# Gamma hedging

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**Gamma hedging** allows investors to mitigate risk related to changes of an option's delta. Delta defines expected change in price of the option related to change of asset value. Delta tends to change very fast, which is not convenient.
Gamma shows rate of delta change in relation to asset value (in fact is twice calculated rate of change of the asset). The gamma changes more smoothly and allows investor to make better decisions.
In order to optimize the strategy, investors usually use both delta and gamma hedging. For example, investor can buy call options and short shares based on delta and add another short at different level to protect against large delta.

**Gamma (Γ)** measure the change of the option’s delta with respect to the underlying asset price. It can be defined by the second part of the price asset price.
Suppose the gamma has a value of 0.8, it means 1 cash unit increase in the stock price will increase the delta of the option by 0.8. A 1 cash unit decrease in the stock price will decrease the delta of the option by 0.8.
If gamma is small, it means the change of the delta is slowly with changes in the underlying asset price, so the adjustments to keep the delta neutral need to be made only relatively infrequently. If gamma is big, that is the delta is very sensitive to the underlying asset price, so it is risky to leave a delta-neutral portfolio unchanged for any length of time^{[1]}.
High gamma means, that variations in delta are high, and hence more frequent balancing to maintain low delta exposure.
Delta hedging is based on small changes during a very short time period, assuming that the relation between option and the stock is linear
locally.
When gamma is high, the relation is more curved than linear, and the hedging error is more likely to be large in the presence of
large moves. The gamma of a stock is zero.
We can use traded options to adjust the gamma of a portfolio, but when we thinking aout large moves, it is better to try something else^{[2]}.

## "Pinning" a stock approaching expiry[edit]

It is possible to "**pin**" by Gamma Hedging a stock approaching expiry.
If an investor with long gamma can delta hedge by sitting on the bid and offer, this trade can pin an underlying to the strike. If we are selling when the stock rises above the strike, and buying when the stock falls below the strike, it can be the negative effect.
Important thing is, that amount of buying and selling has to be compared with traded volume of the underlying, so pinning is the best thing, when stocks are relatively illiquid or when the position is particulary sizeable.
Given the high trading volume of indices, it is difficult to pin a major index. The biggest chance for pinning is when market is calm, because there is no strong trend to drive the stock away from its pin.
Pin risk take place when the market price of the underlier of an option contract at the time of the contract’s expiration is close to the option’s strike price. In this situation, the underlier is said to have pinned. Seller risks, because he can't predict if the option will be exercised,so the seller cannot hedge his position and can loose or gain. An option position can result in a undesired risky position in the underlier immediately after expiration^{[3]}.

## References[edit]

- Andriot, A., Nirascou, P., (2013).
*Market’s gamma hedging absorption capability for barrier options*. 05/12/2013. - Benchaphon, C., Chutima, K., Apiwat, P., Thanasunun, S., (2008).
*Hedging with Options*. October 6, 2008, Västerås. - Jarrow, R. A., & Turnbull, S. M., (1994).
*Applied Mathematical Finance*. Delta, gamma and bucket hedging of interest rate derivatives, 1(1), 21-48. - Wu, L., (2005). Options Markets"P&L Attribution and Risk Management".

## Footnotes[edit]

**Author:** Maja Rogalska