Greek Letters

[SamuelMartin]

I've seen the following question in different books and they provide with different answer so now I'm confused.

One said B is correct, the other said that D is correct.

Thanks

Which of the following statements is correct?

I. The Rho of a call option changes with the passage of time and tends to approach zero as expiration approaches, but this is not true for the Rho of put options.

II. Theta is always negative for long calls and long puts and positive for short calls and short puts.

a. I only

b. II only

c. I and II

d. Neither

 

[David Harper CFA FRM]

HI @SamuelMartin I like these questions although neither is simple

For I. we can use the formula for greek (percentage) rho, rho(c) = K*T*exp(-rT)*N(d2) and the rho(p) = -K*T*exp(-rT)*N(d2). Both are conveniently the product of four variables and the question concerns the tendency of Rho as T --> 0

For II. The issue is that the question means "Position Theta" rather than "Percentage Theta." Position Greek = Qty * Percentage Greek, where a short position in, say, 10 options is given by: Position Greek = -10 * Percentage Greek. In this way, percentage theta is almost always positive (i.e., the value of an option declines as maturity approaches) such that the following are true statements:

• Percentage theta is generally (usually) negative such that a long position has negative position theta (e.g., +10 options * -3.0 % theta = -30 position theta) and a short position has positive position theta (e.g., -10 options * -3.0 theta = +30 position theta; intuitively, short option position is increasing in value as maturity nears/decreases)
• However, statement II is technically false due to the "always:" a deeply in-the-money European put can have positive percentage theta, so it's an exception to the general phenomenon of time decay, it's a rare instance where the march of time (maturity decreasing) can actually increase the value of an long option. If a put is deep OTM, there is little upside possible but significant downside (which volatility might realize) and the holder of a deep ITM put wants to exercise immediately for the gain, but a European option forces him to wait and the time is mostly a chance for volatility to work against the long deep ITM put holder such that a shorter maturity counter-intuitively increases the value of the deep ITM put (a rare case of positive %Theta which itself implies positive position theta for the long or negative position theta for the short. Each atypical) ... of course for this question, we don't need to understand position vs. percentage Greek: the use of "always" guarantees II is false

 

[SamuelMartin]

So answer D is the correct answer?

Thanks David

 

[David Harper CFA FRM]

In my opinion, yes D. (I) is wrong because "Rho of put option" does also approach zero because rho(p) = -K*T*exp(-rT)*N(d2) approaches zero as T --> 0. Thanks,