Greeks

[Hend Abuenein]

Hello David,

1-Are we expected to remember the formulas for the Greeks?

2- In the history of the exam , were there ever questions that asked you to calculate theta, for example?
And if so, were the embedded factors given (d1, d2, N'(d1)...etc) ?

3- Would you please explain the sense behind the increase in absolutevalue in an option's sensitivity towards time to expiration when it approaches. (The tetha- Time to expiration curve)

Thank you, Hend

 

[David Harper CFA FRM]

Hi Hend,

(1)
I just emailed GARP's attention on this exact point and pointed them to this thread, because this question definitely comes up every year.
So here are the two AIMs that clearly imply the strict answer to your question is "yes:"
"Define and compute delta for an option"
"Define, compute and describe theta, gamma, vega, and rho for option positions."

And delta has a decent chance of being tested. But the CALCULATION (formulas) of the other Greeks are, to various degrees, tedious and are unlikely to materialize as formulas, frankly, on the exam; e.g., the formula for theta will simply will not be asked (way too tedious). I have no special foresight but if it were me:

* Delta: I would memorize d1 and N(d1), and while we are here d2; d2 is not delta but d2 is a nearby memorization and d2 is essentially Merton model's distance to default but with asset drift replacing riskfree rate, so it is worth your time to memorize the "cluster" of d1, d2 and to understand what N(d1) means and how to get it if you are given a normdist lookup table
* I would understand the other Greeks conceptually/visually and be prepared to handle qualitative questions about them; you probably will be asked about them conceptually
* If a L1 candidate were conserving time, i frankly would skip trying to memorize the formulas for gamma, theta, vega and rho (but i could be wrong!)

(2)
Not that i am aware (I could be wrong, i don't have a perfect memory). This is IMPORTANT: N(d1) can not be calculated with a calculator; the calculator cannot return the normal CDF or its inverse. For this reason, the exam has tended to provide the N(d1) and N(d2) as solved inputs. Otherwise, it must give you a normdist lookup table, and you must know the how and why of this lookup. The normal CDF is the essence of our introduction to VaR

(3)
It's a little dicey because it isn't necessarily true, but if we consider the numbers in the THETA tab of worksheet http://www.bionicturtle.com/how-to/spreadsheet/2011.t4.b.6.-option-greeks/
This in ATM call option (S = K = $100) so, in this case, the ABS(theta) is increasing as term to expiration decreases.
As theta is change in call price w.r.t. a change in passage of time, keep in mind that time passes daily and that each day (i.e., the denominator) is the same exact length

At 10 years to expiration, the call option in the learning XLS has a very large price @ 49.38; a whopping 49% of face value!
As a SINGLE day passes, the time decay lowers the option value, but proportionally it is not a big share

Go down to 1 year, the option price has dropped to $13.75. As a single day passes here, the time unit is the same length, now it's proportional impact the much smaller price is relatively larger.
This is how i look at it: a single day is not much of the relative impact on the time value of a 10-year option; but it has a larger relative impact on a 1-year option.

The effect is most acute for an ATM option: as option value = time value + intrinsic value, an ATM option has no intrinsic value and depends entirely on the time value.

I hope that is helpful, David

 

[Hend Abuenein]

Thank you David
Last 4 lines about theta clarified things greatly.

I have so far memorized ALL the formulas of the Greeks, and solved questions to calculate them. But they're so similar and easy to get scrambled together.

If N distribution tables are given to us in the exam for use of other quants questions, it won't be a problem solving for delta as N(d1) , or both Nd1, Nd2 for BSM model.

--------------------------------------------------
New question about Greeks:

When we're constructing a gamma-neutral hedge, we are bound to affect the delta-neutrality of the position, because of the change in the number of options in the portfolio.
I don't understand how it goes after deciding on the number of options to buy/sell to gamma neutralize the portfolio in order to correct the change in delta of portfolio.

What tells us to buy more shares or sell in order to maintain delta-neutrality?
Does "maintain delta-neutrality" mean bring it back to a one to one ratio, or back to the delta of the portfolio before the gamma hedging happened? (This second delta is not given in the questions I found and solved) So how do I decide what to do with the shares?

Would you please explain?

Thanks a lot for all the work you do here,
Hend

 

[David Harper CFA FRM]

Hi Hend,

Great, you point about the N distribution is well-taken: that's got to be one of the exam's most testable skills.

Re: delta neutrality: I don't think Hull explains this well. I prefer Carol Alexander who distinguishes between position Greeks and percentage Greeks; e.g., Hull p 371:
portfolio has a position gamma = -3,000
(position Greek = percentage Greek * quantity)

to neutralize is to get to ZERO. We then need to add +3,000 POSITION GAMMA; we need:
+3,000 position Gamma = percentage Gamma * quantity; in his example
+3,000 position Gamma = 1.5 percentage Gamma * 2,000 options; ie, long 2,000 options neutralizes gamma by getting position gamma to zero.

The issue then is, what did we do to position delta?
position delta = percentage delta * quantity, in this case:
our additional quantity of long +2,000 * 0.64 percentage delta = +1,240 position delta.

to neutralize is to bring the position Greek (1,240 position delta) to zero, we need -1,240 position delta; so if we use share we need: -1,240 position delta
we need: -1,240 = 1.0 percentage delta per whole share * Q;
so Q must = -1,240; so, must go SHORT 1,240 shares each with 1.0 delta
(note the elegance of: a short position is handled with a negative in the quantity; a long is a positive in the quantity)

It is a long way to say that, I follow the C.A. approach (yes, i understand "percentage" is a slight misnomer as delta is unitless!):
1. to neutralize is to bring the Position greek to zero, and
2. the Position Greek = percentage Greek * Quantity, where short positions are negative quantities

And here is an example:
the position delta of a short position in 100 puts with (percentage) deltas of -0.4 is:
-0.4 * -100 = +40 position delta.

(it's the same as Hull, I don't think it's anything profound, it's just that Hull does not distinguish that i am aware between the position and the percentage Greek, which makes it harder to follow the neutralization i think)

David

 

[Hend Abuenein]

Thank you
About the position and the percentage it clarifies a lot.
It's much easier than I thought

 

[Hend Abuenein]

Hello David,

Please consider this question:

Portfolio of stock A and options on stock A is currently delta neutral , but has a positive gamma.
Which of the following actions will make the portfolio both delta and gamma neutral :
a.Buy call options on stock A, and sell stock A.
b. Sell call options on stock A and sell stock A.
c. Buy put options on stock A and buy stock A.
d. Sell put options on stock A and sell stock A.

Would you please help me decipher the wording here:
1- Positive gamma means we're long the options, right? But are they call or put options?
2- Answer says correct action is d . I knew we should sell options on the stock to neutralize the positive gamma. But how do I know whether to sell PUT options, not call?
3- If not given, I wouldn't know what to sell/buy to neutralize which Greek.
For example, if one of the choices was: sell stock A (to neutralize gamma) ans sell put options (to neutralize delta), I'd probably confuse this for the correct answer.
Would you please help me through this confusion.

 

[David Harper CFA FRM]

Hi Hend,

Here is my thought process (Can I ask you the question source? curious because I think it's a good FRM question..)

Long stock is long position delta. To be position delta neutral, options must be negative position delta to offset. Ergo, options can be short calls or long puts.
(check that with intuition, if stock goes up, do short calls and long puts hedge? yes)
But the long stock has zero gamma, and short calls have negative position gamma, so it must be long puts; i.e.,
long stock + long puts can be position delta neutral and position positive gamma.

So, let's neutralize the gamma: shares won't help, must short calls or puts.
1. If we short calls, we can neutralize gamma but create negative position delta. Need to buy shares to restore position delta neutral. So, short calls + buy shares
2. If we short puts, we can neutralize gamma but create positive position delta (i.e., -QTY * -%delta = + position delta). Need to short shares to restore position delta neutral. So, short puts + short shares, works too.

So, as (1) doesn't appear, answer (d) is only valid. As above, I rely entirely on an understanding of POSITION delta, and really, the order of neutralization within each trade doesn't matter; e.g., we start with delta neutral and positive gamma" ... the only thing to do here is start by neutralizing the POSITION gamma, which as percentage gamma is always positive, we know can be done with either short call or short put.

I hope that helps, I think these take a little practice but one you get the hang of it, you'll see we can keep re-using the same ideas, thanks, David

 

[Hend Abuenein]

Thank you
I'll keep re-reading here until I get the hang of it.

Source of question: Schweser study notes 2011 book 3 end of book practice exam. (Says it's a 2003 actual FRM exam question)

 

[caramel]

I got this question
Suppose stock xyz is trading at $50 and there is a call option that trades on XYZ with an exercise price of $45 which expires in 3 months . The risk fee rate is 5% and SD of returns is 12% annualised. Determine value of the call options gamma. Assume d1 is 1.99 and N( d1) is 0.9767

the solution given
N'(d1)/S0 SD sqrt T=2.89/50*0.12*sqrt 0.25=2.89/3=0.96

Not sure how they arrived at N'd1 as 2.89 Using the formula I am getting N'd1 as 0.055

 

[ShaktiRathore]

Yes N'd1 is .055 while i calculated delta as .96. gamma is .01818 .
please verify here :http://public.sheet.zoho.com/public/btzoho/hull-15-14
I think there is some mistake in the Q.

 

[NNath]

Hi David, The formulas for greeks i.e. Theta (of call and put), Vega, Gamma, Rho (of call and put) are covered in Hull book, however they are not given weight-age in the study notes. Is it that these formulas are of low relevance with respect to the Exam. Is there a way to remember single formula and derive the others.