# P/L table for a delta hedged position

I am trying to replicate the table at pag. 119 of Dynamic Hedging by N. Taleb with no success. In the example called "A misleading delta" an operator has the following position:

• long \$1 million of the 96 call(delta .824, so \$824k long)
• short \$1 million of the 104 call (delta .198, so \$198k short)
• hedge the position by selling \$626k of the forward where all options are European on the same underlying asset and with the same maturity. Plus, we assume a flat yield curve. I found in this answer (Question on an example from "Dynamic Hedging" by Nassim Taleb) the fact that, probably, there is an error in the book and that the positions should be reversed. Anyway, I am not able to calculate the P/Ls and deltas reported and I would like some help. For example, from what I understand, the market is trading at \$100 the underlying so we have P/L=0 and delta=0. Suppose that the underlying is at $103. I would compute the P/L by summing (I consider here the "reversed position" as suggested in the answer above): • gain from long the asset: \$626k*(103-100)/100
• gain from long the 104 call: $1million*(103-100)*0,198 • loss from short the 96 call: -\$1million*(103-100)*0,824

What I obtained is very different from the P/L of \$12k reported. How is this last number calculated? How the delta of \$106k is calculated?

Thanks for the help. Please, let me know if more details are needed.

• Unfortunately I do not have my version of Dynamic Hedging at hand, but isn't this the point of the example? You can not just calculate the P/L over your delta as the delta changes as the asset price moves Commented May 9 at 16:01
• There are no explicit calculations in the book. I am trying with no success to replicate the table and I reported here an example to check if my calculations are correct. Do you see any error committed by me? I can’t find it. Thank you. Commented May 9 at 20:57
• I looked up the table now and it seems like the table is just based on reevaluating the options (in a BS framework). The problem with your calculation is that delta does not stay constant over asset prices. (Otherwise your gamma would be 0). You just assume that delta stays the same between 100 and a 103. I will write a full answer later Commented May 10 at 9:00
• Looking forward for the answer! Commented May 10 at 21:00

delta of the $$96$$ call: $$0.978$$ (calculated in Black-Scholes framework)
delta of the 104 call: $$0.593$$ (calculated in Black-Scholes framework)
As you keep the the forward you still have a delta 1 position of \$$$626,000$$. So your net delta becomes $$0.593-0.978+0.626 = 0.241$$. You can do this for every asset price to get the table just like in the book. To calculate the P/L you just use the prices in a Black-Scholes framework instead of the Delta. • Thanks! Why are you summing directly the deltas? I mean, shouldn't they be "transformed" in dollars in some way given the fact that the asset price is changed? In particular, the 626k\$ should be like 626k\$x 1.05 no? Commented May 15 at 8:09