How Interest Rate and Correlation Changes Can Reverse a Delta Hedge
Summary
The question asks whether a contingent claim’s stock hedge can be negative for every stock price at one time and positive for every stock price later. The accepted explanation says this is theoretically possible, even if unlikely in practice. Its example uses a claim paying the future stock value: changing relationships between the stock and interest rates can make the stock’s forward value move opposite to spot at one time, then move in the same direction later. The hedge’s sign can therefore change across dates without depending on the current stock price.
The explanation is conceptual rather than a worked derivation. It does not provide a model, parameter conditions, or evidence that such a pattern occurs in market data, and the proposed relationship between forward value and hedge sign is left largely intuitive. The useful lesson is that a hedge’s sign need not remain fixed over time when rates and the underlying asset interact; intuition based only on buying low or selling high does not establish whether a hedge formula is possible.
Key ideas
- A stock hedge can theoretically be negative for all spot values at one time and positive for all spot values later.
- Changes in the relationship between stock prices and interest rates can affect the direction of forward-value movements.
- The example claim pays the stock’s value at maturity.
- The answer describes a theoretical possibility, not a common market pattern or a fully derived pricing result.
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Full text
# Can a delta hedge be negative for all values at one time, and positive for all values at another time? # 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? ## Answer by dm63 (score 1, accepted) https://quant.stackexchange.com/a/51706 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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