Summary of Foreign Exchange Option Rules

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Option Premium Quotation Rules

Table 1.2 Standard market quotation types for option values. In the example, we take FOR$= $EUR$, DOM$= $USD, S_0= 1.2000, r_d= 3.0\%, r_f= 2.5\%, \sigma= 10\%, K= 1.2500, T=1$ year, $\phi=+1$ (call option), with a notional amount of 1,000,000 EUR = 1,250,000 USD. For pips, the quote of 291.48 USD per EUR is sometimes expressed as 2.9148% USD per 1 EUR. Similarly, the quote of 194.32 EUR per USD can also be expressed as 1.9432% EUR per 1 USD.

Name	Symbol	Unit	Example
Domestic Cash	d	DOM	29,148 USD
Foreign Cash	f	FOR	24,290 EUR
Domestic Percentage	%d	DOM per unit DOM	2.3318% USD
Foreign Percentage	%f	FOR per unit FOR	2.4290% EUR
Domestic Pips	d pips	DOM per unit FOR	291.48 USD pips per EUR
Foreign Pips	f pips	FOR per unit DOM	194.32 EUR pips per USD

The Black-Scholes formula uses d pips for quotation. Other quotation methods can be calculated using the following formula:

$$\text{d pips}\stackrel{\times\frac{1}{S_{0}}}{\longrightarrow}\%f\stackrel{\times\frac{S_{0}}{K}}{\longrightarrow}\%d\stackrel{\times\frac{1}{S_{0}}}{\longrightarrow}\textbf{f pips}\stackrel{\times S_{0}K}{\longrightarrow}\text{d pips}$$

Delta and Premium Convention

If the option premium is paid in USD, then $\phi e^{-r_f\tau}\mathcal{N}(\phi d_+)$ represents Delta.
If the premium is paid in EUR, then $\phi e^{-r_d\tau}\mathcal{N}(\phi d_+)$ represents Delta. The spot Delta for European options, assuming the premium is paid in DOM (i.e., USD), is well-known. Now referred to as raw spot delta $\delta_{raw}$. It can be quoted in either of the two currencies involved. Their relationship is:

$$\delta_{raw}^{reverse}=-\delta_{raw}\frac{S}{K}.$$

Delta is used to buy or sell the corresponding amount of spot to hedge the option to the first order. The raw spot delta multiplied by the FOR notional amount represents the amount of FOR currency the trader needs to buy to hedge a short option.

How do we get the reverse delta? It strictly follows the symmetry of currency options. A FOR call is a DOM put. Therefore, buying FOR in delta hedging is equivalent to selling DOM multiplied by the spot S. The negative sign reflects the change from buying to selling. This explains the negative sign and the spot factor. The right to buy 1 FOR (and pay K units of DOM) is equivalent to the right to sell K DOM and receive 1 DOM. Therefore, treating a FOR call as a DOM put and applying its delta hedge to one unit of DOM (instead of K units of DOM) requires dividing by K.

For consistency, the option premium needs to be included in delta hedging because the premium in foreign currency has already hedged part of the option's delta risk. In the equity option environment, such issues never arise because equity options are always paid in cash, not in stock. In FX, both currencies are cash, and it is entirely reasonable to pay currency options in DOM or FOR currency. For clarity, let's consider EUR-USD. In any financial market model, $\upsilon$(x) represents the dollar value or premium of an option with a face value of 1 EUR when the spot is x, and the raw delta $\upsilon_\mathrm{y}$ represents the amount of EUR to be purchased to hedge a short option.

DeltaString to Strike Conversion Rules

DeltaString	Strike Calculation Rule
ATM, ATMF	Equal to the forward price F
ATMD	strike = forward * exp(-0.5 * volatility * volatility * timeToExpiry)
ATMS	Equal to the spot price S
ITM %1	1% in-the-money value of the option, if put, then Sx(1+1%); if call, then Sx(1-1%)
OTM 1%	1% out-of-the-money value of the option, if put, then Sx(1-1%); if call, then Sx(1+1%)
ITMF %1	1% in-the-money value of the option, if put, then Fx(1+1%); if call, then Sx(1-1%)
OTMF 1%	1% out-of-the-money value of the option, if put, then Fx(1-1%); if call, then Sx(1+1%)
25D	Calculate the strike corresponding to 25Delta separately for Call and Put

Note: All calculations need to distinguish between bid, mid, and ask directions.

Market Conventions for Option Premium Quotation

In the FX market, there are multiple ways to express the option premium when a trade is executed. We refer to Wystup[2006], Castagna[2010], and Clark[forthcoming], and use the example data provided in the previous section to explain common methods. The most common premium convention is the standard Black-Scholes quotation described earlier. The corresponding premium is called the foreign/domestic price, denoted by $\nu^{f-d}$. Another method is to call it the domestic pip price after multiplying by a currency-related factor (e.g., 10,000 for EUR-USD). In our example, we have $\nu^\mathrm{EUR-USD}=0.1024$ USD. If a notional amount of N$_f$ foreign currency units is specified, the actual amount paid/received will be $N_f\nu^{f-d}$. In the previous example, this amount is 102,400 USD. The corresponding domestic pip price is $\nu^\mathrm{EUR-USD}\times10,000=1024$ USD pips.

For the exchange of one unit of foreign currency for K units of domestic currency in the FX market, it can be analyzed from the perspective of a foreign investor. In our example, this is the party paying the foreign currency unit and receiving the domestic currency unit. For the foreign investor, their premium $\nu^{f-d}/S_t$ will not be recorded as $\nu^{d-f}$, due to the embedded inconsistency, which will be explained later. The standard case is that the domestic investor receives one unit of foreign currency and pays K units of domestic currency. This indicates that the foreign investor receives K units of another currency, not one unit. To maintain consistency, the symbol $\nu^{\kappa-f}$ is used for an option denominated in currency $\gamma$ with a notional amount of 1 in currency X. To obtain $\nu^{d-f}$, we need to adjust the amount exchanged so that the foreign investor receives one unit of domestic currency instead of K units. This can be achieved by paying $1/K$ units of foreign currency instead of one unit, which is equivalent to adjusting the foreign notional amount to $(1/K)$ instead of one unit. Since the domestic party receives $(1/K)$ units of foreign currency, the corresponding payment in domestic currency is $(1/K)\times K=1$. This shows that, after adjustment, the foreign investor receives one unit of domestic currency as expected. Then, $\nu^{d-f}$ can be represented by resetting the foreign notional amount to $N_f=1/K$ and converting the premium to foreign currency. This results in

This price is called the domestic/foreign price (or foreign pip price after multiplying by the appropriate factor); in our example, we calculate

$\nu^\mathrm{USD-EUR}=0.1024/(1.3500\times1.3900)=0.054570$

This quotation is specified in domestic currency amount, not foreign currency amount. The total premium amount in foreign currency units will be $\nu^{d-f}N_\mathrm{d}$, where $N_d$ is the notional amount in domestic currency units. For example, a notional amount of 1,000,000 USD can be specified, making the premium in foreign currency 54,570 EUR.

Finally, the adjustment of the notional amount can be ignored, and the price $\nu^{f-d}$ can be expressed in foreign currency units. The price $\nu^{f-d}/S_t$ is called the foreign percentage price. We define

$$\nu^{f\%}=\frac{\nu^{f-d}}{S_t}$$

In our example, the foreign percentage price is 0.073669 EUR. This is how some standard vanilla options and barrier options are quoted in the interbank market (see Castagna[2010]). We will explain why this name is used. For an option where the holder receives 1 unit of foreign currency, the premium $\mathcal{V}^{f\%}$ is expressed in foreign currency units. This indicates that $\mathcal{V}^{f\%}$ is a percentage value relative to the foreign notional amount: 0.073669 EUR can be interpreted as the option premium in foreign currency units equivalent to 7.3669% of the foreign notional amount. If the notional amount is changed to any value, this interpretation still holds. Once this value is known, the factor can be multiplied by any foreign notional amount to obtain the total premium in foreign currency.

Alternatively, $\nu^{d-f}$ can be used to achieve the same effect. This premium needs to be multiplied by $S_t$ to calculate the price of the option in domestic currency. Therefore, the domestic percentage price can be specified as $\nu^{d-f}\times S_t$. We define

$$\nu^{d/o}=\nu^{d-f}S_t=\frac{\nu^{f-d}}K$$

where we use equation (4) to calculate the last equation. In our example, this yields 0.075852 USD. The domestic percentage quotation is the standard quotation method for exotic path-dependent options (single touch, double no-touch) in FX options, where the payoff is denominated in domestic currency units (see Castagna[2010]).

Currency Pair	Option Premium Rule
USD-CNY
USD-CNH
EUR-CNY
GBP-CNY
JPY-CNY
HKD-CNY
EUR-USD	$\nu^{EUR-USD}$Pips
EUR-CAD	$\nu^{EUR-CAD}$Pips
EUR-CHF	$\nu^{EUR\%}$Pips
EUR-GBP	$\nu^{EUR-GBP}$Pips
EUR-JPY	$\nu^{EUR\%}$Pips
EUR-ZAR	$\nu^{EUR\%}$Pips
GBP-CHF	$\nu^{GBP\%}$Pips
GBP-JPY	$\nu^{GBP\%}$Pips
GBP-USD	$\nu^{GBP-USD}$Pips
USD-CAD	$\nu^{USD\%}$Pips
USD-CHF	$\nu^{USD\%}$Pips
USD-JPY	$\nu^{USD\%}$Pips
USD-ZAR	$\nu^{USD\%}$Pips

Source: Catagna[2010].

The table above provides the premium conventions for selected currency pairs. For example, USD-JPY spot is traditionally quoted with JPY as the domestic currency because a number like 87.00 USD-JPY is easier to quote than 0.01149 JPY-USD, but the premium is usually expressed in USD. The market standard is to quote the premium in the currency of the more frequently traded currency (Clark[forthcoming]). Almost all currency pairs involving USD will be expressed in USD as the premium currency. Similarly, contracts involving EUR but not USD will be expressed in EUR. The basic hierarchy of premium currencies is as follows (Clark): $\mathrm{USD\succ EUR\succ GBP\succ AUD\succ NZD\succ CAD\succ CHF}$
$\mathrm{NOK, SEK, DKK \succ CZK, PLN, TRY, MXN\succ JPY\succ \ldots}$
To summarize, currency option trading always has two sides. Depending on the details of the trade, investors may be interested in standardized notional amounts, whether in foreign or domestic currency. Domestic investors calculate the premium in domestic currency and may be interested in foreign or domestic notional amounts. The corresponding premium conventions are $\nu^{fa}$ and $\mathcal{V}^{f\%}$. Foreign investors express the premium in foreign currency and may also be interested in domestic or foreign notional amounts. The corresponding premium conventions are $\nu^{d-f}$ and $\mathcal{V}^{f\%}$.