Correcting a Call Spread and Forward Hedge in a Delta Example
Summary
The document resolves a reader’s confusion about a delta-hedged option example in a chapter on dynamic hedging. The described position combines calls at two strikes with a forward hedge, but the stated long and short directions do not match the accompanying table and payoff graph. The accepted response says the text has reversed the position and hedge instructions: the consistent interpretation is short the lower-strike call, long the higher-strike call, and long forwards to offset the position’s negative delta at the reference price.
The response frames delta as an equivalent position in the underlying. At the reference price, the hedge makes that equivalent position approximately neutral. When the underlying moves, the option position’s delta changes because of its curvature, so the portfolio is no longer neutral and can lose value as the move continues. The example is useful for connecting delta, hedging, and changing exposure, but its interpretation depends on correcting what the response identifies as a textual typo. It is not a general trading recommendation.
Key ideas
- Delta can be interpreted as an equivalent position in the underlying security.
- A forward position can offset an option portfolio’s delta at a chosen underlying price.
- As the underlying moves, option delta changes and the initial hedge may no longer be neutral.
- The example’s stated option directions and hedge direction must be reversed to match its table and graph.
- The document illustrates changing exposure from option curvature rather than laying out a trading recommendation.
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# Question on an example from "Dynamic Hedging" by Nassim Taleb # Question on an example from "Dynamic Hedging" by Nassim Taleb 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 19.8 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? ## Answer by zer0hedge (score 6, accepted) https://quant.stackexchange.com/a/32822 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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