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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:

  1. 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.

  2. 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!

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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.

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  • $\begingroup$ I am still confused: why is it even possible to have the option of getting \$100 in cash when the value of the option is only \$95??? In other words, your argument is only valid if the amount of cash that one can retrieve at this moment is greater than/equal to the final value of the option at maturity after the increase. $\endgroup$ – Richard Oct 1 '19 at 3:39
  • $\begingroup$ Don't mix up the options. The european is at 95 and will appreciate to 100. The American gives you 100 by exercising it now (which the european does not allow). $\endgroup$ – Alex C Oct 1 '19 at 3:49
  • $\begingroup$ @Alex C Yes, of course American options are worth more than European options. What I do not understand is why a positive theta implies early exercise, when there is an anticipated future rise in the value of portfolio??? $\endgroup$ – Richard Oct 1 '19 at 4:00
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    $\begingroup$ @Richard A positive theta of the European put option implies early exercise of the corresponding American put option. That is not to say the American option will increase in value over time. It won’t as it is already optimal to exercise it at that point. $\endgroup$ – Ivan Oct 1 '19 at 7:11

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