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QF5203 Foreign Exchange Derivatives
创建日期：2024-05-02 09:45:38
标签： QF5203
Foreign Exchange Derivatives

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QF5203

Foreign Exchange Derivatives and
their Risk Measures – Part 2
1. FX Spot
2. FX Forwards
3. FX Swaps
4. Non-Deliverable Forwards (NDFs)
5. Vanilla FX Options
6. FX Option Structures
7. FX Option Volatilities
8. FX Option Risk Sensitivities
9. Exotic FX Options
10. Term Project 2
1. References
• Option, Future and Other Derivatives, John Hull
• Interest Rate Option Models, Riccardo Rebonato
• The Volatility Surface: A Practitioner’s Guide, J. Gatherall
• FX Options and Smile Risk, Antonio Castagna
1. FX Spot
• The foreign exchange spot market is the market for delivery of a unit of one
currency in exchange for a specific amount of another currency on the
settlement
• The settlement date is usually two working days after the transaction date
(T+2). An exception is USDCAD where the settlement date is T+1.
• For example, if on Friday 3 Apr 2020 I agree to sell $100m USD/JPY to another
dealer at an agreed FX spot rate of 100, then on the settlement date, namely
Tuesday 7 April, my USD bank account will be debited by USD 100mn and my
JPY bank account will be credited with JPY 10bn
• Note that strictly speaking, an FX spot transaction leads to two future cash
flows which should be discounted to today
1. FX Spot
• The top 10 most popular currency pairs are:
1. EUR/USD (Euro/US Dollar)
2. USD/JPY (US Dollar/Yen) – nickname ‘the gopher’
3. GBP/USD (British Pound/US Dollar) – nickname ‘cable’
4. AUD/USD (Australian Dollar/US Dollar) – nickname ‘Aussie dollar’
5. USD/CAD (US Dollar/Canadian Dollar) – nickname the ‘loonie’
6. USD/CNY (US Dollar/Chinese Renminbi)
7. USD/CHF (US Dollar/Swiss franc) – nickname ‘dollar Swissie’
8. USD/HKD (US Dollar/Hong Kong Dollar)
9. EUR/GBP (Euro/British Pound)
10. USD/KRW (US Dollar/South Korean Won)
1. FX Spot
• There are two conventions for quoting FX spot rates:
➢ European Convention – the number of foreign currency units per 1 USD
➢ American Convention – the number of USD per 1 unit of foreign currency
• Most FX spot rates are quoted according to the European convention.
• These include USDJPY, USDCHF, USDCNY, etc.
• A few currency pairs are quoted using the American quotation, notably,
GBPUSD, AUDUSD, NZDUSD.
• The first tag of the currency pair is the base (or foreign) currency and the
second tag is the numeraire (or domestic) currency.
• So USDJPY mean the number of JPY per 1 USD.
• GBPUSD means the number of USD per 1 GBP.
1. FX Spot
• When a quote between two currencies is not available, one can compute a
cross-rate using existing quotes.
• Example: Suppose
USDJPY FX spot = 100.00
GBPUSD FX spot = 1.300
• Suppose now I want to know the FX spot rate for GBPJPY:
= ∗ () = 130.00
• Note that these 3 currency pairs are linked through a triangle relationship,
• A/B x B/C x C/A = 1, where A is the base currency, and B and C are the two
counter-currencies to be used in the arbitrage trade. If the equation does not
equal one, then an opportunity for an arbitrage trade may exist.
2. FX Forward
• The FX forward (or FX outright) market is the market for delivery at a fixed future date of a
specified amount of foreign currency to be exchanged at a pre-determined exchange rate for
the domestic currency.
• Note that an FX forward contract only differs from an FX spot contract only by the settlement
date, which is longer than T+2 for FX spot.
• The pre-determined FX rate that is agreed upfront is called the FX forward rate.
• Common terms for forward contracts are 1 month, 2 months, 3 months, 6 months and 12
months.
• The payout
(in domestic currency) of an FX forward contract is:

= −
where is the FX spot rate on the maturity date (expressed in terms of units of domestic
currency per 1 unit of foreign currency), and K is the strike.
• An FX forward contract allows one to lock in (or hedge) the delivery of a foreign currency
against the domestic at an initially agreed rate of K.
2. FX Forward Rate
• Recall that the payout of the USD interest rate payer FRA was
2 () = () 1, 2 − 1, 2
• In order to understand the payout of the FX forward it is useful to be specific as
to the currency of the payout by considering the USDJPY example
() = ()


−


• If I enter into this contract (implicitly ‘long’ the FX forward), then at maturity I
will receive N (USD) of value NX(T) (JPY) and pay NK (JPY).
• Whereas the LIBOR random variable and strike of the interest rate forward rate
agreement are dimensionless, the random variable and strike of the FX forward
have dimensions of JPY per USD. However, since the notional of the FX forward
contract is expressed in USD, the final payout must be expressed in JPY.
2. FX Forward Rate (cont’d)
• In the same way as we calculated the LIBOR forward rate in the context of
interest rate derivatives, the FX forward rate , can also be determined
through no-arbitrage arguments.
, =
,
,
where is the FX spot rate, , is the foreign discount factor, and ,
is the domestic discount factor.
• The derivation of the above result is best understood through a concrete
example
• Suppose that I am a USD-based investor with USD 1m to invest in Japanese Yen
cash with a one-year investment horizon.
• Assume that the current USDJPY FX Spot rate is 120 and that 1y USD interest
rates are 3% and 1y JPY interest rates are 1%.
2. FX Forward Rate (cont’d)
• The interest rate differential between USD and JPY is 2%
• In an arbitrage-free market, this interest rate differential will be eliminated
through the FX Forward market.
• The FX Forward (exchange) rate is determined by comparing what one would
earn on the USD 1m deposit by investing this for one year at the USD interest
rate, versus exchanging this USD amount into JPY and investing the JPY proceeds
for one year at the JPY interest rate.
• Case 1: USD Investment
➢ Deposit USD 1m at the assumed USD interest rate of 3% for one year
➢ At maturity, the value
of this USD investment is:

= 1,000,000 1 + 3% ∗ 365/360 = 1,030,417
2. FX Forward Rate (cont’d)
• Case 2: JPY Investment
➢ First convert my USD investment amount into JPY at the current (i.e. today)
exchange rate
➢ Using the assumed FX spot rate of 1.20, USD 1,000,000 = JPY 120,000,000
➢ Now deposit JPY 120m at the assumed JPY interest rate of 1% for one year
➢ At maturity, the value

of this JPY investment is:

= 120,000,000 1 + 1% ∗ 365/365 = 121,200,000
• Note that in each case the correct day count convention for deposits in each
currency has been respected (i.e. A/360 for USD and A/365 for JPY).
2. FX Forward Rate (cont’d)
• The FX forward (exchange) rate is defined to be the exchange rate for which
these two (deterministic) investment outcomes are equivalent
• In this specific case of the 1y USDJPY FX Forward rate 1
/
, we have:
1,030,417 =
121,200,000
1

leading to 1
= 117.62
• Replacing the maturity value investment amounts in USD and JPY by their initial
values, along with the corresponding investment interest rates, a slightly more
general expression for the 1y Forward exchange rate can be written:
2. FX Forward Rate (cont’d)
1
=
1 + 1

1 + 1

• Noting that the USD and JPY notional amounts were exchanged at the spot FX rate,
=


the expression at the top can be written
1
= ()
1 + 1

1 + 1

• Finally, re-expressing in terms of discount factors, leads to:
1
= ()
1

1

2. FX Forward Rate (cont’d)
• Recall from the previous lectures on interest rates where we studied the interest
rate FRA.
• The valuation of interest rate FRA was obtained by discounting the expected
payout.
• In the case of the interest rate FRA the expected LIBOR rate is just the current
FRA rate, expressed in terms of discount factors known today.
• Similarly, in the case of the FX forward, the expected FX rate is just the current
FX forward rate, again expressible in terms of discount factors that I know today.
2. FX Forward Rate (cont’d)
The valuation formula for an FX forward is obtained in the same way as for an
interest rate FRA, namely the present value (in the payout currency) of the
expected future payout, namely

= , − (, )
where
is the value of the FX forward contract as of today (t), expressed in units
of the foreign currency (e.g. JPY), is the notional denominated in units of the
foreign currency (e.g. USD), K is the strike of the FX forward, (, ) is the
domestic (e.g. JPY) discount factor observed at time t, for a maturity date T, and
, is the FX forward rate, defined on slide 10.
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