The Greeks (Gamma and Vega)

Can someone explain to me the difference between Gamma and Vega? I understand that Vega measures the sensitivity of the options price to changes in volatility of returns in underlying assets. Gamma measures the rate of change in Delta(so it’s kind of like the 2nd derivative of an options price?), but isn’t the rate of change in delta, pretty much the same thing as volatility? Any help would be much appreciated

Vega is sensitivity of option price to volatility of underlying asset price. Gamma is sensitivity of delta (which is sensitivity of option price to asset price) at different underlying asset prices. For example; if asset price has high volatility, it’s vega is higher. Very differently, it’s gamma is high when the asset price is close to strike price and low when the option is out of the money. An option with high vega may not necessarily have high gamma, and vice versa.

fpersic, look for a thread where Joey explained Vega. if market goes down 10% and then comes back changes due to delta and gamma should be zero. But because of higher volatility prices can change. That would be vega component.

Gamma, as you suppose, is the second derivative of an option price, much like acceleration is the second derivative of speed. As such, you can view Gamma as a measure of how quickly Delta will change when the price of the underlier moves further away from where it is right now. For instance, for an ATM option, Delta = 0.5. Gamma is measured by measuring the change in the option’s price when Delta moves 1%, while holding everything else constant. maratikus’s example explains Vega quite well in that if the price moves around rapidly, but ends in the same place where it started, there is going to be a change in the price of the option, eventhough the underlier hasn’t changed.

Let me give you another example, from fixed income. Imagine you have a non-prepayable mortgage. That mortgage will have positive delta with respect to the underlying which is the interest rate. The mortgage will also have a gamma. Interest rate delta and gamma are oftentimes called duration and convexity. However, that mortgage will not have gamma - if the volatility of the interest rates changes, the valuation of the mortgage does not change. now image you have a prepayable mortgage, meaning the mortgage has an embedded option. This mortgage will have non-zero vega.