Delta is a number, and numbers are not directional. The word "directional" often is used to refer to the effect of market movement on the value of the position. What I am trying to figure out is how to get a position that has a very small delta when the underlying moves in one direction, and a large delta when the underlying moves in the opposite direction? Signs of the delta are not my concerns here. In other words, how to get large positive gamma when underlying moves one way, and nearly zero gamma for the other way. So far I have not been successful. Mathematically possible? Any suggestions?

Not sure I understand your question but delta is directional when looking at your position. A positive delta for a long position can have a bullish directional bias. For example a long call with positive delta, a bull call spread, an OTM butterfly or calendar. What are you trying to achieve?

If I understand correctly then a short-term OTM option fits the bill. In this gamma pic with the stock at ~ 35 or 45 this 40 call or put (respectively) has flatland on one side and a mountain on the other, especially close to expiration (at Time = 100). Take this to the extreme by buying cheap OTM's during expiration week; those with low IV have the steepest gamma curve. But, perhaps not quite what you had in mind?

Thank you all for your replies. Let me try to re-state what I am trying to achieve. Assume there is a set of options which I will denote "S." I am trying to construct S with the following characteristics: (1) when the underlying moves in one direction (e.g., up), no matter how far, S loses little value; this requires S to have delta of small magnitude in this (up) direction; and (2) when the underlying moves in the opposite direction (down), no matter how little, S will gain in value; this requires S to have delta of large magnitude in the down direction. This is what I mean by delta being "directionally biased," and this must be mathematically hard or impossible to do because somehow I feel that it will violate the non-arbitrage principle. Comments?

If I understand you correctly you want the following: 1. a set of options with zero or very small delta AND AT THE SAME TIME 2. the same set of options with large negative delta. So you thus want this set of options to have a delta of near zero and a delta of large negative magnitude simultaneously? Obviously not possible. db

Try an out of the money butterfly. That should fit your risk/reward profile. If the underlying moves AWAY from your short strikes, you have limited losses (sometimes no losses). If the underlying moves TOWARDS your short strike, you will be dreaming of that new Bentley and how you are the smartest inividual in the universe. Put an OTM butterfly in your analysis proggie and smoke it.