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So I'm reading through Dynamic Hedging to start trying to learn option theory better. I hit Chapter 8 on Delta and am completely lost on a certain example he gives.

The example is from Page 119 and is labeled "A Misleading Delta" - He posits the following scenario -

A trader has the following position, yield curve is flat and forward is same as spot. European options with one month maturity:

Long 1M 96x calls delta of 82.4
Short 1M 104x calls delta of 198
Net delta is 62.6

Taleb says that the trader "could hedge it by selling $626,000 of forward" which makes it unclear whether or not this is included in the position (though it makes even less sense if it isn't included).

He then posts a table which shows a flat delta and P&L at 100 (so assuming position was put on at 100 and delta hedged). However, it also shows the delta increasing for price movements in either direction. A graph is also shown that shows the position as hedged to some extent around 100 (the origin) i.e. flat P&L, with positive P&L accruing with higher prices and losses at lower prices.

How is this possible? My limited understanding suggests that this would result in the opposite exposure - decreasing P&L to the upside and gains to the downside as the delta of the net option position would be near or at its max at 100 and decreasing in either direction. So due to the forward hedge, you'd have a net short position.

In fact, from what I can tell the data output provided in the book shows the opposite position - long 104 put, short 96 put, long ~$626,000 fwd.

Can anyone help me understand what I'm missing?

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Taleb's explanations are correct, ONLY if you replace "short" with "long", "sell" with "buy" and vice versa in the text on page 119. In this case the narrative will correspond to pictures and Taleb's explanations will be reasonable.

So the position is actually short 96 calls, long 104 calls, total continuous delta is short. Then "could hedge" means that the position is hedged by "buying" (not selling as in the text) forwards.

You should think of "delta" as an equivalent position in the underlying security. So when the underlying price is 100, your equivalent position is 0.

When price goes down to 98, long 104 calls and long forwards are loosing value faster than short 96 calls are gaining it, so you are incurring some loss (-2).

Furthermore, due to non-convexity, your position is no more delta-neutral, but equivalent of 39 in underlying, i.e. long position. You may expect that you will now be losing value faster if the price continues to go down. In fact, this is confirmed by the data in the table: price 97 your loss is -9 etc.

Similar logic works if the price goes up from 100. Long calls and long forwards will be gaining value faster than short calls are losing it etc.

It is obviously a typo because already on the next page Taleb is saying "The trader in the example buys \$550,000 cash instead of \$626,000"

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  • $\begingroup$ I agree with what you said but aren't you ignoring the impact of the hedge? $\endgroup$ – somethinghere Mar 5 '17 at 16:13
  • $\begingroup$ Going back to your example - if underlying is decreasing, you've sold 626,000 forward, and the delta of the 96 calls is decreasing faster than the 104 calls then your net delta would be short, not long because you are now over hedged by virtue of the forwards you sold, right? $\endgroup$ – somethinghere Mar 5 '17 at 16:20
  • $\begingroup$ Just re-read what you said a couple times - Are you suggesting that the hedge is not static and that would be "trued-up" based on each corresponding price level? $\endgroup$ – somethinghere Mar 5 '17 at 16:56
  • $\begingroup$ Consider the position static, i.e. long 96 calls, short 104 calls and short forwards - nothing changes. Then consider what will be happening with the position's delta (not P/L!) when the price of underlying is changing. The position's delta (sum of deltas of 96 calls 104 calls and forwards) will change! Individual deltas of components are changing differenlty and together will not give zero any more when price move far enough. $\endgroup$ – zer0hedge Mar 5 '17 at 19:21
  • $\begingroup$ Delta then impacts P/L shown on Figure 7.2. If you had just long forward position (i.e. with some positive delta), the P/L would be straight line from left bottom to top right and could be tangent to the graph on the Figure 7.2 at point (92.5, -125). That's why Taleb is saying that the graph reflects "long" position $\endgroup$ – zer0hedge Mar 5 '17 at 19:28

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