Question

Q3. Consider the following data: S = 100; r = .08; s = .3 on the following two options:

Call 1   Call 2

K = 100;     K = 90

T-t=90 days.      T-t=180 days.

Price: 6.91 16.33

Delta: .58 .78

Gamma: .02 .0138

3.1 A trader just shorted 100 CBOE calls 2.

Calculate the number of shares necessary to create a Delta Neutral position.

3.2 A trader just shorted 100 CBOE calls 2.

Calculate the number of calls 1 and shares of the underlying stock that the trader must hold (long or short) in order to create a Delta-Gamma neutral position

Q4. A trader just shorted 100 of calls 1 and 100 of calls 2 from Q3.

Calculate the delta and the Gamma of this position.

Q5. Consider put 1 and put 2, which are on the same stock and with the same exercise prices and time to expiration as their respective calls from Q3.

Calculate the number of shares covered in put 1 that together with the short calls 1 and 2 from Q3. create a Delta-neutral position.

Q6. Calculate the number of the shares in put 1 and put 2 needed to create a delta-Gamma neutral portfolio consisting of the short calls (call 1 and call 2 from Q3.) and the two puts.

Answer 1

	Call 1	Call 2
S	100	100
K	100	90
T-t	90	180
Price	6.91	16.33
Delta	0.58	0.78
Gamma	0.02	0.0138
r	0.08	0.08
s	0.3	0.3

3.1 Short Position in 100 Call 2

Call 2 delta = 0.78

Position required in shares for delta hedging:

Each call has 100 shares. So total numbers of underlying assets for Long position to be delta hedged is

0.78*100*100 = 7800 Shares to be Purchased

3.2 Short Position in 100 Call 2

Delta Gamma Hedge in Call 1 and Shares:

Delta of Short Position in Call2 is -0.78*100 = -78 (- ve sign due to short position)

Delta of Call 1 is 0.58 and that of shares is 1

Delta Neutrality: -78 + 0.58A + B = 0 (where A is number of Call 1 to be purchased and B is number of Shares to be purchased)

Gamma Neutrality : - (-0.0138*100) + 0.02A +0B = 0 (Gamma of share is 0)

Solving gamma neutrality equation

- (-0.0138*100*100) + (0.02*100)A +0B = 0

-138+2A = 0

A = - 69, Short position in Call 1

Putting A=-69 in delta neutrality equation

-78+0.58*100*(-69)+B=0

-78-4002+B=0

B=4080, Long Position in Share

Q.4) Short position in 100 Call 1 and 100 Call 2

The Portfolio Delta is = 100*(-0.58)+100*(-0.78) = -136

Q.5) Delta of Put 1 =Call 1 Delta - 1 = 0.58-1

Delta of Put 1 = - 0.42  

100 Short Call 1 and 100 Short Call 2 Delta = 100*(-0.58)+100*(-0.78) = -136

Position in Put Required for delta neutral

A*(-0.42) -136 = 0

A = -323.8 = -324

Short Poistion in 324 Put 1