Portfolio Delta calculation

[SaiJyotsna92]

In R27.P1.T4.Hull_Ch13_15 page 46 while calculating the Delta of the portfolio why are we subtracting each option position and it's individual delta with that of other options.

 

[David Harper CFA FRM]

Hi @SaiJyotsna92 I moved this post to this T4 forum.

They are subtracted because they are short positions. [Position Greek] = Quantity * [Percentage Greek] where a short position is captured as negative quantity; for example, short 200,000 call options each with (percentage) delta of 0.468 = (-200,000)*0.468. When neutralizing, we are generally neutralizing position Greeks. We have many discussion about this on the forum, a good search is "percentage delta" (which is Carol Alexander's term) https://forum.bionicturtle.com/threadloom/search?query=percentage delta&tab=237

but here is one of my earlier explanations, I hope it is helpful! https://forum.bionicturtle.com/threads/l1-t4-7-dynamic-delta-hedging.4839/ i.e.,

David's ProTip: I learned from Carol Alexander a useful semantic distinction (not in Hull). Consider a position in 100 call options with per-option delta of 0.6:

• The Percentage Delta is 0.6; this is the unitless first partial derivative, dc/dS
• The Position Delta is 60 because Position Delta = Quantity * Percentage Delta.
• If we are long, we use (+) quantity: Position Delta (long 100 calls) = +100 * 0.6 = +60;
• If we are short , we use (-) quantity: Position Delta (short 100 calls) = -100 * 0.6 = -60
• To neutralize is to get the position Greek to zero

This is robust, for example:

• Selling puts increases position delta because -QTY * -% delta = +position delta; i.e., % delta of puts always negative; % delta of calls is always positive
• Selling calls or puts decreases position gamma because -QTY * +% gamma = - position gamma; i.e., % gamma is always positive for both calls & puts

Just as we use dollar duration (not modified duration) to neutralize duration in the portfolio, we neutralize an option Greek by summing Position Greeks to zero. I often see candidates trying to neutralize with percentage delta directly, but you can't, you need to sum the "Position" Greeks. I hope that's useful! David