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I am trying to hand-price options under the Black-Scholes model.

Given the following parameters:

  • Stock price: $12.53$
  • Strike price: $14.00$
  • Risk-free rate: $0.03$
  • Annualized Volatility: $0.10$
  • Time until expiry in years = $.238095$

The put will have a positive theta of $0.354295$. It has a very high probability of ending up ITM (using delta as an approximation, $\Delta = -0.982251$).

What is the intuition behind this behavior? I thought for long options theta is always negative as a long option loses it's extrinsic value over time. I could see a short option having a positive theta, but a long option? This behavior seems unintuitive.

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    $\begingroup$ Possible duplicate of Negative time value european options $\endgroup$
    – LocalVolatility
    Commented Nov 12, 2018 at 22:50
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    $\begingroup$ @LocalVolatility: The two questions are different as theta and time value are two distinct concepts. $\endgroup$
    – Hans
    Commented Nov 13, 2018 at 1:54
  • $\begingroup$ You neglected the increment, as the time to maturity shortens, of the discounting factor in the theta, while the deep in the money gives a very small time decrement effect. $\endgroup$
    – Hans
    Commented Nov 13, 2018 at 2:05
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    $\begingroup$ I find it strange @LocalVolatility got two upvotes and the two concepts are different. Do people even read the questions? $\endgroup$
    – CL40
    Commented Nov 13, 2018 at 2:32
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    $\begingroup$ The answer that I linked explains why the time value can be negative for deep in-the-money European put options when the interest rate is positive. From the same argument, you could see why the theta would be positive. The accepted answer pretty much repeats this same argument. $\endgroup$
    – LocalVolatility
    Commented Nov 13, 2018 at 17:45

1 Answer 1

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If a european option value becomes lower than intrinsic value it gets negative time value.

In this circumstance the theta becomes positive because as time approaches to expiry the option value has to converge to intrinsic value.

For european options there are 2 circumstances that can lead to the option value being lower than intrinsic value

  1. deep ITM puts in presence of positive interest rates $r>0$
  2. deep ITM calls in presence of positive dividend yield $q>0$

Note that those are the 2 circumstances under which it makes sense for an american option to be exercised early.

For more details you can check the actual formula for theta on the wikipedia page dedicated to greeks

Greeks formulas (wp)

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    $\begingroup$ This answer for call options has charts demonstrating this behaviour and also shows what @LocalVolatility was refering to. $\endgroup$
    – AKdemy
    Commented Oct 28, 2022 at 18:15

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