Greek Questions

[ajsa]

Hi David,

I have found the greek questions are quite interesting and challenging..

1. Which of the following statements about option time value is true?
a. Deeply out-of-the-money options have more time value than at-themoney
options with the same remaining time to expiration.
b. Deeply in-the-money options have more time value than at-the-money
options with the same amount of time to expiration.
c. At-the-money options have higher time value than either out-of-the
money or in-the-money options with the same remaining time to expiration.
d. At-the-money options have no time value.

Answer is c.
I agree theta is higher for ATM. but it only means the price movement is larger with time passing. Does it mean the time value of ATM (which is a static sense) is higher?


2. Which position is most risky?
a. Gamma-negative, delta-neutral
b. Gamma-positive, delta-positive
c. Gamma-negative, delta-positive
d. Gamma-positive, delta-neutral

answer is c.
But for a (like a negative straddle), payoff will be negative no matter S increases or decreases. Is not it more risky?


3. A trader buys an at-the-money call option with the intention of delta-hedging
it to maturity. Which one of the following is likely to be the most profitable
over the life of the option?
a. An increase in implied volatility
b. The underlying price steadily rising over the life of the option
c. The underlying price steadily decreasing over the life of the option
d. The underlying price drifting back and forth around the strike over the
life of the option

anwer is d. "An important aspect of the question is the fact that the option is held to maturity.
Answer a) is incorrect because changes in the implied volatility would change the
value of the option, but this has no effect when holding to maturity. The profit
from the dynamic portfolio will depend on whether the actual volatility differs
from the initial implied volatility. It does not depend on whether the option ends
up in-the-money, so answers b) and c) are incorrect. The portfolio will be profitable
if the actual volatility is small, which implies small moves around the strike price."

I do not follow it. could you please explain?

Many thanks!

 

[David Harper CFA FRM]

Hi ajsa

I agree about the greeks ... maybe I will add a Greek options sub-forum to collect some of the good ones...

1. Excellent observation ... it's possible the question writer confused theta with time value...theta is rate of change of call option with respect to time and therefore (under assumption of constant intrinsic value) could be called rate of change of time value with respect to time; i.e., theta is "time decay" not time value. Therefore, I don't know how to answer intuitively, so i quickly calculated time value at five strike prices, middle column is ATM:

http://sheet.zoho.com/corporate/1978922/btzoho/oct23-timevalue
... see what i did? I subtracted intrinsic value from call value to get time value; after the fact, i can rationalize an intuition but i could not before
...and you'll note time value "peaks" ATM ... so (c) does seem to be true

2. I don't like the question immediately: risky with respect to what? ...imprecise for an FRM ... but what it means is: the delta-neutral is generally hedged for small stock price movements, but the delta positive will lose value with a small price decline (i.e., change in option portfolio = delta * change in stock, so +delta implies loss in option portfolio when stock drops) ... so, the gamma-negative is exposed to high realized volatility (large moves either way) but buffered against small changes (delta neutral) whereas the portfolio (c) has all the gamma risk *plus* the additional risk to *small decreases* in stock price (not increases, right? small increases are gains)

3. You are right not to follow ...
It's their error, should be exactly the opposite.
Please see http://forum.bionicturtle.com/viewthread/1870/#3812
Instead of “The portfolio will be profitable if the actual volatility is small which implies small moves around the strike price”
should be: “will be profitable if actual (realized) volatility is greater than (initial) implied volatility”

David

 

[hsuwang]

Hello David,

Can you please check to see if I'm right with this: delta neutral hedges against small price movements, but once price movements (volatility) increases, negative position gamma will start to bring losses?

and also, per #2 above, if gamma is negative and delta is slightly positive, like you said, the positive delta will add additional risk to small decrease in stock price, but stock prices can move either way, so wouldn't this in a way cancel out the risk of a "small decrease" in stock price, thus making it as risky as if it were delta-neutral? (ie. if stock price slightly increases, then it will add on to the buffer)

Thanks.

 

[David Harper CFA FRM]

Hi Jack,

Re: "delta neutral hedges against small price movements, but once price movements (volatility) increases, negative position gamma will start to bring losses?"
Yes, I agree 100%. (I love the precision of 'negative position gamma' ... as we've established only position gammas are negative...love it!)
...and just to reinforce, this is true in the same way that duration and marginal VaR--being first partial derivatives, like delta--are linear approximations and therefore only accurate for small changes (locally)...this is true b/c delta is the slope of the tangent line and the larger the underlying price moves, the worse the line becomes as an approximation ... and, here gamma (aka, convexity for bonds) is the "gap" that produces losses for the short (gains for the long)

Re #2: but this question (maybe unlike the 3rd question) is assuming neither a delta-hedge nor volatility around the stock price ... I'm still not liking the imprecision of the question (risk with respect to what?) and feel it leaves a gap for some debate, but i think the question refers to this sort of scenario:
* scenario: the underlying stock price drops a little without any corresponding upside volatility: you lose on the negative gamma (maybe) and you lose on the positive delta, whereas with option (b) you don't lose on the positive delta.
* of course, scenario: underlying stock jumps up a little without any corresponding upside volatility: you lose on the neg gamma but you profit on the delta
...so I think option (c) can produce a higher return than (a) but instead of canceling, i think here that just makes it slighly higher risk/return than option (a)

David

 

[David Harper CFA FRM]

For what (little) it's worth, I think i figured out why the imprecision of question (2) bugs me: if you take Hull's equation 17.4:
theta + f[delta] + f[gamma] = riskless rate*portfolio
i.e., the relationship btwn delta, theta, gamma

Comparing (a) to (c), you increase delta going from (a) to (c), which ceteris paribus, implies theta decreases (!). To this position, who is net short options (negative gamma), position theta is positive and a decrease is adding to his/her risk (the passage of time which accrues to the benefit of the short is a built in hedge--he/she wants no volatility-drama and for the options to expire worthlessly...the reduction in theta is a reduction in the built in "hedge" benefit of time decay)...David

...append: oops, less positive theta (higher positive theta to lower positive theta) would support an increase in riskiness in moving from (a) to (c), so this would only apply if the position (a) is negative theta + delta neutral + negative gamma ... then an increase in delta implies: more negative theta + positive delta + negative gamma ... which is also more risk ...
...sorry: nevermind: the implied impact on theta (a decrease) appears to be risk-additive for either long/short .. so this appears to further support (c)...David

 

[hsuwang]

Hello David,

I think the more I think about the delta and gammas, sometimes I really find myself trapped in different concepts. Sometimes I get it, but sometimes I don't..

- Positive position gamma will benefit from large movement (volatility) in stock prices, but this is true only when we are delta hedging it to maturity? (Like you explained in another thread with the "buy low, sell high" concept. What if we are not delta-hedging the position? then with a positive gamma, we gain from stock price up-movements but actually suffer loss when stock price decrease, so in this case, can we say that the statement "positive position gamma will benefit from large movement in stock prices" is not true (since there is a downside)?


I think the idea the I get stuck with is "a position with positive gamma is relatively safe, that is, it will generate the deltas that benefit from an up or down move in the stock. But a position with negative gamma can be dangerous"
I really don't see how this is true because a naked position can have positive gamma too and it will have a downside (suffer) when stock prices fall.

Thank you.

 

[David Harper CFA FRM]

Hi Jack,

Right i struggle with same things...I think you've highlight two things:

1. we want to keep in mind the delta-neutral hedge is two positions: long (short) call + short (long) delta share; i.e., the only naked option that is (position) delta-neutral is a very deeply OTM call OTM put (they are so worthless that a small change in price cannot much help)...this delta hedging business might be considered on its own terms: as a portfolio of these two positions

2. Regarding "Positive position gamma will benefit from large movement (volatility) in stock prices, but this is true only when we are delta hedging it to maturity?"
Not quite, the positive gamma is always helpful to the long call option (like the negative position gamma is always hurtful) but here the key is that we are isolating on gamma and gamma is more challenging b/c it's the only second derivative we study.
...so if you just go back to the call-price/stock price curve (or the price/yield curve for bonds, for that matter, since we may as well be talking about convexity may as well kill both ideas with one stone)
...assume you are long a single call option: postive delta, positive gamma (the counterparty who writes you the option has negative position delta and negative gamma; i.e., -1 short quantity * positive % Greek = negative position Greek for the option writer)

* assume stock price drops
as the long option holder, you *lose* ... say delta = 0.6 such that -$1 stock drop implies a $0.60 option loss
the point about "positive gamma" helping you is specifically about the gamma impact.

first assume (the baseline case) that gamma is 0. What does this mean?
as 2nd derivative, it only means delta is constant. So then if the stock drops another $1, you lose an additional $0.60.

but what if gamma is positive? then after the stock drops $1, the delta is slightly lower(!)
(i myself have a much easier time visualizing: the tangent curve is becoming more shallow as we move the tangent point)
so, sure: because of positive delta, you are losing value as underlying drops
however, in regard to gamma itself: it is the 2nd derivative appoximation that tries to "bend the line" to meet the curve. The curvature itself is entirely favorable to you (forgetting everything else).
... When stock drops, the curvature itself is beneficial: it flattens to mitigate the downside impact. When stock increase, its steepens to accelerate your gains.
...so as the stock drops, positive delta continues to hurt you, but positive gamma (as the bend in the curve) mitigates the damage
...same thing with convexity, right? it always adds on the adjustment because the curvative makes the gap positive on both sides of the line

David

 

[hsuwang]

David thanks so much for the explanation! It really helped tremendously!

Thanks!

 

[hsuwang]

##Nevermind, I got this one, please ignore this post ##
---------------------------------------------------------------------------
Steve, a market risk manager, is analyzing the risk of teh S&P 500 Index options trading desk. His risk report show the desk is net long gamma and short vega. Which of the following portfolios of options show exposures consistent with this report?
a. substantial long expiry long Call positions and substantial short expiry short Put positions
b. substantial long expiry long Put positions and substantial long expiry short Call positions
c. substantial long expiry long Call positions and substantial short expiry short Call positions
d. substantial short expiry long Call positions and substantial long expiry short Call positions

Answer: D: Short expiry long call is long gamma positions. Long expiry short call is short vega positions.

Gamma risk = short expiry position
Vega risk = long expiry position
Long call/put = long gamma/vega
Short call/put = short gamma/vega
---------------------------------------------------------------------------

 

[ajsa]

Hi David,

A related question about the volatility.. I feel for the same security, both long and short position of it should have the same price volatility because they share the same price with just opposite direction. could you advise?

Thanks.

for example
============================
22. With all other things being equal, a risk monitoring system that assumes constant
volatility for equity returns will understate the implied volatility for which of the following positions
by the largest amount:
a. Short position in an at-the-money call
b. Long position in an at-the-money call
c. Short position in a deep in-the-money call
d. Long position in a deep in-the-money call
ANSWER: D
A plot of the implied volatility of an option as a function of its strike price
demonstrates a pattern known as the volatility smile or volatility skew. The
implied volatility decreases as the strike price increases. Thus, all else equal, a
risk monitoring system which assumes constant volatility for equity returns will
understate the implied volatility for a long position in a deep-in-the-money call.
Reference: Options, Futures, and Other Derivatives, Hull, 2006.

 

[frm_daniel]

Hi David,

Quoted "so as the stock drops, positive delta continues to hurt you, but positive gamma (as the bend in the curve) mitigates the damage ".

I am not sure if your statement can be applied in the OTM option. For Out-of-the-money (OTM) option, the stock price drops, delta decrease, but gamma increase will increase the value of delta, which will not mitigates the damage" but increase the damage . Am I correct ?

Daniel

 

[David Harper CFA FRM]

Hi Daniel,

The problem i have is with phrase "but gamma increase will increase the value of delta:" I don't see how that is possible with a long call/put option (where percentage gamma is always positives).

So with an OTM call option (e..g,) delta is positive (0 < delta < 0.5) and gamma is also positive; positive gamma implies:
as stock price increases, delta increases (gamma = change of delta with respect to stock price), and similarly but in the other direction:
as stock price drops, delta drops

the limit, FWIW is a totally OTM option which is converging on delta = 0 and gamma = 0; i.e., the option is so far out of the money that delta is hardly responding to stock price changes, which describes a ~ 0 gamma. But the gamma itself never goes sub-zero here to become per se hurtful. hope that helps...David

 

[frm_daniel]

David,
What if I am referring to the In-the-money Option ? Under the graph plotting of Gamma against stock price. stock price drop, gamma increase in the range of ITM range (say for example, from 0.03 to 0.04), change of delta increases.Therefore, it increases the damages...?

 

[David Harper CFA FRM]

HI Daniel,

As long as we are referring to a long call/put option, it won't matter if OTM/ATM/ITM. I attached graphic here of stock price (X) versus option price (Y):

http://learn.bionicturtle.com/images/forum/0208_optiondelta.png

In regard to "Under the graph of Gamma against stock price. stock price drop, gamma increase in the range of ITM range, delta wil increase.therefore,"
...if ITM, gamma is low. As price drops, you are correct gamma increases as approaches ATM
...but as the price drops, delta is always decreasing (this is what is meant by positive gamma: increase in price increases delta; decrease in price decreases delta). Or, if the price drops and delta increases, then the gamma must be negative (and we must then be referring to a short position)

if you look at graphic above, gamma is like convexity: it is responsible for the curvature compared to the straight red line. For a given tangent point, gamma = 0 would imply constant delta and the red line. But notice the curvature (the gamma) is always helping not hurting: if price goes up (move right on X axis), then curvature creates "additional gains." And if price moves down, the long position loses but the curvature is always helping to mitigate the loss. The graphic shows that compared to the red line (where gamma = 0 b/c the straight line is constant delta), if you are long the option, you prefer the curvature (aka, convexity or gamma) in either direction and regardless of OTM-ATM-ITM.

David