Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

The Greeks and Option Risk Management › Gamma and Vega Risk Management

Michael BrenndoerferNovember 29, 20254 min read

Quantitative FinanceAll chapters

Part 5 of 15Stochastic Calculus and Pricing Theory

Chapter 31The Greeks and Option Risk Management › Gamma and Vega Risk Management

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While delta hedging handles directional risk, managing a professional option portfolio requires controlling gamma and vega exposures as well. These higher-order risks can dominate P&L during volatile market conditions and require specific techniques to manage.

Gamma HedgingLink Copied

You cannot hedge gamma using the underlying asset alone because stock has zero gamma. The delta of a stock is always 1, regardless of the stock price, so its gamma is zero. To adjust gamma, you must trade other options.

Suppose your portfolio has $\Gamma_p$ and you want to make it gamma-neutral. You can add a position in another option with known gamma $\Gamma_h$ . The number of options to trade is:

N = -\frac{\Gamma_p}{\Gamma_h}

where:

$N$ : number of hedging options to trade
$\Gamma_p$ : current portfolio gamma
$\Gamma_h$ : gamma of the hedging instrument

After adding this position, you'll need to readjust delta because the new options have delta too. This sequential hedging process, first neutralizing gamma with options and then neutralizing delta with stock, is standard practice in professional trading.

Vega HedgingLink Copied

Similarly, vega can only be hedged with other options. A portfolio of options on the same underlying will typically have correlated vega exposures; if implied volatility rises, all options tend to gain value.

You often distinguish between:

Parallel vega risk: Uniform shifts in volatility across all strikes and expirations
Skew risk: Changes in the volatility smile (the pattern of implied volatility across strikes)
Term structure risk: Changes in the relationship between short-dated and long-dated implied volatilities

We'll explore these concepts in depth in the next chapter on implied volatility and the volatility smile.

Building a Greek-Neutral PortfolioLink Copied

Let's work through an example of constructing a portfolio that is simultaneously delta-neutral and gamma-neutral. We first calculate the Greeks for our existing position and a hedging instrument. Then, we solve a system of linear equations: first finding the quantity of the hedging option to neutralize gamma, and then using stock to neutralize the remaining delta:

In[26]:

Code

# Portfolio: We're short 100 ATM calls, strike 100
# Want to hedge with stock and another option (strike 95 call)
S = 100
r = 0.05
sigma = 0.20
T = 0.25

K_atm = 100  # Our short position
K_itm = 95  # Hedging option

# Calculate Greeks for each option
delta_atm = black_scholes_delta(S, K_atm, T, r, sigma, "call")
gamma_atm = black_scholes_gamma(S, K_atm, T, r, sigma)

delta_itm = black_scholes_delta(S, K_itm, T, r, sigma, "call")
gamma_itm = black_scholes_gamma(S, K_itm, T, r, sigma)

# Position: short 100 ATM calls
n_atm = -100

# Portfolio Greeks before hedging
port_delta = n_atm * delta_atm
port_gamma = n_atm * gamma_atm

# Step 1: Find number of ITM calls to neutralize gamma
n_itm = -port_gamma / gamma_itm

# Step 2: Calculate new portfolio delta (after adding ITM calls)
new_port_delta = port_delta + n_itm * delta_itm

# Step 3: Hedge remaining delta with stock
n_stock = -new_port_delta  # Shares to buy (stock delta = 1)

Out[27]:

Console

Portfolio Construction: Delta and Gamma Neutral
=======================================================

Original Position: Short 100 ATM calls (K=100)

Option Greeks:
  ATM Call (K=100): Δ = 0.5695, Γ = 0.039288
  ITM Call (K=95):  Δ = 0.7543, Γ = 0.031488

Initial Portfolio Greeks:
  Delta: -56.95
  Gamma: -3.9288

Hedging Trades:
  Buy 124.77 ITM calls (K=95) to neutralize gamma
  Buy -37.16 shares to neutralize remaining delta

Final Portfolio Greeks:
  Delta: 0.000000
  Gamma: 0.000000

The hedged portfolio is now immune (to second order) to moves in the underlying. However, it still has theta and vega exposure. Managing all Greeks simultaneously requires trading multiple options with different characteristics. In practice, we often accept some residual Greek exposure rather than attempting perfect neutrality, which would be prohibitively expensive in transaction costs.