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We have two different products that follow the same price $S(t)$ for all time $t$. The payout for product one is given by $w_1(\frac{100}{S(t)} - \frac{100}{S(t + \Delta t)})$ and the payout for product two is $w_2(S(t + \Delta t) - S(t))$. Where $w_1$ and $w_2$ are the quantities to buy or sell of product one and two respectively.

At time 0 given we buy $w_1$ of product one, to hedge against a change in price we should sell $w_2 = \frac{100w_1}{S(0)}$ units of product two? The only reason I am unsure is that the inverse product is in the inverse units of product two, so does that change how I should hedge product one with product two?

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Discretized Time?

If we take $t=0$ and $t+\delta t=\tau$, you have payouts of: $$ \begin{align} \text{Payoff 1} &= \omega_1 \left(K_1 - \frac{100}{S_\tau}\right) \quad \text{and} \\ \text{Payoff 2} &= \omega_2 (S_\tau - K_2). \end{align} $$

Payoff 1 clearly increases with $S_\tau$ as does Payoff 2. (Imagine the underlier at 100 and then it goes up to 101: Payoff 1 would be $\frac{100}{100} - \frac{100}{101}>0$.) Therefore, to hedge Payoff 1 you would sell some of the contract offering Payoff 2.

Continuous Time?

However... I suspect the $t$ is, as you say, a time index which continues to increase. Then, Payoff 2 is clearly a "delta-1" investment. However, your discretized statement of Payoff 1 is very odd. So, expressing everything in continuous time (no more $t+\Delta t$), I suspect you mean that $$ \text{Payoff 1} = \omega_1 \frac{-100}{S_t}. $$ In that case, we can look at the derivative of the intrinsic value: $$ \frac{\partial \text{Payoff 1}}{\partial S_t} = \omega_1\frac{100}{S_t^2}. $$ In this case as well, you would sell some of the contract offering Payoff 2 to hedge holding the contract offering Payoff 1.

Log-Payout?

Finally, there is a slight chance this discretization is coming from considering a contract paying off $\log(\text{underlier})$, since the derivative of $\log(S_t)$ is $1/S_t$). If that is the case and your discretized payoffs were just a sort of Taylor Series approximation, then Payoff 1 will be related to the volatility of $S-t$.

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