Valuing FX Forwards with Stochastic Interest Rates

# How to price the FX forward contract under stochastic interest rates?


# How to price the FX forward contract under stochastic interest rates?












Imagine that space Z is exposed to the FX risk (i.e., currency exchange rate risk ), and we aim to provide a hedging solution for that. One choice is to consider a currency-forward contract. I wonder how I can derive the value of the forward contract when the spot domestic and foreign rates are a stochastic process, for example, following a Vasicek model. How should I discount the payoff of the forward contract in order to obtain a fair price? I think the final price should be a function of the forward rate.

If my understanding is correct, for the payoff function, we have something like this. Denote $S_T$ the spot FX rate at time T, K the strike rate at which we exchange the currencies. Then we have that

payoff= $S_T - K$

or I should consider

payoff= $S_T -F(t, T)$

where $F(t, T)$ stands for the forward exchange rate.

## Answer by byouness (score 2, accepted)

https://quant.stackexchange.com/a/65396

Let's denote by:

- $P(s,e)$: the zero coupon bond price at $s$ with maturity $e$

- $d$ and $f$ superscripts: the domestic and foreign currency (of your FX rate).

The FX forward contract with strike $K$ and delivery at $T$ pays the following payoff at $T$ (in $d$ currency): $$ Payoff(T) =(S(T) - K) $$ So, its price at $t$ is the discounted payoff under the (domestic) risk-neutral measure: $$ Price(t) = \mathbb{E} \left[ e^{-\int_t^Tr^d(u)du}(S(T) - K) | \mathcal{F}_t \right] $$ Here, it's convenient to switch to the (domestic) T-forward measure $\mathbb{Q}_T^d$ (associated with numéraire $P^d(u,T)$: $$ Price(t) = P^d(t, T) \mathbb{E}^T \left[ S(T) - K | \mathcal{F}_t \right] $$

Now, no product inside the expectation, only the FX is left inside. We can write: $$ S(T) = S(T)\frac{P^f(T, T)}{P^d(T, T)} $$

The numerator is a tradeable asset. So, expressed in the numéraire $P^d(u,T)$ it is a $\mathbb{Q}_T^d$-martingale, and we get: $$ Price(t) = P^d (t, T) \left( S(t)\frac{P^f(t, T)}{P^d(t, T)} - K \right)\\ $$

In financial terms, this term is what you call the FX forward rate: $$ F(t, T) = S(t)\frac{P^f(t, T)}{P^d(t, T)} $$ and the price of the forward contract with strike $K$ is the discount difference between this FX forward and the strike: $$ Price(t) = P^d(t, T) \left(F(t, T) - K \right) $$

## Answer by AKdemy (score 2)

https://quant.stackexchange.com/a/65393

Using CCY1CCY2 (e.g. EURUSD quoted in units of domestic currency per unit of foreign currency. Here EUR is foreign, USD domestic). To get FWD rate at initiation: $$f(s,ccy1,ccy2,t) = s*exp^{(r_{ccy2}-r_{ccy1})*t}$$

- Does not matter if r is stochastic or not

- notional in ccy1, and value/premium in ccy2 (all else is a transformation as shown here)

At initiation of the FWD, you have zero value at the prevailing forward rate ($f(s,ccy1,ccy2,t) = K$).

Afterwards, for Mark to Market, you use that rate and compare it to the current FWD rate in the market (or what you model, but unless you are a market maker, I am not sure what the benefit of this will be). In other words, the forward value observed at t of a T maturity FWD contract is simply the PV of the difference in foreign exchange prices. $$N_{EUR}*(F_t -K)*𝑒xp^{−𝑟_{𝑐𝑐𝑦2}}$$

If notional is not in CCY1 (EUR), you multiply by K to get the equivalent CCY2 (USD) notional.