Portfolio Gamma Hedging with Options and Delta Neutrality
Summary
The document clarifies that gamma is a property of a position or portfolio, not a quantity that can be rebalanced by trading the underlying alone. Gamma measures how a position’s delta changes as the underlying price moves. A gamma hedge adjusts option holdings so the portfolio’s aggregate gamma approaches zero; share holdings can then be adjusted to manage aggregate delta.
The example models a portfolio of shares and two options with different strikes or expirations, and sets equations for total delta and gamma. Given the instruments’ sensitivities, the option quantities can be solved to offset the portfolio’s gamma exposure, while stock is used for delta adjustment. This is a simplified local hedge framework: it depends on estimated Greeks and does not guarantee replication of an option payoff or eliminate risks from larger price moves, changing volatility, or time decay.
Key ideas
- Gamma hedging targets the aggregate gamma exposure of a portfolio.
- The underlying shares have delta but zero gamma in the basic model.
- Options with suitable sensitivities can offset portfolio gamma exposure.
- Share quantities are used to adjust delta after selecting the option hedge.
- A Greek-neutral hedge is local and does not remove all market or model risk.
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# Meaning of Rebalancing the Gamma in Options?
# Meaning of Rebalancing the Gamma in Options?
What does rebalancing the gamma mean?
In the Book: Dynamic Hedging at the beginning says:
- Rebalancing the gamma corresponds to buying and selling the underlying security in order to replicate the payoff of the option.
Gamma shows the rate of change of delta. How exactly can i buy the underlying security (in what quantity?) so that i rebalance the gamma? Is the gamma only for me? No it is for the stock just as the price. So how can i buy or sell to rebalance the gamma of a security? How buying and selling at the same time will replicate the payoff?
## Answer by Giogre (score 2, accepted)
https://quant.stackexchange.com/a/60189
Achieving gamma neutrality refers to your whole portfolio situation.
If you have a portfolio $P$ made up of $n_S$ shares of stock $S$, and of $n_1$, $n_2$ option calls $C_1$, $C_2$ on $S$ (options 1 and 2 differ in strike price or expiration), pursuing a gamma hedging strategy would imply to achieve neutrality on the Delta $\Delta_P$ and Gamma $\Gamma_P$ overall values of your portfolio:
$$ \begin{align} \Delta_P &= n_S \Delta_S + n_1 \Delta_{C1} + n_2 \Delta_{C2} = 0\\ \Gamma_P &= n_S \Gamma_S + n_1 \Gamma_{C1} + n_2 \Gamma_{C2} = 0 \end{align} $$
Say you know $\Delta_S$, $\Gamma_S$, $\Delta_{Cx}$, $\Gamma_{Cx}$ from the markets, and you want to achieve Gamma neutrality for $n_S = 100$ stocks $S$. Then you solve the system above to find the quantities $n_{1,2}$ of options you should own in order to hedge your portfolio against brusque variations in the price of the underlying you would not be able to prevent relying only on Delta hedging.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.