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QF5203 FX Derivatives and their Risk
创建日期：2024-04-08 15:00:37
所属分类：经济Economics
标签： QF5203
FX Derivatives and their Risk

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QF5203 Lecture 8

FX Derivatives and their Risk
Measures – Part 1
1. FX Spot
2. FX Forwards
3. FX Swaps
4. 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
• QuantLib Python Cookbook, Gautham Balaraman, Luigi Ballabio
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
• 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
• 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.
• Example:
3-month USDJPY FX forward
Suppose USDJPY FX spot = 100 and 3-month deposit rates in USD are 3%
and three-month deposit rates in JPY are 1%. Then from the earlier lectures
on interest rates,
2. FX Forward
, =
1
1 +
= 0.992556
, =
1
1 +
= 0.997506
where we have assumed a simple day count fraction of 0.25
• So the FX forward rate is therefore 99.50372.
• Market convention is to quote FX forwards in terms of forward points defined as:
, = , − ()
• In the example above the FX forward points are -0.49628.
• Note that FX forward points can be both positive and negative and are driven by
the interest rate differential.
2. FX Forward
• 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
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.
2. FX Forward
Fx Forward Details
FX Fwd Expiry Strike Foreign Ccy Domestic Ccy Foreign Amount Domestic Amount Spot Fx Fwd Fx Prem (Foreign Ccy) Prem (Domestic Ccy)
3M 93.00 USD JPY 10,000,000 930,000,000 100.0000 99.49 647,855 64,785,485
3M 93.00 USD JPY 10,000,000 930,000,000 100.0000 99.49 647,855 64,785,485
3M 90.00 USD JPY 10,000,000 900,000,000 100.0000 99.49 947,109 94,710,872
4M 92.00 USD JPY 10,000,000 920,000,000 100.0000 99.31 728,734 72,873,378
4M 92.00 USD JPY 10,000,000 920,000,000 100.0000 99.31 728,734 72,873,378
4M 90.00 USD JPY 10,000,000 900,000,000 100.0000 99.31 928,056 92,805,629
4Y 44.41 USD JPY 10,000,000 444,100,000 100.0000 92.39 4,609,855 460,985,475
4Y 44.41 USD JPY 10,000,000 444,100,000 100.0000 92.39 4,609,855 460,985,475
5Y 62.00 USD JPY 10,000,000 620,000,000 100.0000 90.58 2,719,003 271,900,288
5Y 62.00 USD JPY 10,000,000 620,000,000 100.0000 90.58 2,719,003 271,900,288
7Y 75.00 USD JPY 10,000,000 750,000,000 100.0000 87.05 1,124,073 112,407,331
7Y 75.00 USD JPY 10,000,000 750,000,000 100.0000 87.05 1,124,073 112,407,331
10Y 111.99 USD JPY 10,000,000 1,119,886,873 100.0000 82.04 -2,710,272 -271,027,205
10Y 75.00 USD JPY 10,000,000 750,000,000 100.0000 82.04 637,165 63,716,493
12Y 75.00 USD JPY 10,000,000 750,000,000 100.0000 78.85 341,768 34,176,834
12Y 75.00 USD JPY 10,000,000 750,000,000 100.0000 78.85 341,768 34,176,834
15Y 75.00 USD JPY 10,000,000 750,000,000 100.0000 74.30 -59,840 -5,983,965
15Y 75.00 USD JPY 10,000,000 750,000,000 100.0000 74.30 -59,840 -5,983,965
20Y 72.00 USD JPY 10,000,000 720,000,000 100.0000 67.30 -384,577 -38,457,688
20Y 72.00 USD JPY 10,000,000 720,000,000 100.0000 67.30 -384,577 -38,457,688
25Y 78.50 USD JPY 10,000,000 785,000,000 100.0000 60.96 -1,366,458 -136,645,761
25Y 78.50 USD JPY 10,000,000 785,000,000 100.0000 60.96 -1,366,458 -136,645,761
30Y 62.50 USD JPY 10,000,000 625,000,000 100.0000 55.22 -539,855 -53,985,534
30Y 62.50 USD JPY 10,000,000 625,000,000 100.0000 55.22 -539,855 -53,985,534
3Y 68.90 USD JPY 10,000,000 689,000,000 100.0000 94.23 2,457,914 245,791,408
3Y 68.90 USD JPY 10,000,000 689,000,000 100.0000 94.23 2,457,914 245,791,408
20Y 39.00 USD JPY 10,000,000 390,000,000 100.0000 67.30 2,318,067 231,806,709
20Y 39.00 USD JPY 10,000,000 390,000,000 100.0000 67.30 2,318,067 231,806,709
3. FX Swaps
• An FX forward contract has FX spot exposure, and an FX swap is created by
combining an FX forward with an FX spot trade.
• Specifically, two counterparties entering into an FX swap contract agree to the
execution of an FX spot trade for a given amount of the base currency, and at the
same time they agree to reverse the trade through an FX forward (outright) with
the same base currency amount at a given time in the future.
• The quoted price of an FX swap contract is simply the FX forward points.
• Note that whereas an FX forward contract has exposure to both FX spot and the
domestic and foreign interest rates, and FX swap contract is only exposed to the
domestic and foreign interest rates.
4. Non-Deliverable Forwards
• Recall that with an FX forward contract, at maturity one party pays (or receives) a fixed
amount of cash in the domestic currency in exchange for receiving (or paying) the
previously agreed upon amount of the foreign currency
• Due to government restrictions in certain countries, their respective currencies are not
freely convertible, leading to separate onshore and offshore markets.
• Examples include the Korean Won (KRW), Taiwan dollar (TWD) and Chinese Renminbi
(CNY), and Indian Rupee (INR).
• A non-deliverable FX forward allows an efficient way to hedge a FX exposure against
non-convertible currencies.
• An NDF is similar to a regular FX forward contract, except at maturity the NDF does not
require physical delivery of currencies, and is typically settled in an international
financial center in U.S. dollars.
• The financial benefits of an NDF are similar to those of an FX forward linked to
deliverable currencies.
5. FX Options - Vanilla
• Recall that in the case of FX forwards, the parties commit to delivering or
receiving fixed amounts of a domestic and foreign currency at some pre-
specified future date, and therefore the payout can be both positive and
negative.
• An FX option eliminates the downside risk but at the expense of paying an
upfront premium.
• Generally speaking, options as hedging instruments are recommended if one is
not sure about the magnitude, the timing or even the existence of the exposure
(for example, where one is bidding for a contract but the outcome pay-out of
the bidding process is unknown).
• As with interest rate caplets (call options on LIBOR) and floorlets (put options
on LIBOR), there are both FX call and FX put options.
5. FX Options - Vanilla
• An FX call option grants the holder the right (but not the obligation) to buy a fixed
amount of the domestic currency in exchange for the foreign currency at a pre-agreed
(strike price) FX rate.
• Similarly an FX put option grants the holder the right (but not the obligation) to sell a
fixed amount of the domestic currency in exchange for the foreign currency at a pre-
agreed (strike price) FX rate.
• The payout of an FX call option is:

() = 0, −
and

() = 0, −
for an FX put option, and where () denotes the payout expressed in domestic currency
units, denotes the foreign currency amount (e.g. USD in the case of USDJPY), denotes
the option strike and denotes the FX spot rate observed on the maturity date .
5. FX Options - Vanilla
• The valuation formula for an FX call option is

() = , 1 − 2
,
and

() = −2 − , −1
,
for an FX put option, where
1 =

(,)

+
1
2
2 −
−
; 2 =

(,)

−
1
2
2 −
−
and , is the domestic discount factor observed at time corresponding to a
maturity date .
• Note that the valuation formulae above implicitly assume that the FX forward
rate is lognormally distributed.
• Unlike the current situation with interest rates, FX forward rates are always
assumed to be positive.
5. FX Options - Vanilla
• There are specific quotation conventions for FX options which are used in the
market.
• Firstly vanilla FX options are usually quoted for standard expiry dates (e.g. 1W,
1M, etc.), although it is always possible to obtain a price for any expiry date.
• Secondly FX options are quoted in terms of implied (Black) volatilities, which is
to say, the volatility parameter number that enters the valuation formulae on
slide 17.
• Thirdly, strike prices are quoted in terms of the FX option delta (e.g. 1y 25 delta
put) which means that the strike of the option is not initially agreed but only
finalised once the trade is finalised.
• The advantage of this way of quotation is that the dealers don’t need to focus
on the small movements in the underlying markets during the trading process.
• Finally, there is the assumption that the option is traded ‘delta hedged’.
6. FX Option Structures
• There are 3 main FX option structures which are quoted in the market and form
the building blocks for the construction of the FX volatility surface
1. ATM Straddle
• an at-the-money (ATM) straddle is an option structure based on the
simultaneous trade of a call option and a put option for the same expiry and
strike combination where the strike is chosen such the FX delta of the
straddle is zero (ZDS). The definition of ATM can be:
i. Strike of the option is set to the current FX spot rate.
ii. Strike of the option is set to the FX forward rate.
iii. Strike of the option is chosen so that the FX delta of the resulting straddle is
zero. This is commonly referred to as the zero delta straddle (ZDS) and is
the basis for quoting ATM volatilities in the market.
6. FX Option Structures
2. Risk Reversal
• this is an option strategy whereby one buys an out-of-the-money (OTM) call
and simultaneously sells an OTM put with the same FX delta.
• The two common delta quotations are 10 delta and 25 delta.
• The risk reversal is quoted as the difference between the two implied
volatilities that are used directly in the valuation formulae on slide 17.
• A positive number for the risk reversal means that the implied volatility of
the call is higher than the implied volatility of the put, whereas a negative
number for the risk reversal means that the implied volatility of the put is
higher than the implied volatility of the call.
, ; 25 = 25 , − 25(, )
, ; 10 = 10 , − 10(, )
• Where 25 , and 25 , are the implied volatilities of the 25 delta
call and put, with similar definitions for the 10 delta volatilities.
6. FX Option Structures
3. Butterfly
• The vega weighted butterfly is constructed by selling an ATM straddle and
simultaneously buying a symmetric delta strangle.
• By symmetric delta strangle, we mean the delta of the OTM put and call are
the same (modulo the sign).
• Recall that the a strangle differs from a straddle in that a straddle involves
buying a put and a call at the same strike but a strangle involves buy a put
and a call at two different strikes.
• As with the risk reversal, the two most common butterfly ‘deltas’ to trade are
the 10 delta and 25 delta.
BF , ; 25 = 0.5 [25 , + 25(, )] − (, )
BF , ; 10 = 0.5 [10 , + 10(, )] − (, )
• Where 25 , and 25 , are the implied volatilities of the 25 delta
call and put, with similar definitions for the 10 delta volatilities.
7. FX Option Volatilities
• Constructing the FX Volatility Surface
• As mentioned previously, the market quotes ATM volatilities (on a zero delta
straddle basis), as well as OTM volatilities based on risk reversal and butterfly
volatility quotes
USD/JPY Fx Vol Curve USD/JPY Fx Vol Shifts
USD/JPY ATM RR 25 RR 10 BF 25 BF 10
O/N 15.000 -0.850 -1.460 0.1400 1.8000
1W 19.500 -1.750 -3.080 0.2300 1.8000
1M 18.500 -3.150 -5.870 0.3700 1.8000
2M 18.000 -3.700 -6.980 0.4200 1.9500
3M 17.500 -4.250 -8.130 0.4800 2.2000
6M 16.250 -4.950 -9.590 0.5500 2.5000
1Y 15.400 -6.050 -12.020 0.6600 3.5000
2Y 13.800 -6.450 -12.820 0.7000 3.6000
3Y 12.700 -6.650 -13.220 0.7200 3.7000
4Y 12.600 -6.850 -13.610 0.7400 3.7500
5Y 12.500 -7.000 -13.910 0.7500 3.8000
7Y 12.700 -7.350 -14.250 0.6800 3.8000
10Y 14.500 -7.650 -14.400 0.4500 3.5500
12Y 14.500 -7.800 -14.650 1.5500 4.4500
15Y 15.750 -7.800 -14.300 1.5500 4.3000
20Y 18.050 -7.850 -14.300 1.6000 4.3000
25Y 18.950 -7.850 -14.250 1.6000 4.2500
30Y 20.650 -7.900 -14.250 1.6000 4.3000
7. FX Option Volatilities
• The first step in the FX volatility surface construction is to use the equations
for the 10 and 25 delta risk reversal and butterfly shown on slides 22 and 23,
to solve for the volatilities.
• It is straightforward to show that:
25 , = , +
1
2
2 , ; 25 + , ; 25
25 , = , +
1
2
2 , ; 25 − , ; 25
10 , = , +
1
2
2 , ; 10 + , ; 10
10 , = , +
1
2
2 , ; 10 − , ; 10
• This allows us to transform the initial market data into something that is
usable in our valuation formulae and is this is shown on the next slide

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COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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FN1024 MATH0094 MANG 6554 ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 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