What Susie will want to know is why she made $800. Why not more?
Why not less, for that matter? When trading delta neutral, especially with
more complex trades involving multiple legs, a manual computation of each
leg of the spread can be tedious. And to be sure, just looking at the profit or
loss on each leg doesn’t provide an explanation.
Susie can see where her profits or losses came from by considering the
profit or loss for each influence contributing to the option’s value. Exhibit
12.6 shows the breakdown.
EXHIBIT 12.6 Profit breakdown by greek.
Delta
Susie started out long 0.20 deltas. A $2 rise in the stock price yielded a $40
profit attributable to that initial delta.
Gamma
As the stock rose, the negative delta of the position increased as a result of
negative gamma. The delta of the stock remained the same, but the negative
delta of the 50 call grew by the amount of the gamma. Deriving an exact
P&(L) attributable to gamma is difficult because gamma is a dynamic
metric: as the stock price changes, so can the gamma. This calculation
assumes that gamma remains constant. Therefore, the gamma calculation
here provides only an estimate.
The initial position gamma of −1.6 means the delta decreases by 3.2 with
a $2 rise in the stock (–1.60 times the $2 rise in the stock price). Susie, then,
would multiply −3.2 by $2 to find the loss on −3.2 deltas over a $2 rise. But
she wasn’t short 3.2 deltas for the whole $2. She started out with zero deltas
attributable to gamma and ended up being 3.2 shorter from gamma over that
$2 move. Therefore, if she assumes her negative delta from gamma grew
steadily from 0 to −3.2, she can estimate her average delta loss over that
move by dividing by 2.