How do you hedge a portfolio using delta, gamma, and vega together?

Multi-Greek hedging is one of the most testable topics in the FRM derivatives section. The key principle is: each Greek requires a separate instrument to hedge, and you work from higher-order Greeks down to delta.

The Hedging Hierarchy

1. First, hedge gamma (and/or vega) using options. You cannot hedge gamma or vega with the underlying asset — only options and other non-linear instruments have gamma and vega.
2. Then, hedge delta using the underlying asset or futures. Delta hedging is always the last step because adding options in step 1 changes your delta.

Why This Order?

The underlying has delta=1, gamma=0, vega=0. Adding underlying shares changes only delta. But adding options changes delta, gamma, AND vega. If you hedge delta first, the options you add for gamma will mess up your delta hedge.

Worked Example

Dunewood Trading has the following portfolio position on stock XYZ (currently at $100):

Available instruments:

• Option A (call, strike $105): delta=0.45, gamma=0.03, vega=12
• Option B (put, strike $95): delta=−0.40, gamma=0.025, vega=15
• XYZ stock: delta=1, gamma=0, vega=0

Step 1: Solve for gamma and vega simultaneously.

Let n_A = contracts of Option A, n_B = contracts of Option B.

Gamma: 0.03 n_A + 0.025 n_B = +150 (to offset −150)

Vega: 12 n_A + 15 n_B = −8,000 (to offset +8,000)

Solving the system:

From gamma: n_A = (150 − 0.025 n_B) / 0.03 = 5,000 − 0.833 n_B

Substitute into vega: 12(5,000 − 0.833n_B) + 15n_B = −8,000

60,000 − 10n_B + 15n_B = −8,000

5*n_B = −68,000

n_B = −13,600 (sell 13,600 Option B contracts)

n_A = 5,000 − 0.833(−13,600) = 5,000 + 11,329 = +16,329 (buy 16,329 Option A contracts)

Step 2: Compute new delta after adding options.

New delta = 3,200 + 16,329(0.45) + (−13,600)(−0.40)

= 3,200 + 7,348 + 5,440 = +15,988

Step 3: Hedge remaining delta with stock.

Sell 15,988 shares of XYZ to bring delta to zero.

Exam Tip: The number of non-delta Greeks you need to hedge determines the number of option instruments required. For delta+gamma only: 1 option + stock. For delta+gamma+vega: 2 options + stock. Always solve the system of equations for higher-order Greeks first.

Practice more Greeks problems in our FRM derivatives question bank.