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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.

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  • $\begingroup$ 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 $\endgroup$
    – MrLCh
    Commented May 9 at 16:01
  • $\begingroup$ 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. $\endgroup$
    – Enrico
    Commented May 9 at 20:57
  • $\begingroup$ 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 $\endgroup$
    – MrLCh
    Commented May 10 at 9:00
  • $\begingroup$ Looking forward for the answer! $\endgroup$
    – Enrico
    Commented May 10 at 21:00

1 Answer 1

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Given the delta for the two options I would assume that the volatility is assumed to be about 0.16. If you (delta) hedge your position you will have a P&L close to 0 (as your change in PnL can be approximated by your net delta, which is 0), but as you move further away from the asset price 100 your net delta will change. For example if your price goes to 105 the following applies (assuming 0.16 volatility):

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.

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  • $\begingroup$ 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? $\endgroup$
    – Enrico
    Commented May 15 at 8:09
  • $\begingroup$ Also, could give me an example of P/L? From BS i get that the 96call price is 9.04, the 104 call is 2.42. Then I sum these two, with the proper sign of the position, to 0.626 x 1.05. I get -5, very different from the table. Thanks for the patience! $\endgroup$
    – Enrico
    Commented May 15 at 8:45
  • $\begingroup$ I think you are correct with your comment that the delta has to be adjusted for position size, but it does not seem like Taleb has done it in this table. I just calculated everything using python (the correct vol seems to be 0.1556977536374234) and the delta in the presumably wrong calculation matches up with what is in the table. I have to think about, why this could make sense $\endgroup$
    – MrLCh
    Commented May 15 at 9:50
  • $\begingroup$ I also could not replicate the PnL column, I will have to double check my calculations, although I would not worry that much about this column. For the real insight it is just important to note that your delta becomes positive if you are not close to 0, so the further away you move from the origin, the more your "delta hedged position" becomes just a long position. $\endgroup$
    – MrLCh
    Commented May 15 at 9:55
  • $\begingroup$ Well, thanks for the insight! Anyway, I just want to learn how to calculate the numbers, this is why I asked the question. It's bad that we cannot replicate the table or, even more bad, that it contains errors! $\endgroup$
    – Enrico
    Commented May 15 at 11:44

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