Why hedge gamma risk?
From the last study, we learned that Gamma refers to the ratio of Delta change to the price change of the underlying assets in the trading portfolio. So the value of Gamma is related to the profit and loss of the whole portfolio. When the absolute value of Gamma is large, it shows that the change of Delta will change very quickly with the price of the underlying assets, and investors need to adjust the Delta value frequently to avoid the non-neutral risk of Delta. When the Gamma value is negative, if the underlying asset price changes in a favorable direction, the option position will slow down its appreciation; If the price of the underlying asset changes in an unfavorable direction, the option position will accelerate the depreciation rate. In addition, when Gamma is positive, the situation is contrary to the above conclusion, but the Theta value of time loss is negative, which means that time becomes the enemy of investment income again.
Therefore, it is risky for investors to build a portfolio with any value of Gamma. Only when the gamma neutrality is 0, can we really avoid the gamma risk and reduce the risk of trading portfolio. When the option is equal to or close to maturity, the gamma risk of the option is the greatest. The above figure shows the relationship between the Gamma of call options and the underlying asset price.
How to hedge gamma risk?
Since the Delta of the underlying asset is always 1, the Gamma reflecting the Delta change rate is always 0. If you want to hedge the gamma of the trading portfolio, you can't start with the underlying assets, but only with the help of products with nonlinear relationship between the price and the price of the underlying assets, such as options. In general, investors can obtain the gamma information of option contracts directly from trading software without having to calculate it themselves. However, as an investor who needs to hedge gamma risk, it is necessary to understand the calculation process of gamma value. For stock options or put options without dividends, the Gamma value can be obtained by the following formula:
In the formula, d 1 comes from BS model, and N(x) is the density function of standard normal distribution. S0 is the underlying asset price, σ is the volatility of the underlying asset price, and t is the term of the option. It is worth noting that as a buyer of options, the value of Gamma is greater than 0, while as a seller of options, the value of Gamma is less than 0.
When we hold a Delta neutral portfolio, γ is γ (γ ≠ 0). We need to find an option contract for gamma hedging. Assuming that the Gamma of the contract is γ t, adding wt option to the portfolio, the Gamma of the new trading portfolio thus obtained is WT γ t+γ. In order to keep the new Gamma value neutral, investors need to trade positions with wt =-γ/γ T.
Here is an example to further illustrate how to hedge Gamma risk with options. Suppose investors hold a set of Delta neutral portfolios, but the Gamma value is -300. Investors decided to use the X option contract to hedge the gamma risk. Assuming that the Delta value of the X option contract is 0.5 and the Gamma value is 1.5, in order to keep the Gamma value neutral, we need to add -(-300/ 1.5)=200 options to this trading portfolio. However, since the Delta value has increased from 0 to 200×0.5= 100, investors must sell 100 copies of the underlying assets in order to continue to ensure the neutrality of the trading portfolio.
Through this example, we can find that adding new options to the original Delta neutral portfolio will lead to the change of the portfolio Delta. After Gamma hedging with options, investors must readjust the number of underlying assets to maintain Delta neutrality. Therefore, hedging gamma risk is basically divided into two steps. First, buy/sell a certain number of options to hedge the gamma of existing positions; Second, buy/sell a certain number of underlying assets to offset the new Delta.
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