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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?

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  • $\begingroup$ 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. $\endgroup$
    – nbbo2
    Commented Mar 19, 2020 at 10:28

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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.

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  • $\begingroup$ 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? $\endgroup$
    – Timmy
    Commented Mar 19, 2020 at 21:51
  • $\begingroup$ You wouldn’t. That’s why it’s only a theoretical possibility. $\endgroup$
    – dm63
    Commented Mar 20, 2020 at 2:29

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