# Hedging an Inverse Product

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?

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 $$S-t$$.