Hedging an FX Vanilla Option with Forward Delta

Summary

The document explains how forward delta for a vanilla foreign-exchange option relates to a hedge using a forward contract. Under the stated Black pricing setup, differentiating the call value with respect to the outright forward rate gives a discounted sensitivity. Dividing that sensitivity by the discount factor gives the forward delta, expressed as the option’s exposure to the present value of a same-maturity forward.

For an option on one currency against another with payoff in the second currency, the answer interprets this delta as the number of forward contracts needed to hedge the option’s forward-rate sensitivity. The forward may have any agreed strike because forwards with the same maturity share the same sensitivity to the underlying outright forward rate. This addresses the hedge ratio question, but the explanation is limited to the stated vanilla option setup and does not discuss other risk sensitivities, transaction costs, or practical hedge adjustments.

Key ideas

• The option’s forward delta is its sensitivity to the outright forward rate, adjusted by the discount factor.
• Dividing the price sensitivity by the discount factor gives the forward delta.
• For the stated currency pair and payoff convention, forward delta gives the number of forwards for a sensitivity hedge.
• The forward hedge can use any pre-agreed strike when its maturity matches the option’s.

Tags

# fx vanilla option's forward delta in single currency


# fx vanilla option's forward delta in single currency












According to Black formula , a vanila fx call option's pricing is

$$C(F,\tau) = D[N(d_+)F - N(d_-)K]$$ , where $\tau$ is the time to expiry, $D =e^{-r\tau}$ the discount factor, $F=S/D$ the outright forward rate, and $d_\pm =\frac{1}{\sigma\sqrt{\tau}}\left[\ln\frac{F}{K}\pm\frac12\sigma^2\tau\right]$.

If we look at the forward delta , it's $$\frac{\partial C}{\partial F}=DN(d_+)$$

Can I interprete that, with such an option shorted, if there's a outright forward rate deal at the same maturity, with $N(d_+)$ unit of currency 1, and $-N(d_-)F$ unit of currency 2, the delta will be fully hedged? Of course the ratio $\frac{N(d_-)}{N(d_+)}F$ is not at-the-money, but never mind that.

## Answer by Antoine Conze (score 2)

https://quant.stackexchange.com/a/40497

Actually the forward delta is the option's sensitivity to the PV of the forward contract with same maturity so it is $$ \frac{1}{D}\frac{\partial C}{\partial F} = N(d_{+}) $$ For an option on CCY1CCY2 with payoff in CCY2 the forward delta gives you the number of forwards on CCY1CCY2 required to hedge the option. The forwards can be struck at any pre-agreed rate since all forwards have the same sensitivity to $F$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.