A portfolio has the following position Greeks: delta = -300, gamma = -150, and vega = -3,000. A trader wants to neutralize all three Greeks and, in addition to the underlying shares, can use the following two options: Call option with the following percentage Greeks: delta = 0.60, gamma = 0.20, and vega = 10.0 Put option with the following percentage Greeks: delta = -0.40, gamma = 0.30, and vega = 20.0 Along with the underling shares, which set of trades will make the total position delta-gamma-vega neutral? (hint: first make the portfolio gamma-vega neutral, then use shares to neutralize delta) a. Short 800 of the calls; long 150 of the puts, and short 500 of the underlying shares b. Short 1,500 of the calls; short 680 of the puts, and long 770 of the underlying shares c. Long 2,100 of the calls; short 900 of the puts; and short 1,320 of the underlying shares d. Long 3,000 of the calls; short 1,750 of the puts; and long 540 of the underlying shares ANS - C Plz can anyone explain howcome the ans. is c
So you obviously have to trade the options in a way that neutralises all 3 Greeks…
Starting with delta = -300, gamma = -150, and vega = -3,000:
If you buy 2100 calls, you add (0.6 x 2100) = 1260 delta, (0.2 x 2100) = 420 gamma and (10 x 2100) = 21000 vega.
So now we have (1260 - 300) = 960 delta, (420 - 150) = 270 gamma and (21000 - 3000) = 18000 vega
If we then short 900 of the puts, we get (-0.4 x - 900) = 360 delta, (0.3 x -900) = -270 gamma and (20 x -900) = -18000 vega
So now we have:
(960 + 360) = 1320 delta, (270 - 270) = 0 gamma and (18000 - 18000) = 0 vega
Shorting the 1320 shares cancels out the 1320 delta position, so you are left with 0 delta, 0 gamma and 0 vega.
Vega is 10 and 20, not 0.10 and 0.20
Fix that and you’re good to go.
Hah…eesh - that’s what you get when you try to calculate this and type it all out on a phone screen. I’ll edit now to reflect…
I’ll help you with the calculations from the scratch. In the exam of course, doing the first part below is enough because it will determine the number of calls and puts required which gives option “C” as the answer. But this is for your understanding!
_ Part 1 _
Portfolio Call Put
Delta -300 0.6 -0.4
Gamma -150 0.2 0.3
Vega -3000 10 20
You can neutralize only two things at a time. You’ll have to neutralize gamma-vega first since if you neutralize delta with either of the others, then you will have to redo the delta which will again affect the others and become a never ending process.
Do it in simulataneous equations. You need “C” Calls and “P” Puts to find out the number of calls and puts to long/short.
So, 0.2C + 0.3P = -150 -----------> Gamma
and, 10C + 20P = -3000 -----------> Vega
Solving the above simultaneous equations, you get “900P” and “2100C”.
Part 2
2100 Calls and 900 Puts with their respective greeks gives:
Delta Gamma Vega
2100C = 1260 420 21000
900 P = -360 270 18000
Portfolio= -300 -150 -3000
Thus, to get an overall “Zero” Vega and Gamma, you need to sell/short the 900 Puts (multiply the “-” sign to the greeks corresponding to the 900 P above which gives +360, -270, -18000) and buy/long 2100 Calls. Summing the Vega and Gamma colums up with the Portfolio’s greeks gives a zero vega and zero gamma overall thus neutralizing these two.
_ Part 3 _
The Delta goes to 1260 - (-360) = 1620 and the Portfolio has a -300 Delta. Thus, 1620 - 300 = + 1320 is the remaining delta implying that you are now combinedly Long 1320 Delta. Since shares have a Delta of “1” we have to sell 1320 shares which gives a Zero delta overall.
Hope this helps!