Chapter 10
Vanilla FX Derivatives Miscellaneous Topics
Several FX derivatives pricing technicalities have thus far been brushed under the carpet. In this chapter, present and future valuing, tenor expiry date calculations, and premium conversions are examined.
Present Valuing and Future Valuing
It is well understood that $10 received today is worth more than $10 received in the future due to the time value of money. The core of the argument is that money received now has interest-earning potential if it is placed on deposit (of course, assuming interest rates are positive).
- Present valuing involves bringing a cash value in the future back to its equivalent value at the present day. In derivatives, option values are often calculated at maturity and must then be present valued.
- Future valuing is the opposite operation: taking a cash value today and pushing it to some future date.
The calculation for present and future valuing uses a discount factors 
to a given maturity in the currency in which the cash value in denominated. Discount factors are generally less than 1 and:
Discount factors are calculated from interest rates but the exact calculation depends on how interest compounds on the deposited cash balance. If interest is all paid in one payment at the deposit maturity, the interest rates are called zero interest rates and the discount factor is calculated using:
where 
is the time to deposit maturity measured in years and 
is a zero interest rate.
If interest is compounded on an annual basis, the discount factor is calculated using:
where 
is an annually compounded interest rate. The market instrument called deposits (also called “depos”) follow this convention.
In general, given a regular compounding frequency of m times a year, the discount factor is calculated using:
where 
is an interest rate in the given compounding rate.
As m gets larger, in the limit the rate becomes a continuous compounded interest rate: the rate used within the Black-Scholes mathematical framework. In this case:
where 
is a continuously compounded interest rate.
In practice, interest rate curve building and calculations are far more complicated than this. Different interest rate curves are used and they must be bootstrapped together using a variety of quantitative techniques. These methods are important for interest rate traders but they are not day-to-day concerns for most FX derivatives traders.
Market Tenor Calculations
A well-defined logic exists for calculating expiry and delivery dates for market tenors within the FX derivatives market. Four dates are defined:
- Horizon: the date on which the trade originates (i.e., today)
- Spot date: the date on which the initial transfer of funds (the premium) often takes place and the date on which any spot hedge settles.
- Expiry date: the date on which the contract expires and any final transfer of funds is known.
- Delivery date: the date on which the final transfer of funds generated from the contract usually takes place and the date on which forward hedges usually settle.
If the final transfer of funds takes place after the natural delivery date, the option is described as late delivery (see Chapter 27 for examples of different late delivery vanilla options). All these dates can only ever be weekdays since the FX market is not open over the weekend.
These four dates are summarized on the timeline shown in Exhibit 10.1. This timeline may be different for overnight options since the expiry date can be before the spot date.
Exhibit 10.1 Timeline of the four key dates within market tenor calculations
The term “business day” is used to describe a day that is not on a weekend and is also not a holiday in either currency within the relevant currency pair. A stylized version of these tenor calculations is implemented in Practical D.
Calculating Spot Dates
The spot date is calculated from the horizon (T). There are two possible cases:
- If a currency pair has T+1 settlement (e.g., USD/CAD), the spot date is one day after the horizon. In this case, T+1 must be a business day and also not a U.S. holiday. If an unacceptable day is encountered, move one further day into the future and test again.
- If a currency pair has T+2 settlement, the spot date is two days after the horizon. The calculation of T+2 must be done by considering each currency within the pair separately. For USD there must be one clear working day between the horizon and the spot date and for all non-USD currencies there must be two clear working days between the horizon and the spot date.
In addition, for most currencies, no money can clear (settle) on U.S. holidays, meaning that the spot date cannot occur on a U.S. holiday even if USD is not a currency within the currency pair.
Calculating Expiry and Delivery Dates
Market tenors for FX option contracts are quoted either as “overnight” or in terms of a number of days, weeks, months, or years. In general, the expiry date can be any weekday even if it is a holiday in one or both of the currencies, except January 1. There are differing conventions for calculating expiry and delivery dates depending on the tenor.
Overnight
For overnight trades, the expiry date is the next weekday after the horizon. The delivery date is then calculated from the expiry date in the same way as the spot date is calculated from the horizon.
Days and Weeks
For a trade with a v days tenor, the expiry date is the day v calendar days after the horizon (unless this expiry date is a weekend or January 1, in which case the tenor is invalid) and for a trade with an x weeks tenor, the expiry date is 7x calendar days after the horizon (unless this expiry date is a weekend or January 1, in which case the tenor is invalid). The delivery date is then calculated from the expiry date in the same way as the spot date is calculated from the horizon.
Months
For a trade with a y months tenor, the expiry date is found by first calculating the spot date, and then moving forward y months from the spot date to the delivery date. If the delivery date is a non-business day or a U.S. holiday, move forward until an acceptable delivery date is found. Finally, the expiry date is calculated from the delivery date using an “inverse spot date” operation (i.e., find the expiry date for which the delivery date would be its spot date).
Years
For a trade with a z years tenor, the expiry date is found by first calculating the spot date, and then moving forward z years from the spot date to the delivery date. If the delivery date is a non-business day or a U.S. holiday, move forward until an acceptable delivery date is found. Finally, the expiry date is calculated from the delivery date using an “inverse spot date” operation (i.e., find the expiry date for which the delivery date would be its spot date).
Special Cases
There are two special cases involving trades that take place around the end of the month and have a tenor defined in month or year multiples. Defining the target month to lie x months forward from the spot date month if the tenor is x months, for example, if the spot date month is February and the tenor is 3M (three months), the target month is May.
- If the spot date falls on the last business day of the month in the currency pair, then the delivery date is defined by convention to be the last business day of the target month. For example, assuming all days are business days: If the spot date is April 30, a one-month time to expiry will make the delivery date May 31. This is described as trading “end-end.”
- If the spot date falls before the end of the month but the resultant delivery date is beyond the end of the target month, then the delivery date is defined by convention to be the last business day of the target month. For example, assuming all days are business days: If the spot date is January 30, a 1 month time to expiry implies a delivery date of February 30; however, this doesn't exist and the expiry date becomes February 28 (in a non-leap year, obviously).
Also, expiry date and delivery date calculations sometimes adjust in different time zones. For example, when trading USD/JPY for Tokyo cut in Asia time, the expiry date may be adjusted to avoid JPY holidays, but once London comes in and starts trading USD/JPY for NY cut, the expiry date will change. Therefore, expiry dates for market tenors can change not only from day to day but within the trading day.
Finally, a quick word for anyone wondering why this section looks similar to the Wikipedia entry on this subject: Documenting this process was one of my first jobs on the trading desk many years ago. Some kind soul obviously took the document and put it up on Wikipedia.
Option Premium Conversions
FX option premiums can be quoted in four ways: CCY1%, CCY2 pips (meaning a number of CCY2 for one CCY1, as spot is quoted), CCY2%, and CCY1 pips (the number of CCY1 for one CCY2).
- % prices have the same notional and premium currency. Example: Notional USD10m, premium 0.40 USD% implies a cash premium of USD40k.
- Pips prices have different notional and premium currencies. Example: Notional USD10m, premium 52 JPY pips implies a cash premium of JPY520m.
Prices on vanilla FX derivatives are usually quoted in CCY1% or CCY2 pips terms, depending on the market convention in the currency pair. For example, in EUR/JPY the notional will usually be quoted in EUR and the premium will be quoted in EUR terms (i.e., CCY1%) while in EUR/USD the notional will usually be quoted in EUR but the premium will be quoted in USD terms (i.e., CCY2 pips). Pairs where the premium is paid in CCY1 are called left-hand side (LHS) pairs, while pairs where the premium is paid in CCY2 are called right-hand side (RHS) pairs.
When quoting premiums in %, the term basis point is often used to mean one-hundredth of a percent (i.e., 0.01%). For example, if the price of a contract is 0.25 EUR%, it might be verbally described as “twenty-five beeps.”
Exhibit 10.2 shows how to convert options premiums quoted in different terms. It is important to note that these conversions are only possible if the option contract has a strike.
Exhibit 10.2 Formulas for converting options premiums