Option pricing: Relationship between Theta and early exercise

Question

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!

Answer 1

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 2

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.