# 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.