Option Theta – is it always negative?

Theta can be positive if the option is in a net short position, so if you’re creating a position by buying one option and writing a more expensive option simultaneously, you would be in a net credit position (i.e. you owe people a certain value). So your theta would be positive, i.e. the value of your position increases as time goes by.

Hey @rahul12, a very good question, let me attempt to answer this as I remember mulling over it for a while.

Using your definition of theta, as time passes (i.e. from t=0 to t=1, meaning as time increases from inception t=0), options usually have negative theta like you said given the decay in time value and hence option value. 

However, there is a rare scenario that a positive theta can occur for a deep in-the-money European put option. Quick reminder: European put option value (p) >= max (0, X/(1 + risk free rate)^t – S), i.e. p is the max of 0 and the PV of strike price (X) less Stock price at maturity (S).

Using an extreme example where S (stock price) is 0 (e.g. bankruptcy scenario), the put value (p) is PV of X (strike price). In this case, the closer it is to maturity, the greater the put value (p), because as time goes by, that discounted strike value increases towards X ==> therefore implying a positive Theta.

So, positive theta situations only arise when option values are less than intrinsic values, and this happens in the rare case of deep ITM European puts. It doesn’t happen with European calls because the value European call options can only be greater than its intrinsic value by definition: c >= max( 0, S – X / (1 +risk free rate)^t), i.e. S – X / (1 +risk free rate)^t is always greater than S-X.

Hope this is helpful!