# MtM of FX Forward

I had a look at pnl calculation of FX forward but it didn't quite match my question.

Say $X_{t,\tau}$ is the USDJPY FX Forward Rate as seen at time $t$ for expiry $t+\tau$. So $X_{t}^{spot} := X_{t,0}$ can be understood as "Spot" at time $t$.

• At time $t=0$ (today), I enter into a 12M FX Forward on USDJPY at the fair strike of $$K=X_{0,12M}=110$$ That is, in 1 year I receive 1 USD and pay 110 JPY. No money changes hands, because the trade's PV is zero at inception.

• At time $t=3M$ (i.e. 3 month later), FX rates have moved and I want to know what my trade's PV is. Which one is correct:

$$\text{PV in USD} = \left(1 - \frac{K}{X_{3M}^{spot}}\right) \cdot D_{9M}^{USD}$$ $$\text{or}$$ $$\text{PV in USD} = \left(1 - \frac{K}{X_{3M,9M}}\right) \cdot D_{9M}^{USD}$$

Two questions:

1. Do I use Spot or the 9M-Fwd to compute my PV/PnL? The 9M-Fwd seems more correct to me, because that's the rate I'd use to close out my position.
2. Is my use of the USD discount factor $D_{9M}^{USD}$ correct? I am asking because it seems like I have no direct exposure to JPY rates when calculating my PV this way.

## 1 Answer

You are correct on both questions. 1 you answered yourself. It is the correct rate to close out the trade. 2 you use a dollar discount rate because you are discounting dollars. The (1-K/X) term represents one dollar from the first trade and K/X dollars from the close out trade.

• Thanks, so does an FX Forward naturally only have one IR Rho (to USD rates in this example)? Commented Jun 2, 2018 at 12:51
• No. If you want to describe your risk in terms of spot fx and rates only, there should also be some Rho coming from the determination of the forward fx rate, both yen Rho and dollar Rho.
– dm63
Commented Jun 2, 2018 at 12:57
• But the way I am calculating my PV right now does not give me direct exposure to JPY rates, correct? That is, I only get Rho to JPY rates when I "decompose" the Fwd value into Spot+USD Rates+JPY Rates? Commented Jun 3, 2018 at 10:22