Stock prices often jump rather than move continuously and smoothly (which is against BSM model assumptions), which creates “gamma risk”.
Can I understand this as if stock price abruptly jumps, it means delta is big, it gives more errors in delta measuring call value. however, gamma can capture the error.
Am I understood right?
then, why this should be gamma risk not delta risk?
I found this context from schweser saying Consider a delta hedge involving a long position in stock and short positions in calls. If the stock price falls abruptly, the loss in the long stock position will not equal the gain in the short call position. This is the gamma risk of the hedge.
why it would be different between gains in short call and loss in long stock?
When you delta-hedge, you are approximating the change in the option price as a linear function of the stock price. This is shown as a line in the graph. The problem with this approach is that delta is also a function of the stock price, and as a result it will also change. The book seems to call this tendency of delta to change as a ‘gamma risk’. Note that delta will almost always change. You just don’t know by how much a priori. With a delta-hedge you are hoping that the stock will not change by much.
For small changes in the stock price, the actual and the predicted option price using the linear approximation is pretty close (but always below the actual price), but as the change in the underlying becomes greater, the difference between the predicted price and the actual price increases.
If you took Calculus, you’ll notice that delta-hedging is just an application of Taylor series. With a delta hedge, you approximate the option price using the first-order Taylor series. You can get a better approximation using a second-order Taylor series. A delta-gamma hedge is a quadratic, rather than linear, approximation. You can see in the graph that the delta-gamma hedge performs much better than just the delta-hedge when the stock price moves significantly. There’s still a difference in the option price and the price predicted using the delta-gamma hedge. This is because gamma itself also changes. You can get a better approximation by taking the 3rd order Taylor series, the 4th order, and so on.
I actually wrote a priori with a space, but for some reason the text editor marked it as wrong, so I second-guessed myself and removed the space. Strangely, the editor will sometimes mark ‘a priori’ as being a wrong spelling. Other times it marks it as correct. It will also change its mind after a random number of seconds. Forum bug, maybe?
