly) long time to expiry, the time value can be negative. The reason is that the value of a put option is constrained because the value of the underlying (generally) cannot go below zero, so if the price is near zero there’s a much greater chance that it will rise (decreasing the value of the put option) than that it will fall.