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QF5203-程序代写案例
时间：2022-04-03
QF5203 Lecture 9
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
• QuantLib Python Cookbook, Gautham Balaraman, Luigi Ballabio
• https://www.quantlib.org/quantlibxl/
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.
2. FX Forward Points
• Note that it is the interest rate differential between the domestic and foreign currencies
which determines the FX Forward rate
• Market convention is to quote FX Forwards in terms of forward points defined as:
, = , − ()
• In order to obtain the FX Forward rate one adds the (market quoted) Forward points to
the FX Spot rate
• In the USDJPY example shown previously,
1
= 117.62 − 120 = −2.38
• Note that FX Forward points can be both positive and negative and are driven by the
interest rate differential.
• In the USDJPY example, USD interest rates were higher than JPY interest rates, hence
leading to negative forward points (i.e. a forward FX rates lower than spot)
2. FX Forward Rate Example
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 −
−
; =
1
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 spot and 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 (ATMS).
ii. Strike of the option is set to the FX forward rate (ATMF).
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 (note that for longer
dated FX options the convention can change from ZDS to ATMF.
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 two implied
(lognormal) volatilities.
• 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 Volaility Inputs USD/JPY Fx Vol Shifts
USD/JPY ATM RR 25 RR 10 BF 25 BF 10
O/N 7.000 0.300 0.600 0.3500 1.1000
1W 8.750 -0.050 -0.050 0.2500 0.7500
1M 7.650 -0.750 -1.400 0.3500 1.1000
2M 7.450 -1.000 -1.850 0.3500 1.1500
3M 7.350 -1.050 -2.400 0.3500 1.2500
6M 7.300 -1.250 -2.650 0.4000 1.3500
1Y 7.500 -1.350 -2.850 0.4000 1.5000
2Y 7.450 -1.400 -2.900 0.4000 1.7500
3Y 7.500 -1.450 -2.800 0.4500 2.0000
4Y 7.550 -1.500 -2.850 0.5000 2.2500
5Y 7.550 -1.500 -2.700 0.5500 2.5000
7Y 7.650 -2.000 -3.400 0.5500 2.7500
10Y 8.500 -4.000 -7.000 0.2500 2.8000
12Y 8.750 -4.500 -7.750 0.5000 2.1000
15Y 9.500 -5.000 -8.500 0.5500 2.1500
20Y 9.750 -5.500 -9.250 0.6000 2.2000
25Y 10.000 -5.750 -9.500 0.6500 2.2500
30Y 10.250 -6.000 -9.750 0.7000 2.3000
7. FX Option Volatilities (cont’d)
• 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.
7. FX Option Volatilities (cont’d)
USD/JPY -10D Put Vol -25D Put Vol ZDS/ATM Vol 25D Call Vol 10D Call Vol
O/N 7.800 7.200 7.000 7.5000 8.4000
1W 9.525 9.025 8.750 8.9750 9.4750
1M 9.450 8.375 7.650 7.6250 8.0500
2M 9.525 8.300 7.450 7.3000 7.6750
3M 9.800 8.225 7.350 7.1750 7.4000
6M 9.975 8.325 7.300 7.0750 7.3250
1Y 10.425 8.575 7.500 7.2250 7.5750
2Y 10.650 8.550 7.450 7.1500 7.7500
3Y 10.900 8.675 7.500 7.2250 8.1000
4Y 11.225 8.800 7.550 7.3000 8.3750
5Y 11.400 8.850 7.550 7.3500 8.7000
7Y 12.100 9.200 7.650 7.2000 8.7000
10Y 14.800 10.750 8.500 6.7500 7.8000
12Y 14.725 11.500 8.750 7.0000 6.9750
15Y 15.900 12.550 9.500 7.5500 7.4000
20Y 16.575 13.100 9.750 7.6000 7.3250
25Y 17.000 13.525 10.000 7.7750 7.5000
30Y 17.425 13.950 10.250 7.9500 7.6750
7. FX Option Volatilities (cont’d)
• The next step of the volatility surface construction is to then associate each of
the (Black) volatilities shown on the previous slide to a strike.
• We begin with the zero delta straddle volatilities and recall that this is the
volatility for which the FX delta of the straddle is zero.
• Note that depending on the maturity of the volatility quotation, either FX Spot
• Now it is straightforward to calculate both the FX Forward delta and FX Spot
delta from the valuation formulae shown earlier.
∆ =
, 1
∆ =
, 1
where =1 for a call and =-1 for a put.
• This allows us to directly solve for the ATM (ZDS) strike
= ,
1
2
2 −
7. FX Option Volatilities (cont’d)
• We then move on to the 25 delta put and call strikes, again using the definitions
for FX spot delta, ∆ , shown on the previous slide
• Following Castagna, we find by straightforward algebra that
25 = , exp 25 − Φ
−1 0.25
(,)
+ 1225
2 −
25 = , exp −25 − Φ
−1 0.25
(,)
+ 1225
2 −
with similar formulae for the 10 delta put and call strikes
10 = , exp 10 − Φ
−1 0.10
(,)
+ 1210
2 −
10 = , exp −10 − Φ
−1 0.10
(,)
+ 1210
2 −
where Φ−1 is the inverse normal distribution
7. FX Option Volatilities (cont’d)
• In practice there are a number of subtleties associated with the concept of FX
option volatility quotations.
• First, there are two possible definitions of ATM, namely Zero Delta Straddle (ZDS)
and At-The-Money Forward (ATMF). It is common practice for FX volatilities with
expiries up to and including 10y to be quoted based on ZDS, and beyond 10y FX
volatilities are quoted based on ATMF.
➢ In the case of ZDS one solves for the strike for which FX delta (see later
comments on this) of the corresponding straddle is zero.
➢ In the case of ATMF the strike is (by definition) equal to the FX Forward rate
calculated based on the formula shown on slide 10.
7. FX Option Volatilities (cont’d)
• Secondly, there are two choices of FX Delta, namely FX Spot Delta and FX
Forward Delta.
• This is important because the ZDS ATM convention as well as the 10 and 25 Delta
Risk Reversal and Butterfly quotations all make reference to FX Delta.
• The specific FX Delta convention one uses is based on the corresponding FX
Option expiry, and typically FX Options with expiries less than 2y are quoted
based on FX Spot Delta, whereas FX Options with expiries 2y and above are
quoted based on FX Forward Delta.
• Currency pairs which follow the above conventions include EURUSD, USDJPY and
GBPUSD.
• Note that there are exceptions to this, notably USDCNY where the FX Forward
Delta convention is typically used for all expiries. Other exceptions exist as well
and so one needs to exercise caution when building FX volatility surfaces to
ensure that one is using the correct conventions.
7. FX Option Volatilities (cont’d)
• A final nuance associated with the quotation of FX Options, and again relates to
Delta, is whether the Delta in question is premium adjusted.
• This is relevant because you will see from the FX Option (Black-based) valuation
formulae shown on slide 23 are naturally expressed in units of the domestic
currency (e.g. JPY in the case of the USDJPY currency pair).
• The implication of this is that it introduces an additional (second order)
sensitivity to the FX Spot (or FX Forward), that needs to be incorporated into the
spot (or forward) hedge that would need to be executed.
• This leads to the concept of a premium-adjusted FX Spot Delta and a premium-
adjusted FX Forward Delta.
7. FX Options Volatilities (cont’d)
• We now have two separate grids that we can combine and interpolate volatilities
from
USD/JPY -10D Put Vol -25D Put Vol ZDS/ATM Vol 25D Call Vol 10D Call Vol USD/JPY 10D Put Strike 25D Put Strike ZDS/ATM Strike 25D Call Strike 10D Call Strike
O/N 7.80 7.20 7.00 7.50 8.40 O/N 99.48 99.75 100.00 100.26 100.56
1W 9.53 9.03 8.75 8.98 9.48 1W 98.73 99.36 100.00 100.64 101.28
1M 9.45 8.38 7.65 7.63 8.05 1M 96.53 98.36 99.99 101.49 103.04
2M 9.53 8.30 7.45 7.30 7.68 2M 95.21 97.76 99.97 101.96 104.00
3M 9.80 8.23 7.35 7.18 7.40 3M 93.85 97.18 99.89 102.34 104.76
6M 9.98 8.33 7.30 7.08 7.33 6M 91.23 95.92 99.75 103.15 106.61
1Y 10.43 8.58 7.50 7.23 7.58 1Y 87.28 94.00 99.41 104.27 109.48
2Y 10.65 8.55 7.45 7.15 7.75 2Y 81.78 91.09 98.35 104.96 113.00
3Y 10.90 8.68 7.50 7.23 8.10 3Y 77.62 88.81 97.30 105.18 116.07
4Y 11.23 8.80 7.55 7.30 8.38 4Y 73.49 86.28 95.50 104.18 117.62
5Y 11.40 8.85 7.55 7.35 8.70 5Y 70.01 83.98 93.54 102.67 118.86
7Y 12.10 9.20 7.65 7.20 8.70 7Y 63.79 80.07 89.83 98.76 118.17
10Y 14.80 10.75 8.50 6.75 7.80 10Y 55.38 76.44 85.99 92.61 112.06
12Y 14.73 11.50 8.75 7.00 6.98 12Y 53.32 75.66 83.68 89.31 106.01
15Y 15.90 12.55 9.50 7.55 7.40 15Y 50.49 76.91 81.44 84.75 104.85
20Y 16.58 13.10 9.75 7.60 7.33 20Y 50.58 84.63 80.20 78.29 102.13
25Y 17.00 13.53 10.00 7.78 7.50 25Y 52.57 97.48 78.67 70.26 98.76
30Y 17.43 13.95 10.25 7.95 7.68 30Y 56.18 119.87 75.82 59.21 92.05
7. FX Option Example
General Inputs
Quote Date 14-Mar-22
Fx Spot Shift 0.00
Fx Vol Shift 0.00%
Market Inputs
Dom Yield Curve IDJPY Yield Curve#0000
For Yield Curve IDUSD Yield Curve#0000
Fx Option Details
Option Expiry Strike Ccy 1 Put/Call Ccy 2 Put/Call Amount 1 Amount 2 Spot Fx Fwd Fx Fwd Points Flat Vol d1 d2 Prem (Ccy1)
3M 93.00 USD CALL JPY PUT 10,000,000 930,000,000 100.0000 99.82 0.18 20.00% 0.75541446 0.65500434 818,053
3M 93.00 USD PUT JPY CALL 10,000,000 930,000,000 100.0000 99.82 0.18 20.00% 0.75541446 0.65500434 136,705
3M 90.00 USD PUT JPY CALL 10,000,000 900,000,000 100.0000 99.82 0.18 20.00% 1.08197341 0.98156329 74,436
4M 92.00 USD CALL JPY PUT 10,000,000 920,000,000 100.0000 99.76 0.24 20.00% 0.75776933 0.64214121 931,762
4M 92.00 USD PUT JPY CALL 10,000,000 920,000,000 100.0000 99.76 0.24 20.00% 0.75776933 0.64214121 157,805
4M 90.00 USD PUT JPY CALL 10,000,000 900,000,000 100.0000 99.76 0.24 20.00% 0.94785204 0.83222391 111,711
4Y 44.41 USD CALL JPY PUT 10,000,000 444,100,000 100.0000 94.42 5.58 20.00% 2.08630764 1.68644465 4,860,332
4Y 44.41 USD PUT JPY CALL 10,000,000 444,100,000 100.0000 94.42 5.58 20.00% 2.08630764 1.68644465 28,211
5Y 62.00 USD CALL JPY PUT 10,000,000 620,000,000 100.0000 92.21 7.79 20.00% 1.11141837 0.66432732 3,213,813
5Y 62.00 USD PUT JPY CALL 10,000,000 620,000,000 100.0000 92.21 7.79 20.00% 1.11141837 0.66432732 326,698
7Y 75.00 USD CALL JPY PUT 10,000,000 750,000,000 100.0000 88.00 12.00 20.00% 0.56674796 0.03739063 2,258,938
7Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 100.0000 88.00 12.00 20.00% 0.56674796 0.03739063 1,048,315
10Y 111.99 USD CALL JPY PUT 10,000,000 1,119,886,873 100.0000 82.93 17.07 20.00% -0.1585699 -0.791112 1,081,212
10Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 100.0000 82.93 17.07 20.00% 0.47523728 -0.1573049 1,403,447
12Y 75.00 USD CALL JPY PUT 10,000,000 750,000,000 100.0000 79.92 20.08 20.00% 0.43815589 -0.2549017 1,999,958
12Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 100.0000 79.92 20.08 20.00% 0.43815589 -0.2549017 1,582,170
15Y 75.00 USD CALL JPY PUT 10,000,000 750,000,000 100.0000 76.11 23.89 20.00% 0.40624844 -0.3684897 1,859,056
15Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 100.0000 76.11 23.89 20.00% 0.40624844 -0.3684897 1,771,025
20Y 72.00 USD CALL JPY PUT 10,000,000 720,000,000 100.0000 72.92 27.08 20.00% 0.46155834 -0.4331139 1,803,816
20Y 72.00 USD PUT JPY CALL 10,000,000 720,000,000 100.0000 72.92 27.08 20.00% 0.46155834 -0.4331139 1,738,567
25Y 78.50 USD CALL JPY PUT 10,000,000 785,000,000 100.0000 69.42 30.58 20.00% 0.37731203 -0.6230167 1,529,036
25Y 78.50 USD PUT JPY CALL 10,000,000 785,000,000 100.0000 69.42 30.58 20.00% 0.37731203 -0.6230167 2,107,693
8. FX Option Risk Sensitivities
• Recall that interest rate options were sensitive to the yield curve (i.e. interest
rate delta) as well as caplet/floorlet or swaption volatilities (interest rate vega).
• Since two yield curves are required to calculate the FX forward rate, FX options
have interest rate delta risk to both the domestic and foreign yield curves.
• In a model which fits the smile (e.g. SABR), the vega risk report will show
sensitivities to all the parameters used to fit the model to the smile.
• For FX options, the volatility information is captured through the explicit marking
of ATM volatilities, 10 and 25 delta risk reversals, and 10 and 25 delta butterflies.
• The sensitivity to the ATM volatility is referred to as vega.
• The sensitivity to the risk reversal is referred to a rega.
• The sensitivity to the butterfly (or strangle) is referred to a sega.
9. FX Options - Exotic
• Exotic FX Options are divided into first generation and second generation.
• First generation exotic options include:
➢ Digitals
➢ Knock-in/out barriers
➢ Double-knock in/out barriers
➢ One-touch/no-touch/double-no-touch/double-touch
➢ Asian
➢ Basket
• Second generation exotics include:
➢ Window knock-in/out barriers
➢ First-in-then-out barriers
➢ Quanto barriers
➢ Etc.
9. FX Options - Exotic
• A European style FX digital option has the following payout on the expiry date :
➢ = 1 ≥ , = 0 ℎ ( )
➢ = 1 ≤ , = 0 ℎ ( )
• A European style FX barrier option has a payout which is identical to a vanilla FX option
apart from the fact that its terminal value is contingent upon whether or not the
underlying FX spot rate hits a predefined value during the life of the contract.
➢ Knock-in FX options only pay out when the underlying FX spot has hit the barrier.
➢ Knock-out FX options only pay out as long as the underlying FX spot does not hit the
barrier.
➢ The position of the barrier with respect to the initial FX spot rate determines
whether the barrier option is an up or down barrier.
➢ Four possibilities exist, namely up-and-in, down-and-in, up-and-out, down-and-out.
9. FX Options - Exotic
• Combinations of FX barrier options can replicate vanilla FX options
➢ , , + , , = ,
➢ , , + , , = ,
• One-touch options pay one unit of the notional amount if at any time until the
option expiry, the underlying FX spot rate breaches a given barrier level, with the
payment occurring immediately or alternatively on the expiry date.
• A double-no-touch option pays one unit of the notional amount at expiry,
contingent on the event that neither the upper or lower barrier has been
breached during the life of the contract.
10. Term Project 2
1. Using QuantLib Python/Excel and the volatility market data provided on slide
28, along with a USDJPY FX spot rate of 120, a flat USD yield curve of 3% and a
flat JPY yield curve of 1%, calculate the strikes corresponding to the ATM, 10
delta put, 25 delta put, 25 delta call and 10 delta call, and produce a table,
similar to what was shown on slide 36 of these lecture slides.
2. Using QuantLib Python/Excel develop a function FXVolatility which takes as
arguments the FX Forward, the time to expiry and the strike, and returns a
volatility. As a check you should demonstrate that when you pass your function
the same strikes as calculated in question 1 above, that you are able to
reproduce your input volatilities.
3. Using QuantLib Python/Excel calculate the premium of a 3y Down and Out USD
Call JPY Put on USD 25m, with barrier 115 and strike 125. Assume a Black
Scholes Merton process (defined in QuantLib) and use a constant volatility of
20% with the same FX spot and yield curve assumptions as in question 1 above.
10. Term Project 2 (cont’d)
4. Assuming you have bought this option, what FX Spot trade would you put on as
an initial hedge? Produce a vega, rega and sega risk report showing the
sensitivity of this FX option to your original FX volatility inputs.
5. Calibrate a Heston Model to your FX volatility surface and reprice the same
barrier option as in question 3 above.
Hint: See chapters 22 and 23 in the QuantLib Python Cookbook
Please submit by 11 April 2022
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
• QuantLib Python Cookbook, Gautham Balaraman, Luigi Ballabio
• https://www.quantlib.org/quantlibxl/
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.
2. FX Forward Points
• Note that it is the interest rate differential between the domestic and foreign currencies
which determines the FX Forward rate
• Market convention is to quote FX Forwards in terms of forward points defined as:
, = , − ()
• In order to obtain the FX Forward rate one adds the (market quoted) Forward points to
the FX Spot rate
• In the USDJPY example shown previously,
1
= 117.62 − 120 = −2.38
• Note that FX Forward points can be both positive and negative and are driven by the
interest rate differential.
• In the USDJPY example, USD interest rates were higher than JPY interest rates, hence
leading to negative forward points (i.e. a forward FX rates lower than spot)
2. FX Forward Rate Example
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 −
−
; =
1
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 spot and 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 (ATMS).
ii. Strike of the option is set to the FX forward rate (ATMF).
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 (note that for longer
dated FX options the convention can change from ZDS to ATMF.
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 two implied
(lognormal) volatilities.
• 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 Volaility Inputs USD/JPY Fx Vol Shifts
USD/JPY ATM RR 25 RR 10 BF 25 BF 10
O/N 7.000 0.300 0.600 0.3500 1.1000
1W 8.750 -0.050 -0.050 0.2500 0.7500
1M 7.650 -0.750 -1.400 0.3500 1.1000
2M 7.450 -1.000 -1.850 0.3500 1.1500
3M 7.350 -1.050 -2.400 0.3500 1.2500
6M 7.300 -1.250 -2.650 0.4000 1.3500
1Y 7.500 -1.350 -2.850 0.4000 1.5000
2Y 7.450 -1.400 -2.900 0.4000 1.7500
3Y 7.500 -1.450 -2.800 0.4500 2.0000
4Y 7.550 -1.500 -2.850 0.5000 2.2500
5Y 7.550 -1.500 -2.700 0.5500 2.5000
7Y 7.650 -2.000 -3.400 0.5500 2.7500
10Y 8.500 -4.000 -7.000 0.2500 2.8000
12Y 8.750 -4.500 -7.750 0.5000 2.1000
15Y 9.500 -5.000 -8.500 0.5500 2.1500
20Y 9.750 -5.500 -9.250 0.6000 2.2000
25Y 10.000 -5.750 -9.500 0.6500 2.2500
30Y 10.250 -6.000 -9.750 0.7000 2.3000
7. FX Option Volatilities (cont’d)
• 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.
7. FX Option Volatilities (cont’d)
USD/JPY -10D Put Vol -25D Put Vol ZDS/ATM Vol 25D Call Vol 10D Call Vol
O/N 7.800 7.200 7.000 7.5000 8.4000
1W 9.525 9.025 8.750 8.9750 9.4750
1M 9.450 8.375 7.650 7.6250 8.0500
2M 9.525 8.300 7.450 7.3000 7.6750
3M 9.800 8.225 7.350 7.1750 7.4000
6M 9.975 8.325 7.300 7.0750 7.3250
1Y 10.425 8.575 7.500 7.2250 7.5750
2Y 10.650 8.550 7.450 7.1500 7.7500
3Y 10.900 8.675 7.500 7.2250 8.1000
4Y 11.225 8.800 7.550 7.3000 8.3750
5Y 11.400 8.850 7.550 7.3500 8.7000
7Y 12.100 9.200 7.650 7.2000 8.7000
10Y 14.800 10.750 8.500 6.7500 7.8000
12Y 14.725 11.500 8.750 7.0000 6.9750
15Y 15.900 12.550 9.500 7.5500 7.4000
20Y 16.575 13.100 9.750 7.6000 7.3250
25Y 17.000 13.525 10.000 7.7750 7.5000
30Y 17.425 13.950 10.250 7.9500 7.6750
7. FX Option Volatilities (cont’d)
• The next step of the volatility surface construction is to then associate each of
the (Black) volatilities shown on the previous slide to a strike.
• We begin with the zero delta straddle volatilities and recall that this is the
volatility for which the FX delta of the straddle is zero.
• Note that depending on the maturity of the volatility quotation, either FX Spot
• Now it is straightforward to calculate both the FX Forward delta and FX Spot
delta from the valuation formulae shown earlier.
∆ =
, 1
∆ =
, 1
where =1 for a call and =-1 for a put.
• This allows us to directly solve for the ATM (ZDS) strike
= ,
1
2
2 −
7. FX Option Volatilities (cont’d)
• We then move on to the 25 delta put and call strikes, again using the definitions
for FX spot delta, ∆ , shown on the previous slide
• Following Castagna, we find by straightforward algebra that
25 = , exp 25 − Φ
−1 0.25
(,)
+ 1225
2 −
25 = , exp −25 − Φ
−1 0.25
(,)
+ 1225
2 −
with similar formulae for the 10 delta put and call strikes
10 = , exp 10 − Φ
−1 0.10
(,)
+ 1210
2 −
10 = , exp −10 − Φ
−1 0.10
(,)
+ 1210
2 −
where Φ−1 is the inverse normal distribution
7. FX Option Volatilities (cont’d)
• In practice there are a number of subtleties associated with the concept of FX
option volatility quotations.
• First, there are two possible definitions of ATM, namely Zero Delta Straddle (ZDS)
and At-The-Money Forward (ATMF). It is common practice for FX volatilities with
expiries up to and including 10y to be quoted based on ZDS, and beyond 10y FX
volatilities are quoted based on ATMF.
➢ In the case of ZDS one solves for the strike for which FX delta (see later
comments on this) of the corresponding straddle is zero.
➢ In the case of ATMF the strike is (by definition) equal to the FX Forward rate
calculated based on the formula shown on slide 10.
7. FX Option Volatilities (cont’d)
• Secondly, there are two choices of FX Delta, namely FX Spot Delta and FX
Forward Delta.
• This is important because the ZDS ATM convention as well as the 10 and 25 Delta
Risk Reversal and Butterfly quotations all make reference to FX Delta.
• The specific FX Delta convention one uses is based on the corresponding FX
Option expiry, and typically FX Options with expiries less than 2y are quoted
based on FX Spot Delta, whereas FX Options with expiries 2y and above are
quoted based on FX Forward Delta.
• Currency pairs which follow the above conventions include EURUSD, USDJPY and
GBPUSD.
• Note that there are exceptions to this, notably USDCNY where the FX Forward
Delta convention is typically used for all expiries. Other exceptions exist as well
and so one needs to exercise caution when building FX volatility surfaces to
ensure that one is using the correct conventions.
7. FX Option Volatilities (cont’d)
• A final nuance associated with the quotation of FX Options, and again relates to
Delta, is whether the Delta in question is premium adjusted.
• This is relevant because you will see from the FX Option (Black-based) valuation
formulae shown on slide 23 are naturally expressed in units of the domestic
currency (e.g. JPY in the case of the USDJPY currency pair).
• The implication of this is that it introduces an additional (second order)
sensitivity to the FX Spot (or FX Forward), that needs to be incorporated into the
spot (or forward) hedge that would need to be executed.
• This leads to the concept of a premium-adjusted FX Spot Delta and a premium-
adjusted FX Forward Delta.
7. FX Options Volatilities (cont’d)
• We now have two separate grids that we can combine and interpolate volatilities
from
USD/JPY -10D Put Vol -25D Put Vol ZDS/ATM Vol 25D Call Vol 10D Call Vol USD/JPY 10D Put Strike 25D Put Strike ZDS/ATM Strike 25D Call Strike 10D Call Strike
O/N 7.80 7.20 7.00 7.50 8.40 O/N 99.48 99.75 100.00 100.26 100.56
1W 9.53 9.03 8.75 8.98 9.48 1W 98.73 99.36 100.00 100.64 101.28
1M 9.45 8.38 7.65 7.63 8.05 1M 96.53 98.36 99.99 101.49 103.04
2M 9.53 8.30 7.45 7.30 7.68 2M 95.21 97.76 99.97 101.96 104.00
3M 9.80 8.23 7.35 7.18 7.40 3M 93.85 97.18 99.89 102.34 104.76
6M 9.98 8.33 7.30 7.08 7.33 6M 91.23 95.92 99.75 103.15 106.61
1Y 10.43 8.58 7.50 7.23 7.58 1Y 87.28 94.00 99.41 104.27 109.48
2Y 10.65 8.55 7.45 7.15 7.75 2Y 81.78 91.09 98.35 104.96 113.00
3Y 10.90 8.68 7.50 7.23 8.10 3Y 77.62 88.81 97.30 105.18 116.07
4Y 11.23 8.80 7.55 7.30 8.38 4Y 73.49 86.28 95.50 104.18 117.62
5Y 11.40 8.85 7.55 7.35 8.70 5Y 70.01 83.98 93.54 102.67 118.86
7Y 12.10 9.20 7.65 7.20 8.70 7Y 63.79 80.07 89.83 98.76 118.17
10Y 14.80 10.75 8.50 6.75 7.80 10Y 55.38 76.44 85.99 92.61 112.06
12Y 14.73 11.50 8.75 7.00 6.98 12Y 53.32 75.66 83.68 89.31 106.01
15Y 15.90 12.55 9.50 7.55 7.40 15Y 50.49 76.91 81.44 84.75 104.85
20Y 16.58 13.10 9.75 7.60 7.33 20Y 50.58 84.63 80.20 78.29 102.13
25Y 17.00 13.53 10.00 7.78 7.50 25Y 52.57 97.48 78.67 70.26 98.76
30Y 17.43 13.95 10.25 7.95 7.68 30Y 56.18 119.87 75.82 59.21 92.05
7. FX Option Example
General Inputs
Quote Date 14-Mar-22
Fx Spot Shift 0.00
Fx Vol Shift 0.00%
Market Inputs
Dom Yield Curve IDJPY Yield Curve#0000
For Yield Curve IDUSD Yield Curve#0000
Fx Option Details
Option Expiry Strike Ccy 1 Put/Call Ccy 2 Put/Call Amount 1 Amount 2 Spot Fx Fwd Fx Fwd Points Flat Vol d1 d2 Prem (Ccy1)
3M 93.00 USD CALL JPY PUT 10,000,000 930,000,000 100.0000 99.82 0.18 20.00% 0.75541446 0.65500434 818,053
3M 93.00 USD PUT JPY CALL 10,000,000 930,000,000 100.0000 99.82 0.18 20.00% 0.75541446 0.65500434 136,705
3M 90.00 USD PUT JPY CALL 10,000,000 900,000,000 100.0000 99.82 0.18 20.00% 1.08197341 0.98156329 74,436
4M 92.00 USD CALL JPY PUT 10,000,000 920,000,000 100.0000 99.76 0.24 20.00% 0.75776933 0.64214121 931,762
4M 92.00 USD PUT JPY CALL 10,000,000 920,000,000 100.0000 99.76 0.24 20.00% 0.75776933 0.64214121 157,805
4M 90.00 USD PUT JPY CALL 10,000,000 900,000,000 100.0000 99.76 0.24 20.00% 0.94785204 0.83222391 111,711
4Y 44.41 USD CALL JPY PUT 10,000,000 444,100,000 100.0000 94.42 5.58 20.00% 2.08630764 1.68644465 4,860,332
4Y 44.41 USD PUT JPY CALL 10,000,000 444,100,000 100.0000 94.42 5.58 20.00% 2.08630764 1.68644465 28,211
5Y 62.00 USD CALL JPY PUT 10,000,000 620,000,000 100.0000 92.21 7.79 20.00% 1.11141837 0.66432732 3,213,813
5Y 62.00 USD PUT JPY CALL 10,000,000 620,000,000 100.0000 92.21 7.79 20.00% 1.11141837 0.66432732 326,698
7Y 75.00 USD CALL JPY PUT 10,000,000 750,000,000 100.0000 88.00 12.00 20.00% 0.56674796 0.03739063 2,258,938
7Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 100.0000 88.00 12.00 20.00% 0.56674796 0.03739063 1,048,315
10Y 111.99 USD CALL JPY PUT 10,000,000 1,119,886,873 100.0000 82.93 17.07 20.00% -0.1585699 -0.791112 1,081,212
10Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 100.0000 82.93 17.07 20.00% 0.47523728 -0.1573049 1,403,447
12Y 75.00 USD CALL JPY PUT 10,000,000 750,000,000 100.0000 79.92 20.08 20.00% 0.43815589 -0.2549017 1,999,958
12Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 100.0000 79.92 20.08 20.00% 0.43815589 -0.2549017 1,582,170
15Y 75.00 USD CALL JPY PUT 10,000,000 750,000,000 100.0000 76.11 23.89 20.00% 0.40624844 -0.3684897 1,859,056
15Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 100.0000 76.11 23.89 20.00% 0.40624844 -0.3684897 1,771,025
20Y 72.00 USD CALL JPY PUT 10,000,000 720,000,000 100.0000 72.92 27.08 20.00% 0.46155834 -0.4331139 1,803,816
20Y 72.00 USD PUT JPY CALL 10,000,000 720,000,000 100.0000 72.92 27.08 20.00% 0.46155834 -0.4331139 1,738,567
25Y 78.50 USD CALL JPY PUT 10,000,000 785,000,000 100.0000 69.42 30.58 20.00% 0.37731203 -0.6230167 1,529,036
25Y 78.50 USD PUT JPY CALL 10,000,000 785,000,000 100.0000 69.42 30.58 20.00% 0.37731203 -0.6230167 2,107,693
8. FX Option Risk Sensitivities
• Recall that interest rate options were sensitive to the yield curve (i.e. interest
rate delta) as well as caplet/floorlet or swaption volatilities (interest rate vega).
• Since two yield curves are required to calculate the FX forward rate, FX options
have interest rate delta risk to both the domestic and foreign yield curves.
• In a model which fits the smile (e.g. SABR), the vega risk report will show
sensitivities to all the parameters used to fit the model to the smile.
• For FX options, the volatility information is captured through the explicit marking
of ATM volatilities, 10 and 25 delta risk reversals, and 10 and 25 delta butterflies.
• The sensitivity to the ATM volatility is referred to as vega.
• The sensitivity to the risk reversal is referred to a rega.
• The sensitivity to the butterfly (or strangle) is referred to a sega.
9. FX Options - Exotic
• Exotic FX Options are divided into first generation and second generation.
• First generation exotic options include:
➢ Digitals
➢ Knock-in/out barriers
➢ Double-knock in/out barriers
➢ One-touch/no-touch/double-no-touch/double-touch
➢ Asian
➢ Basket
• Second generation exotics include:
➢ Window knock-in/out barriers
➢ First-in-then-out barriers
➢ Quanto barriers
➢ Etc.
9. FX Options - Exotic
• A European style FX digital option has the following payout on the expiry date :
➢ = 1 ≥ , = 0 ℎ ( )
➢ = 1 ≤ , = 0 ℎ ( )
• A European style FX barrier option has a payout which is identical to a vanilla FX option
apart from the fact that its terminal value is contingent upon whether or not the
underlying FX spot rate hits a predefined value during the life of the contract.
➢ Knock-in FX options only pay out when the underlying FX spot has hit the barrier.
➢ Knock-out FX options only pay out as long as the underlying FX spot does not hit the
barrier.
➢ The position of the barrier with respect to the initial FX spot rate determines
whether the barrier option is an up or down barrier.
➢ Four possibilities exist, namely up-and-in, down-and-in, up-and-out, down-and-out.
9. FX Options - Exotic
• Combinations of FX barrier options can replicate vanilla FX options
➢ , , + , , = ,
➢ , , + , , = ,
• One-touch options pay one unit of the notional amount if at any time until the
option expiry, the underlying FX spot rate breaches a given barrier level, with the
payment occurring immediately or alternatively on the expiry date.
• A double-no-touch option pays one unit of the notional amount at expiry,
contingent on the event that neither the upper or lower barrier has been
breached during the life of the contract.
10. Term Project 2
1. Using QuantLib Python/Excel and the volatility market data provided on slide
28, along with a USDJPY FX spot rate of 120, a flat USD yield curve of 3% and a
flat JPY yield curve of 1%, calculate the strikes corresponding to the ATM, 10
delta put, 25 delta put, 25 delta call and 10 delta call, and produce a table,
similar to what was shown on slide 36 of these lecture slides.
2. Using QuantLib Python/Excel develop a function FXVolatility which takes as
arguments the FX Forward, the time to expiry and the strike, and returns a
volatility. As a check you should demonstrate that when you pass your function
the same strikes as calculated in question 1 above, that you are able to
reproduce your input volatilities.
3. Using QuantLib Python/Excel calculate the premium of a 3y Down and Out USD
Call JPY Put on USD 25m, with barrier 115 and strike 125. Assume a Black
Scholes Merton process (defined in QuantLib) and use a constant volatility of
20% with the same FX spot and yield curve assumptions as in question 1 above.
10. Term Project 2 (cont’d)
4. Assuming you have bought this option, what FX Spot trade would you put on as
an initial hedge? Produce a vega, rega and sega risk report showing the
sensitivity of this FX option to your original FX volatility inputs.
5. Calibrate a Heston Model to your FX volatility surface and reprice the same
barrier option as in question 3 above.
Hint: See chapters 22 and 23 in the QuantLib Python Cookbook
Please submit by 11 April 2022
