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Option Theta - is it always negative?

hi all, i've a derivative question/concept that puzzled me a bit: can option Theta be positive? if so, when and why?

i get that the definition of option theta as a change in the value of an option for a given change in time, assuming all else constant. as time passes (e.g. from t=0 to t=1), the time value of an option reduces, therefore option value reduces if all else constant, i.e. theta is negative.

but in the curriculum books, it states "typically, theta is negative for options", which for the life of me i can't figure out what scenario an option theta would be positive? 

any help would be much appreciated, thanks!
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Comments

  • 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.
    This is CFA!!!
    rahul12
  • SophieSophie Posts: 1,958 Jr Partner
    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!
    rahul12
  • thanks so much sophie! it took me a few times to read your explanation and make sense of it - definitely one of those tricky questions that trips ppl up!
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