The Greeks

[ahnnecabiles]

Hi David,

Got difficulty analyzing these problems:

If risk is defined as a potential for unexpected loss, which factors contribute to the risk of a (1) long put option position: (2) short call position: (3) long at the money straddle:

a. delta, vega, rho
b. vega, rho
c. delta, vega, gamma, rho
d. delta, vega, gamma, theta, rho

the answer for no. 1 is a. delta, vega, rho. But then I read from Jorion that vega has the same position as gamma that it is positive for long positions. how come vega is included?

the answer for no. 2 is c. delta, vega, gamma and rho, and I agree with that since vega and gamma now here is negative.

the answer for no. 3 is b. vega and rho. Again, I would like to know why do we have to include vega considering that vega is positive for long positions. Likewise, delta now is not included, because the solution said that the position is delta neutral, but, long straddle is a long call and a long put, why would it become delta neutral when in fact the value of a call is not equal to the value of a put based on put-call parity?

Thanks so much.

 

[David Harper CFA FRM]

Hi Chinquee,

I empathize with your difficulty; I think these questions are dubious (and, further, these sorts of questions would not quite IMO be justified by the Hull cirriculum). And you are right about (3), (3) plainly has an error. I feel you have a quasi-argument in regard to the seeming inconsistency btwn (1) and (2); similarly, I could try an argument for why both (1) and (2) should include gamma and vega. The use of the term "unexpected" is bit off-putting, but here is what Jorion means:

(1) If I buy a put from you, I am long the put. I am long volatility: if volatility decreases, my put is less valuable. It is tricky why gamma is excluded. The put delta is negative, the gamma is always positive, and so the delta is always increasing with the stock price. But as i am long the put, my loss concerns an increase in stock price; i.e., as the stock price increases, my put is becoming less valuable. But my positive gamma means that the delta is *changing* from something nearer to -1.0 to something nearer to 0.0. This dynamic itself helps me; it "mutes" the impact. The positive gamma in the context of a negative delta is a helpful change as the put loses value; it means my put is becoming less and less responsive to the stock price increase. Or, as Jorion compares, Gamma is here like convexity/curvature: it is always beneficial. The reason I think this is difficult to analyze is that you cannot really look at them in terms of positive/negative (which leads to the seeming inconsistency); the question refers rather to, "which Greeks exacerbate on the downside only." Gamma is here a risk factor (in the sense it signifies instability), but in long position it benefits.

(2) vega is here again, all positions (long/short) on both call and put are implicitly taking a position on volatility. And again, this proves the positive/negative test is not particularly useful.

(3) Re Vega, if i write the straddle to you, I want to see higher volatility; I am twice long volatility.
You are correct about the delta neutral. As
delta of call = N(d1) and
delta of put = N(d1) - 1,
delta of straddle = 2*N(d1) - 1
Therefore, the straddle is only delta neutral if the call delta = 0.5. So, (3) looks wrong to me.

David

 

[ahnnecabiles]

Hi David,

Thanks so much for your enlightening reply, especially on your concepts on volatility, this was not discussed thoroughly in Hull. I have however encountered another ambiguous problem:

1. A delta-neutral position exhibits a gamma of -3,200. An existing option with a delta equal to 0.5 exhibits a gamma of 1.5. Which of the following will generate a gamma-neutral position for the existing portfolio?

a. buy 4,800 of the available options
b. sell 4,800 of the available options
c. buy 2,133 of the available options
d. sell 2,133 of the available options

2. Which of the following actions would have to be taken to restore a delta neutral hedge to the gamma neutral position created in Question 2?

a. buy 1,067 shares of underlying stock
b. sell 1,067 shares of underlying stock
c. buy 4,266 shares of underlying stock
d. sell 4,266 shares of underlying stock

First, a delta neutral position that has a negative gamma may either be a short call with long a stock or a short put with short a stock. If we want to have a gamma neutral position then we may either buy a call or a put, so the answer is c. However, in the next question, we are asked to have a delta-neutral hedge position with regard to the gamma neutral position that we have previously created, then the answer here must be either a or b depending on what we have bought in the first question?!

 

[David Harper CFA FRM]

Hi Chinquee,

I think you have made yet another winning observation. I agree, while the question expects a long call followed by short stock sale, you can achieve delta-gamma-neutral with long puts following by long shares because:

-3,200/1.5 gamma = 2,133 long options (calls or puts) will neutralize gamma.
This itself adjust delta by 2,133*(+ or - 0.5 delta) = adds or subtracts +/- 1,067 delta
so, delta neutralize with +/- 1,067 shares.

But, okay, technically the question is precise because it says delta = 0.5 and therefore the option must be a call (put delta < 0). So, i think you deserve partial credit

David

 

[ahnnecabiles]

Ah, yes, it must be a call. You're really great David! Thanks!