Solution

I will give two examples. The first is quite contrived. The second is less so. It actually arises in practice.

For the first example, imagine combining long and short options of various strikes but the same expiration to achieve a portfolio that has the payoff indicated on the left in Exhibit S1. You might do so by shorting two butterflies and a straddle. Prior to expiration, this might have a market value that depends on the underlier value as indicated on the right. In that graph, the current underlier value is marked by the gold triangle and dashed line. At that value, the position positive gamma. However, the position would likely lose value if the underlier's implied volatility rose. Hence, the position is short volatility.

Example: Long Gamma, Short Vega Position Exhibit S1

Combining long and short positions in vanilla options with different strikes but the same maturity, a position with the payoff on the left can be achieved. Prior to expiration, this might have a market value that depends on the underlier value as indicated on the right. In that graph, the current underlier value is marked by the gold triangle and dashed line. At that value, the position positive gamma. However, the position would likely lose value if the underlier's implied volatility rose. Hence, the position is short volatility.

For the second example, note that gamma tends to be highest for at-the-money options that are close to expiration. Vega tends to be highest for options that have more time to expiration. If we construct a portfolio that is long at-the-money options close to expiration and is short at-the-money options that have a long time to expiration, two things will happen:

The positive gamma of the options close to expiration will dominate the negative gamma of the options far from expiration, so the portfolio's net gamma will be positive.

The negative vega of the options far from expiration will dominate the positive vega of the options close to expiration, so the portfolio's net  vega will be negative.

The result is a positive gamma, negative vega portfolio.

[Return]