How to effectively hedge a Fixed-Term deal in a foreign currency?

Question

Assume my firm is based in USD and agrees with some counterparty to buy, at time $T$, some quantity $Q$ of asset $A$ for a fixed price $K$.

Assume also that $A$ prices and $K$ are denominated in EUR.

Assume also that we have forward prices:

• For $A$, denotated by $F^A(t,T)$
• For the FX rate, denotated by $F^{FX}(t,T)$, expressed in USD per EUR.

Now my mark-to-market at time $t$ for this agreement is:

$$\text{MTM}_t = DF(t,T)\cdot Q\cdot( F^A(t,T) - K ) \cdot F^{FX}(t,T)$$

I would like to hedge my position for both FX and market effects, assuming I can get perfectly matching contracts for asset and FX in order to hedge.

I would sell $Q$ "lots" of to $F^A(t,T)$ hedge the dynamics of the hedge.

However, I'm unclear how much to hedge for the FX. Indeed, my FX exposure at time $t$ is $F^A(t,T) - K$, which can vary very much from $F^A(t-1,T) - K$ and hence results in an hedge completely off. So, what should be the amount to hedge in FX? I thought about $\mathbb{E}[ F^A(t,T) - K | \mathcal{F_{t-1}}]$ but my backtest doesn't give me good results at all.

Is there a common way of hedging this king of "spread" exposure?

Answer 1

You have already agreed to pay $QK$ EUR at $T$ to receive $Q$ units of A. If you sell $Q$ lots of $F^A(t,T)$ then you will receive $Q F^A(t,T)$ EUR and deliver $Q$ units of A. The combined flow is now just in EUR: at $T$ you receive a net of $Q(F^A(t,T)-K)$ EUR. You can hedge that by selling $Q(F^A(t,T)-K)$ of $F^{FX}(t,T).$ Then with both hedges, the net flow is just you receive $Q(F^A(t,T)-K)F^{FX}(t,T)$ USD at $T$.

You are confusing yourself by trying to calculate the FX hedge without assuming the asset hedge is in place. Indeed, if you didn't hedge the asset move but did hedge the FX move, the FX hedge has to be rebalanced every time the asset price moves, which is what you are seeing.