I’m having trouble understanding the idea behind portfolio delta and gamma. As I understand, call option have delta that depends on factors (such as ITM, ATM, OTM between 0 and 1) and underlying also have delta which fortunately is 1, means if the underlying goes up by $1, then value of the underlying goes up by $1, sounds stupid but the point is the the underlying also has delta, it just happens to be 1 and gamma of 0.
In curriculum example, an investor take action for covered call (long stock and short calls for stock held), where he writes call that have delta of 0.516 and gamma of 0.156. Portfolio delta (gamma) is underlying delta (gamma) minus call option delta (gamma) for covered call. So if underlying price increase of 1 will result in portfolio gain of approx. 0.484 (1 - 0.516) that make portfolio gamma to -0.156 (0 - 0.156).
My question is, how can call option delta affect portfolio price below the increase of underlying (which is only 0.484) where the investors already received premium when he writes the calls (and also decreasing portfolio gamma).
Since portofolio Delta being 0.5, it means increase of $1 in underlying will increase portfolio value for $0.5. my question is why value only increase 0.5 meanwhile investor already receive premium when write a call. Isn’t it supposed to be higher than $1?
Delta gives the information you need to create a replicating portfolio (see the derivatives portion of CFA level 2 for more details). Every portfolio can be broken down into a long/short position in shares and borrowing or lending money at the risk-free rate. The fair price of the portfolio incorporates the information of Delta. It should be noted that Delta only gives a rough, approximate measure of how much the fair price of the portfolio changes when the price of the underlying changes. Sometimes you want a more accurate estimate for the change in price. To accomplish this and get a better approximation to the change in price, you use Gamma in addition to Delta.
Lets say I have a covered call portfolio with delta 0.484. When underlying price +1, my option value will increase 0.484. does that value disregard premium received?
One more thing. Can I have portfolio delta greater than 1? let’s say I have portfolio of an underlying stock and long call option. Does it mean my portfolio delta is greater than 1?
Forget the premium (the initial fair price of the portfolio before the stock price changed) for now. There are at least two interpretations for delta.
The first one is that delta is the number of shares you need to create a replicating portfolio (a portfolio with identical cash flows) (note that you typically also need to have borrow/lend some amount in risk-free securities as well when creating a replicating portfolio.
Another interpretation is mathematical: delta is the rate of change of the fair price of the portfolio with respect to the stock price. That is, if the stock price increases by $1, the fair price of the Portfolio increases by approximately delta.
Graphically, imagine the fair price of the portfolio as a curve, where the x-axis is the price of the stock and the y-axis is the fair price of the portfolio. Take a particular point P in that graph. With Delta, you try and approximate the change in the fair price of the portfolio with a tangent through P.
You could say that the premium received tells you the point P in this graph, while delta is the slope of the tangent through that point, if that makes sense.
Yes, delta can be greater than 1 (or lower than -1 for that matter). When delta is greater than 1, it means that you need to buy more than 1 share to replicate the option. Also yes, if you are both long the stock and long a call, your net delta is greater than 1.
