Why Positive Theta Can Support Early Exercise of Deep-In-The-Money Puts
Summary
The discussion explains why a deep-in-the-money European put can have positive theta when interest rates are positive. As the stock price approaches zero, the put behaves roughly like a zero-coupon bond paying its strike at expiration; the passage says this bond-like value can rise as maturity approaches. It also distinguishes the European option’s changing value from the American holder’s exercise choice: exercising early makes the strike available sooner to invest at the risk-free rate, while holding the European option means waiting for payment.
The answers use a near-zero stock price as an intuitive limiting case and a comparison between cash today and an option expected to be worth more later. They do not derive the Black–Scholes formula or give a general exercise boundary. The second answer notes that, without dividends, an American call is exercised at maturity, while its value can still decline as time passes because less time remains for large price moves. These points illustrate that theta alone does not determine whether early exercise is optimal; the comparison depends on the value of receiving the strike sooner and on the option’s remaining time value.
Key ideas
- A deep-in-the-money European put can have positive theta when positive rates make its strike-like payoff more valuable as expiration approaches.
- Early exercise of an American put can provide the strike earlier for investment in a risk-free asset.
- An option’s time decay does not by itself determine the optimal exercise time.
- The near-zero stock price example offers intuition, not a complete rule for American put exercise.
- Without dividends, an American call is generally held until maturity even though its value may fall as time passes.
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# Option pricing: Relationship between Theta and early exercise
# Option pricing: Relationship between Theta and early exercise
I am confused about the following:
For a European put option, the parameter $\Theta$ is given by $$ \Theta= \frac{d V}{dt} = -\frac{SN'(d_1) \sigma}{2 \sqrt{T-t}} + rK e^{-r(T-t)}N(-d_2).$$
My textbook claims the following:
> For deep-in-the-money European puts, $S<<K$. Hence, $d_1, d_2 \approx - \infty$, which implies that $N'(d_1) \approx 0$ and $N(-d_2) \approx 1$. Hence, $\Theta >0$. This shows that it can be optimal to exercise a deep-in-the-money American put before maturity.
I have two questions regarding this statement:
- I have a deep confusion. $\Theta$ measures the rate of change of $V$ with respect to $t$ by its definition. Hence, a positive value of $\Theta$ should imply that the put option increases in value over time. This means that we should wait further for the future increase in the value of the option and therefore wait and not exercise at the moment.
- The formula of $\Theta$ comes from the Black-Scholes formula of $V$, which is only valid for European options, from what I know. Therefore, I am puzzled about the conclusion regarding the American counterpart in my textbook.
Any ideas? Thanks!
## Answer by Charles Fox (score 3)
https://quant.stackexchange.com/a/49006
Let's say the company was bankrupt (ie, stock price is 0). A put option effectively becomes a bond with face value equal to the strike and maturity equal to the expiration.
With positive interest rates, zero coupon bonds generally become more valuable as time passes.
In this extreme case, an American option is worth more because you could early exercise and invest the proceeds in the risk free asset, while the European option would (setting aside any special rules due to the bankruptcy) require that you wait. The European option would have positive theta (expected to increase in value).
Would you rather have \$100 today or an option that is currently worth \$95 but is expected to be worth \$100 in one year? You would probably choose the former despite the expected increase in the latter's value.
## Answer by User9601 (score 0)
https://quant.stackexchange.com/a/73537
To answer the first question, if there are no dividend payments, it is known that an American option needs to be exercised at maturity. That means that the payoff (for example $(S_T - K)^{+}$ for stock $S_t$ and strike K in the case of american call) is bigger if you wait until maturity. Nevertheless, as much as you approach maturity, the value, the price of your option worth less, because there are less chances for up/down huge movements. So that explains why the Theta is negative, but the optimal exercise time is maturity.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.