# Can a delta hedge be negative for all values at one time, and positive for all values at another time?

I have a problem that states there was a formula for the hedge $$\delta(t, S_t)$$ for a contingent claim whose value depends on only the stock value when $$T=20$$. In this hedge, $$\delta(t, S_t)<0$$ at $$t=11$$ for all possible values of the stock, and $$\delta(t, S_t)>0$$ at $$t=14$$ for all possible values of the stock, and we are asked whether such a formula is possible.

I'm not entirely sure how to go about answering whether such a hedge is possible or not. My intuition is that such a hedge is not possible as it would not makes sense to be selling stock at a time regardless of the stock's price, only to buy more regardless of its price at a later time, as the stock's price could be lower at $$t=11$$ than at $$t=14$$, which would result in a loss. Is this along the right lines of thinking, or am I a bit off here?

• Perhaps you should argue not in terms of profit or loss, but whether a rational prediction of $S_T$ can be be based only on time until $T$ and no other information. It would not appear that changing the prediction in this way accomplishes anything. Surely $S_T$ is either completely unpredictable (for ex. it jumps at the last moment) or depends in some way on $S_t$ (or other outside information), it cannot depend just on what day it is. That is not how a rational expectation works. Mar 19 '20 at 10:28

This is one of those situations that is not practically possible but is possible in theory. For example , the contingent payoff at $$T=20$$ is just $$S_(20)$$. But the world is such that at t=11, the stock is negatively correlated with interest rates to such an extent that the forward price $$S(11,20)$$ observed at t=11 for the stock at T=20 actually moves in the opposite direction to the spot stock price. However at t=14 the correlation is no longer present , so that $$S(14,20)$$ moves in the same direction as the stock at t=14. An unlikely situation , but possible.