Relative truths: the world of FX valuation

Valuing an FX portfolio can be tricky, and nigh on impossible without an intuition for how the market thinks

Valuing an FX portfolio is a bit like shooting a moving target. Most assets are priced in one obvious currency. Equities, bonds and oil futures, for example, are all priced in one established currency which depends on the market. FX instruments don't have an obvious currency unit. Especially when it comes to crosses that don't involve USD like NOK/SEK.

That means people tend to rely on market conventions. But they can often be confusing and lacking obvious rationale, especially as conventions evolve over time. Currency exchange itself is ancient (the Byzantine government held a monopoly on it in the 4th century AD), but the conventions used today are far more recent.

The numeraire depends on the purpose

In addition, the numeraire used tends to depend on the purpose of the valuation. Speculative trading firms tend to have only one reporting currency they care about internally, usually USD. But for an FX desk that needs to hedge and control their exposure, the numeraire often needs to be something else.

For example again, NOK/SEK. If you buy spot NOK/SEK, i.e. receiving NOK and handing over SEK in exchange, knowing the value of your newly acquired NOK in USD is not particularly useful. Especially if you're trading spot NOK/SEK to hedge exposure from options on NOK/SEK, in which case the thing you are most likely interested in is how much spot NOK/SEK to trade to offset the directional risk ("delta") of the option. In that scenario, you want to know how much spot to trade to offset your option delta, so that you can isolate the things that affect the option's value from directional dependency on spot rates.

Value, premium and delta

Options models such as Garman-Kohlhagen give a value in the right-hand currency, known in FX as the "domestic" currency: USD in EUR/USD, for example. Options are only around 7% of FX volume, according to the BIS, so why start here? Because they're where the question of currency is hardest to avoid. A spot trade is just one currency exchanged for another, but an option has a value, a premium and a delta, and each comes with its own convention. Understand those, and you can see why FX desks measure things the way they do.

The models give us the mathematical picture, which doesn't always reflect how the market thinks. The market often thinks in terms of premium payments. Options involve premium payments, and the currency in which premiums are paid has its own convention, often being the left-hand ("foreign currency"). USD/JPY option premiums are paid in USD, for example.

Meanwhile, delta tends to be measured in the foreign currency, for reasons beyond the scope of this article¹.

Hedging with spot

So if you're trying to trade spot to offset your option delta, which as we know is typically in the foreign currency, measuring spot exposure (again "delta", but for spot this time) in USD is not particularly useful. Hence you want to measure your spot delta in the foreign currency too. Luckily, by definition, this is just the amount of foreign currency you're buying (for a discussion on this see Reiswich and Wystup (2010), sections 1.2–1.3). $N$ units of foreign currency are worth $NS$ in domestic, so:

V = NS \quad\Rightarrow\quad \frac{dV}{dS} = N

One wrinkle, however: if the premium is paid in the foreign currency, the amount to hedge is adjusted by the premium, a convention known as premium-adjusted delta².

By no means straightforward

By no means straightforward is it? This is not even the full picture, as an FX desk will almost certainly be trading more than one currency pair. And that means standardising measurements across those pairs. Again, that's beyond the scope of discussion here³.

But by remembering the foreign/domestic terminology and the standard option valuation currency convention for the two, you have a way to understand why people tend to follow particular numeraires for FX trading.

No single right currency

So there's no single right currency for an FX portfolio. It depends on the question you're asking:

What does the model say this option is worth? Domestic.
How much spot do I trade to hedge it? Foreign.
What is the whole book worth? Your reporting currency, often USD.

Next time someone quotes you an FX number, the first thing to ask isn't how big it is, but what currency it's in and why.

Standard FX option pricing formulae derive the value of an option in the domestic currency. The standard definition of delta (the one that an FX desk might hedge with), is that it is the rate of the change of the value of the option with respect to a unit change in spot rate. Hence you have:
$\Delta = \frac{dV}{dS}$
where $V$ is value in the domestic currency and $S$ is the spot rate. The spot rate is the amount of domestic currency you can buy for one unit of foreign. Hence, using DOM for domestic, and FOR for foreign, its units are DOM/FOR. With $V$ in DOM, dimensional analysis tells us that $dV/dS$ has units:
$\frac{\text{DOM}}{\text{DOM}/\text{FOR}} = \text{FOR}$
Hence the delta of an option is typically measured in the foreign currency. ↩
The premium-adjusted delta is:
$\frac{dV}{dS} - \frac{V}{S}$
In Reiswich and Wystup's example, a 60% delta on a EUR 1m option would mean buying EUR 600,000, but with a EUR 73,669 premium received, the hedge is EUR 526,331. ↩
The way this is typically done is to keep deltas in foreign currency and group instruments by portfolio so that delta across the portfolio has one unit and can be aggregated. That allows for a desk to control their risk. And then for obtaining a valuation across portfolios, the desk can convert valuations to dollars at currency to USD spot rates so that a single USD number shows the value of all portfolios together. ↩