37. An equity options trader is short a call option of a stock with strike at $104. The maturity of the option is within half an hour and the current price is $103.75. Which of the following Greeks poses the highest risk to his position?

a. Delta
b. Gamma
c. Rho
d. Theta

To answer this question I referred to the graphs that describe the relationship between time to maturity and the greeks (Time versus Delta, Time versus Gamma, etc.) The graphs show that gamma rises exponentially with short-term at-the-money options. However, this option is out-of-the money by .25, but gamma was still the correct answer. Is it reasonable to say that since it is close to at-the-money, gamma would still be correct?

Answer Explanations:
‘A’ is incorrect. Delta is the rate of change of the option price with respect to the price of the underlying. Delta is greatest for in-the money options.

‘B’ is correct. Gamma is the rate of change of the option’s delta with respect to the price of the underlying security. The magnitude of gamma is greatest for short-term at-the-money options. Given the traders is short the option, the
gamma poses the highest risk to his position.

‘C’ is incorrect. Rho is the rate of change of the option with respect to the interest rate. The longer the time to expiration, the more sensitive is the option value to changes in the interest rate.

‘D’ is incorrect. Theta measures the change in an option price with respect to the passage of time. Time decay is more severe for short-term options that are close to the money.

Hi David,
I would have selected Theta because ATM option has highest extrinsic value and hence will loose more value over time. Also Theta is higher if option is closer to expiration. So why not Theta?
Ravi

According to the Time to Maturity versus Theta graphs, Theta decreases exponentially as the time to maturity approaches zero for at-the-money options.

I believe this tells us that time becomes a smaller factor of risk as the option is about to expire?

I read this once .Theta gives a measure of an options extrinsic value getting reduced. Hence all things equal an option with more days to expiration will have more extrinsic value then one nearer to expiration. Since the above option is closer to expiration it extrinsic value gets reduced more and more. Is there something wrong in my understanding? Quiet confused

Theta, or “time decay,” measures sensitivity to the passage of time, the time value.Time Value = Option Value - Intrinsic Value.

Theta = -dV/dT.

“ Even a deeply out of the money put will be worth something as there is some chance the stock price will fall below the strike. However, as time approaches maturity, there is less chance of this happening, so the time value of an option is decreasing with time. Thus if you are long an option you are short theta: your portfolio will lose value with the passage of time (unless there is enough volatility to offset this).”
Source wikipedia.

So if you short the option an time approaches muturity it will be less possible the te intrisic value will be positief.

Here’s another question that seems related. Does “highest time value” here mean highest theta? If so, then based on the “Time to maturity versus Theta” relationship (see market notes and Hull), the at-the-money call has the lowest theta (decreases exponentially) as time to maturity approaches 0.

Since the question refers to a put option, I guess the “Time to maturity option versus Theta” relationship gets flipped, which is why the answer is C? (At-the-money put has an exponentially increasing theta as time to maturity approaches 0). The explanation below mentions that theta is highest at-the-money.

67. A European put option on a non-dividend paying stock has a remaining life of 6 months with a strike of USD 50 and the risk-free rate of 1%, after 3 months which of the following stock prices has the highest time-value of the option (in % of stock price)?
a. USD 10
b. USD 40
c. USD 50
d. USD 60

ANSWER: C
A is incorrect. A deep in-the-money option has virtually no time value.

B is incorrect. An in-the-money option has smaller time value than an at-the-money option.

C is correct. The at-the-money option has the highest time value, given its highest gamma and theta.

D is incorrect. The out-of-the-money option has smaller time value than at-themoney option.

In regard to #37, Dennis you are right about this: “since it is close to at-the-money, gamma would still be correct?” I plugged these assumptions into the Greeks XLS calculator on member page and the near-term Gamma starts plunging (i.e., Gamma starts acting like the out of the money option) when stock price is < 70. With 70-75 or above, gamma is acting like an at the money; i.e., increasing as maturity nears zero.

In regard to the second question, time value isn’t theta.  Intrinsic value + time value = option value. If option value is $3 and intrinsic value (stock - strike for a call option) = $2, then time value =$1. Time value is what you forgo/sacrifice when you exercise the option to collect the intrinsic value. So, it tends toward zero as option nears maturity. However, this would be an unfair question (in addition to, IMO, being a very difficult question) against our assignments. I don’t think time value is mentioned in any of Hull.

I tweaked the Gamma page here to illustrate.

Time value in red (option value - intrinsic value), it does peak ATM but i think the intuition is more difficult than the answer suggests.

David