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I would like to understand why the Black and Scholes greek letter theta for european call option behave in the following way:

  • as time to maturity is far away (right part of the x-axis in the the graph) theta is small for all the call options (ATM, ITM e OTM). Therefore this means that the call value decrease by a small amount as time passes when time to maturity is far away.

  • as time to maturity approach zero, i.e. close to the expiry, (left part of the x-axis in the graph) ITM and OTM call option theta get close to zero (i.e. theta decrease in absolute value) while ATM call option theta get bigger and bigger in absolute value. Therefore, when we are close to maturity, ATM call option decrease in value much more than ITM and OTM call option due to passage of time.

Can someone explain me why is that? I would like to understand the underlying concepts.a

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With a long time to maturity, your options have a low theta because their time value decays quite slowly. If there are many months to go, the passage of one day does not change the exercise probabilities too much, whereas short life options with only a few days left have a much higher time value decay. Hence, the larger the time to maturity, the lower theta.

Deep ITM/OTM options basically ``lose'' a lot of their optionality prior to their exercise day. If you're well above the strike price and have only a few days left, what's the probability that your call can lose much of its value? So, again, the passage of a day has little influence on the option price and thus, theta is low. ATM options are more interesting. Here, it is not quite clear yet whether they will run into the money or not. Hence, every day matters a lot for the value of options with short time to maturity. You have a lot of time value decay and hence, a large theta.

Note that the Black Scholes model assumes continuous sample paths, so you can't argue with the possibility of sudden news occurring days before the expiration. This is true in the real world and motivates jump diffusion models (and changes your theta) but does not apply to the Black Scholes model.

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  • $\begingroup$ Can you please explain a little more the behaviour of deep ITM/OTM options near expiration? $\endgroup$ – luca dibo Nov 18 '19 at 10:21
  • $\begingroup$ @lucadibo Say you only have one week left before the option expire. Suppose you consider a deep ITM call, i.e. the stock price is far above the strike price. In the continuous Black-Scholes model, it is very likely that the option is exercisedd (i.e. is in the money at expiry). Thus, $N(d_1)\approx N(d_2)\approx1$ and the call option price looks like $S_te^{-qT}-Ke^{-rT}$. So, there is no real ``optionality'', there is no question on whether the option will be exercised. The option is too far ITM. So, the influence of a day shorter time to matuirty is low, i.e. theta is low. $\endgroup$ – KeSchn Nov 18 '19 at 19:25

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