# Valuation of open FX-Forward

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?

• I feel it could be some sort of average between "closed" FX Fwd quotes. Say we enter into an "open" FX Fwd with two exchange dates: at the 3M & 6M mark. Then in the absence of arbitrage the buyer should be projected to be indifferent to exchanging money at 3M or 6M (as seen from today), since the FX Fwd quotes represent exactly this equilibrium. Hence it should be 50-50 which date he chooses. So the closed quote is the average of the two. Thoughts? – Phil-ZXX Jul 10 '18 at 16:50
• I wonder whether one could use some dominance argument similar to American options on non-div paying socks which have the same price as European options. – Ric Jul 10 '18 at 17:02
• This could depend on the interest rate differential between the two currencies @Phil-ZXX – Ric Jul 10 '18 at 17:03
• Thinking about it more, you probably need a volatility term-structure model since each "open" date (on which the buyer is allowed to exchange money at the predetermined fx "strike rate") essentially represents an option. So american/bermudan pricing can probably be applied. Perhaps longstaff schwartz or binomial trees are applicable? – Phil-ZXX Jul 10 '18 at 19:15
• @Phil-ZXX thank you for your comments. So far in the methods you mention the amount at each of the Bermudan dates does not matter ... maybe it should? I wonder whether there exists some market practice - a rule of thumb? – Ric Jul 11 '18 at 6:25

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.

• Thank you for this remark. By "slightly greater than 1.1679" you mean in order to reflect the option? If so then this would be the new content. The rest is contained in "my answer" below. Do you agree? If you bring up new aspects then I would be happy to read about them. Thanks! – Ric Sep 11 '18 at 11:00
• Yes, the excess over 1.1679 is the value of this switch option. It would be complex to value it precisely. – dm63 Sep 12 '18 at 4:24

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.