Question

A portfolio of derivatives on gold (G0=1910/ounce) has a delta of 1500, a gamma of 200, and a vega of -400 (per 1%). Options on gold are available, where one option on 100 ounces, and with the following Greeks and premium (all per ounce):

Delta Gamma Vega Premium

Call 0.5 0.6 0.5 18.00

Put -0.6   0.4 0.5 24.00

Using these options, long/short positions in gold, and long/short positions in USD risk free assets (for investing/borrowing cash), create a hedge to eliminate this portfolio’s delta, gamma and vega.

Answer 1

Creating a hedge for the portfolio by eliminating the portfolio's delta, gamma, vega is done first by neutralizing the vega and gamma using options and then neutralizing the delta with the underlying asset which is gold here.

Let say we buy X amount of call and Y amount of put for making the portfolio vega and gamma neutral

=> 0.6X + 0.4Y = 200

&

=> 0.5X + 0.5Y = -400

On solving both the equations we get,

X = 1300

Y = -3400 ( - sign represents we will short)

Now the new delta of the portfolio would be = 1500 + 1300*0.5 + (-3400)*(-0.6) = 4190

The delta of underlying asset is 1 so we will short 4190 gold.

So to make the portfolio hedge we will buy 1300 call option, sell 3400 put option and sell 4190 gold.