Problem
Suppose you have a portfolio that has a delta of 50, a gamma of 1.76, and a vega of 109. You want to delta-gamma-yega hedge your portfolio. Suppose you have access to two call options: A, B. They are on the same underlying stock. Suppose Option A has a delta of 0.547, a gamma of 0.0065, and a vega of 0.78. Option B has a delta of 0.35, a gamma of 0.0043, and a vega of 0.85.
What position size must you have in these options and the underlying asset to delta-gamma-vega hedge your portfolio without changing the original portfolio itself? Assume you can hold fractional units.