# About American Forward Pricing

I just want to know if there is an analytical solution about FX American forward. I recently get a solution that computes price for each τ-maturity forward contract and then take a maximum price. So, I have two questions:

1. Is this analytical solution correct?

2. If it's correct, why can't I do the same for American option? I compute Black-Scholes price for each τ-maturity European Vanilla option and then take a maximum price.They all have stopping time.

I have much confusion about this. Hope to get your help! Thank you !

How exactly do you compute the price of a forward? At initiation, the price of any forward is zero. what do you mean with they "They all have stopping time"?

The solution is incorrect and the exercise boundary can only be found be solving a PDE:

• A (synthetic) forward is just a long call + short put combination: so any price difference between the two would be arbitrage (in American exercise that is not exactly true but the actual price does not deviate too much)
• Generally, if you price options, the longer the tenor (all else equal), the more expensive. Think of it like an insurance premium; the higher the risk, the higher the cost to buy the option and the more time, the riskier it is. That applies to (car insurance), but also options.
• Assuming the underlying is FX (which is the most natural asset class for forwards), IVOL will also increase with time to maturity. Combined with the point before, your approach will always simply result in the longest maturity European option being the most expensive one.

The same as for American call or put option applies for forwards. At any time there is an exercise boundary and when spot is in side this boundary, there’s no motivation of early exercise. When spot is outside, it’s always beneficial to exercise. This boundary can only be determined by solving the PDE.

There are a few generalizations when modelling these:

• With short time to maturity (or near zero domestic rates), the American forward (also called Flexi Forward or Time-Option Forward) price is not affected much be FX vola -> in this case the break-even strike of bein long a forward is approximately the highest forward point on the forward curve throughout the exercise window (which is what you found I suspect).
• However, when the domestic rate is significantly different from zero, and with long time to maturity, vol has a significant impact on the price of the American forward -> in other words, the break-even strike deviates from the vanilla forward curve a lot.

Now, in terms of the actual difference to a normal forward, it is very small in many circumstances.

Edit

I think you misread. I wrote vol has a significant impact. In BBG you just need to type OVML FLEX which loads the Flexi Forward template. I am not aware of this being traded outside of FX.

If you intend to do this for equity, you can load DLIB (provided you are Bloomberg anywhere) and load the flexi forward template there. It will only be for FX again but it is a BLAN template, which means you can re-write the code for equity. These are a bit strange as you need to copy paste the code in a DLIB BLAN template and add the deal parameters manually.

If you are familiar with coding, you can see it models it as a daily Bermudan (since it is Monte Carlo). The regression variables (needed for the implementation) exclude interest rates here but in theory you could also implement this for rates and use HW1F-BS or HW1F-LV in BBG. Because of different discount factors, the NPV of different tenors has a different slope for each forward contract $$F-K$$. As a holder of American FWDs, you benefit from this and the payoff is no longer a straight line. Therefore, vol causes convexity adjustment and the break-even strike deviates from the vanilla forward curve a lot (as explained above).

There is a white paper called Overview of American Monte Carlo Pricing in DLIB  that explains this Longstaff-Schwartz algo used to price in DLIB. A word of caution, the standard error in this exercise will almost certainly be higher than the resulting price which means you are probably better off to simply book a OVME forward deal (unless your expiry is very long and you look at countries with large interest rates - but there aren't many left at the time of writing).

If BBG shows a vega or not is generally not sufficient proof that the contract has no vega. It is a complex PDE solver and they may just simplify things and do not provide an appropriate PDE grid for Greeks. Frequently, this is done via the Crank-Nicolson Scheme as explained for example in Computational Methods Option Pricing from Y. Achdou and O. Pironneau, Society for Industrial and Applied Mathematics, 2005. However, there exist several finite different solver schemes.

In that case however, vega can be positive (I just tested and it works as I described - higher rates and longer expiry makes vega higher).

It is a forward after all, so impact will have to be marginal compared to an option. That said, you can clearly see how the break-even strike is very different from the ATMF here (with some adjustment in rates and vol to make it more obvious). Note that OVML solves for strike here (as a FWD is generally zero cost). You can amend that if desired.

There exists no closed form solution that approximates this (that I know of). The idea is almost identical to an American option problem and no simple rule exists to price this. However, since you have BBG, why re-invent the wheel? Simply use their template (which I think is doing a solid job in this case).

The help desk can for sure provide you details (provide a white paper, show you how to load it and use the template(s), ...).

• Thanks for your reply ! I agree with what you said in part2. In fact, I really study how to price Flexi Forward in Bloomberg. As you said, I have found that volatility can't affect pricing result (vega is zero), and the result is equal to valution of a forward contract. I also get these result by binomial tree. But I haven’t tried to test long-term contract. So, is there an analytical solution for the American forward? What is it? Jun 28 at 2:21
• Thanks for your patience ! I got it that vega is not always equal to zero by constructing a contract like yours. So I decide to use binomial tree to price American forward. Anyway, thank you very much ! I almost went astray ！ Jul 2 at 1:56
• link.Hi, can you see the document? The Exercise 15.12 is about American forward，and the book (at the end) provides a analytical solution. I don't know what is wrong ！You can directly search for "American forward" to find it. @AKdemy Jul 7 at 8:32