Deriving FX Forward Value from the Forward Rate and Discounting

# Present value of an FX Forward contract at each simulation and time point node of a Monte Carlo simulation


# Present value of an FX Forward contract at each simulation and time point node of a Monte Carlo simulation












Recently I started dealing with the xVA and the associated EPE and ENE concepts.

In a numerical example of an FX Forward, after simulating the underlying FX spot $S_t$ (units of domestic per unit of foreign) using GBM ($S_{t} = S_{t-1}e^{(\mu-\frac{\sigma^{2}}{2})dt +\sigma\sqrt{dt}Z_{t-1}}$) the present value of the FX Forward contract for each time point and simulation node is calculated by the following equation:

$V_{t} = S_{t}e^{-r_{for}(T-t)} - Ke^{-r_{dom}(T-t)}$ (1)

In the above equation $S_{t}$ is the simulated underlying FX sport, $r_{for}$ is the interest rate of the foreign currency, $r_{dom}$ is the interest rate of the domestic currency, $T$ maturity, $t$ the time point and finally, $K$ the strike level which was determined at the inception of the contract as the forward exchange rate at maturity (i.e., no-arbitrage opportunities): $K = F_{0,T} = S_{0}e^{(r_{dom} - r_{for})T} $

How equation (1) is derived (I presume that the interest rate parity should be used)?

## Answer by Whitebeard13 (score 1, accepted)

https://quant.stackexchange.com/a/77862

An answer has already been provided in this discussion: How to price the FX forward contract under stochastic interest rates?

Here is a summary of the (backward) derivation of equation (1):

$$ V_{t} = S_{t}e^{-r_{for}(T-t)}-Ke^{-r_{dom}(T-t)} \Leftrightarrow$$ $$ V_{t} = \underbrace{\underbrace{(\underbrace{S_{t}e^{(r_{dom}-r_{for})(T-t)}}_{F_{t,T}\text{: FX forward rate for } T \text{ at } t}-K)}_{\text{FX forward payoff given } F_{t,T}}e^{-r_{dom}(T-t)}}_{\text{FX forward payoff discounted at } t}\Leftrightarrow$$