Valuation of open FX-Forward

Question

So called closed FX-Forwards are well known forward contracts where some amount of foreign currency is bought at a specified date in the future for a price fixed "today". Such contracts can be valuated using the well known cost-of-carry formula.

Recently, I learned about open FX-forward contracts. In this kind of contract the holder has the flexibility to make as many drawdowns as he wants during a specified period as long as the full amount is paid by maturity see e.g. this page.

What is market practice to value such open FX-forward contracts?

Answer 1

The flexible forward contract is very much like an American option: at each exercise date, you have the choice to receive the payoff $(S-K)$ or not. The difference with a regular option is that you must choose a date.

In effect, this is a classical optimal stochastic control problem and may be solved using exactly the same techniques as for an American (or a Bermudan) option: typically a finite difference method applied to the Black-Scholes PDE with the linear complementary constraint $$f(t_i, s_j) \geq s_j - K$$, where $t_i$ is an exercise date and $s_j$ the asset price in the FDM discretization grid.

In particular the price will depend on the asset volatility, and the contract will have a non-zero vega.

See https://www.worldscientific.com/doi/abs/10.1142/S2424786316500109, https://www.globalcapital.com/article/k6b8msb96708/american-currency-forwards.

Answer 2

To answer your answer: Suppose you are the holder of the open contract. You hedge it by executing a vanilla forward at 1.1679 for date 92. You now have an arbitrage, for if the fx forward for one of the dates 88 to 91 becomes higher than that for date 92, you can switch the hedge to that other date, This means that the true price of your open contract must be slightly greater than 1.1679. However, the switch in this case is unlikely, because it would only occur if euro rates exceed usd rates. It is an option on the rate differential. If you created an open FX forward on a currency pair where rates are very similar , the effect would be greater.

Answer 3

I am trying to answer my own question to make discussion possible.

Say we have an open FWD with period $[T_1,T_2]$ in which we can settle it. The strike price $K$ is fixed today.

As a example for EUR USD we have a spot of 1.16 (USD per EUR) and let us assume that the strike price for the above forward is $K = 1.1677$ (we have much higher USD rates than EUR leading to this higher forward price).

Then, on any given day $t$ I can compare this $K$ to the forward prices of forwards that stettle on all the days in the interval $[T_1,T_2]$.

The fair price is $$ F_{T_i} = S_t \exp ( (r_d(T_i)-r_f (T_i))\cdot (T_i-t)/365 ) $$ and a rational agent will settle when the gains are highest thus at $$ T^* = \arg max_{t \in [T_1,T_2]} \{ F_{T_i}-K \}. $$ Thus the price $K$ has to equal this one $F_{T^*}$. If the ir-differential does not changes too much during $[T_1,T_2]$ then I assume that this $T^*$ will be the first of the last day of the period depending on the sign of the differential.

Continuing the example we can calculate the forward prices (crudely) for some $T_i$:

• $T_i = 88$ then $F_{88} \approx 1.1675$ and the gain is $-0.00017$
• $T_i = 90$ then $F_{90} \approx 1.1677$ gain is zero.
• $T_i = 92$ then $F_{92} \approx 1.1679$ gain is approx $0.00017$

Thus if the period is $\{88, 89, 90, 91, 92\}$ the price of the open forward should be $K=1.1679$, which is the forward price for settlement at the last day of the period as the interest rate differential between USD and EUR (USD-EUR ir) is positive.