# 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

## 1 Answer

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.

• This may be a silly question, but given the information we would have when devising the hedging formula at t=0, how would we know how the stock is correlated with the interest rate at later times, and be able to detect changes to this correlation at explicit future times? Mar 19 '20 at 21:51
• You wouldn’t. That’s why it’s only a theoretical possibility.
– dm63
Mar 20 '20 at 2:29