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QF5203-quantlib Python代写
时间：2021-04-09
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
• 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
• 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
7. FX Option Volatilities
Expiry Tenor-10D Put Vol -25D Put Vol ZDS/ATM Vol 25D Call Vol 10D Call Vol
O/N 17.530 15.565 15.000 14.7150 16.0700
1W 22.840 20.605 19.500 18.8550 19.7600
1M 23.235 20.445 18.500 17.2950 17.3650
2M 23.440 20.270 18.000 16.5700 16.4600
3M 23.765 20.105 17.500 15.8550 15.6350
6M 23.545 19.275 16.250 14.3250 13.9550
1Y 24.910 19.085 15.400 13.0350 12.8900
2Y 23.810 17.725 13.800 11.2750 10.9900
3Y 23.010 16.745 12.700 10.0950 9.7900
4Y 23.155 16.765 12.600 9.9150 9.5450
5Y 23.255 16.750 12.500 9.7500 9.3450
7Y 23.625 17.055 12.700 9.7050 9.3750
10Y 25.250 18.775 14.500 11.1250 10.8500
12Y 26.275 19.950 14.500 12.1500 11.6250
15Y 27.200 21.200 15.750 13.4000 12.9000
20Y 29.500 23.575 18.050 15.7250 15.2000
25Y 30.325 24.475 18.950 16.6250 16.0750
30Y 32.075 26.200 20.650 18.3000 17.8250
7. FX Option Volatilities
• 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
• 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
• 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 Options Volatilities
• We now have two separate grids that we can combine and interpolate volatilities
from
Expiry Tenor-10D Put Vol -25D Put Vol ZDS/ATM Vol 25D Call Vol 10D Call Vol Expiry Tenor 10D Put Strike 25D Put Strike ZDS/ATM Strike 25D Call Strike 10D Call Strike
O/N 17.53 15.57 15.00 14.72 16.07 O/N 96.29 97.33 98.25 99.15 100.12
1W 22.84 20.61 19.50 18.86 19.76 1W 94.36 96.37 98.21 99.99 101.77
1M 23.24 20.45 18.50 17.30 17.37 1M 90.58 94.53 98.01 101.30 104.31
2M 23.44 20.27 18.00 16.57 16.46 2M 86.72 92.60 97.67 102.57 106.95
3M 23.77 20.11 17.50 15.86 15.64 3M 84.26 91.42 97.41 103.16 108.22
6M 23.55 19.28 16.25 14.33 13.96 6M 78.98 88.74 96.62 104.23 110.73
1Y 24.91 19.09 15.40 13.04 12.89 1Y 70.92 84.76 95.15 105.21 114.08
2Y 23.81 17.73 13.80 11.28 10.99 2Y 62.86 79.88 92.57 105.14 115.83
3Y 23.01 16.75 12.70 10.10 9.79 3Y 57.59 76.43 90.39 104.27 115.88
4Y 23.16 16.77 12.60 9.92 9.55 4Y 52.65 72.80 87.96 103.87 116.97
5Y 23.26 16.75 12.50 9.75 9.35 5Y 48.61 69.63 85.61 103.21 117.55
7Y 23.63 17.06 12.70 9.71 9.38 7Y 41.94 63.77 80.86 101.83 119.28
10Y 25.25 18.78 14.50 11.13 10.85 10Y 33.43 55.12 72.58 102.30 128.64
12Y 26.28 19.95 14.50 12.15 11.63 12Y 29.04 50.04 68.30 102.93 134.86
15Y 27.20 21.20 15.75 13.40 12.90 15Y 24.22 43.83 60.62 103.33 146.85
20Y 29.50 23.58 18.05 15.73 15.20 20Y 17.87 35.11 47.74 104.49 176.08
25Y 30.33 24.48 18.95 16.63 16.08 25Y 14.15 29.30 38.23 101.00 195.14
30Y 32.08 26.20 20.65 18.30 17.83 30Y 11.08 24.17 28.61 97.09 236.85
7. FX Option Example
General Inputs
Quote Date 3-Apr-20
Fx Spot Shift 0.00
Fx Vol Shift 0.00%
Market Inputs
Dom Yield Curve IDJPY Yield Curve#0004
For Yield Curve IDUSD Yield Curve#0004
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 98.2800 97.78 0.50 20.00% 0.55217252 0.4523096 675,895
3M 93.00 USD PUT JPY CALL 10,000,000 930,000,000 98.2800 97.78 0.50 20.00% 0.55217252 0.4523096 190,395
3M 90.00 USD PUT JPY CALL 10,000,000 900,000,000 98.2800 97.78 0.50 20.00% 0.88052085 0.78065793 108,184
4M 92.00 USD CALL JPY PUT 10,000,000 920,000,000 98.2800 97.60 0.68 20.00% 0.56551976 0.44894772 787,155
4M 92.00 USD PUT JPY CALL 10,000,000 920,000,000 98.2800 97.60 0.68 20.00% 0.56551976 0.44894772 218,885
4M 90.00 USD PUT JPY CALL 10,000,000 900,000,000 98.2800 97.60 0.68 20.00% 0.75406331 0.63749127 159,282
4Y 44.41 USD CALL JPY PUT 10,000,000 444,100,000 98.2800 90.80 7.48 20.00% 1.98686169 1.58658781 4,571,299
4Y 44.41 USD PUT JPY CALL 10,000,000 444,100,000 98.2800 90.80 7.48 20.00% 1.98686169 1.58658781 36,122
5Y 62.00 USD CALL JPY PUT 10,000,000 620,000,000 98.2800 89.02 9.26 20.00% 1.03237474 0.58503864 2,990,989
5Y 62.00 USD PUT JPY CALL 10,000,000 620,000,000 98.2800 89.02 9.26 20.00% 1.03237474 0.58503864 375,211
7Y 75.00 USD CALL JPY PUT 10,000,000 750,000,000 98.2800 85.56 12.72 20.00% 0.51347101 -0.0159898 2,138,830
7Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 98.2800 85.56 12.72 20.00% 0.51347101 -0.0159898 1,137,152
10Y 111.99 USD CALL JPY PUT 10,000,000 1,119,886,873 98.2800 80.63 17.65 20.00% -0.202886 -0.8356014 1,035,577
10Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 98.2800 80.63 17.65 20.00% 0.43074758 -0.2019678 1,530,994
12Y 75.00 USD CALL JPY PUT 10,000,000 750,000,000 98.2800 77.50 20.78 20.00% 0.39384429 -0.2993714 1,980,497
12Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 98.2800 77.50 20.78 20.00% 0.39384429 -0.2993714 1,755,163
15Y 75.00 USD CALL JPY PUT 10,000,000 750,000,000 98.2800 73.03 25.25 20.00% 0.35307232 -0.421878 1,870,052
15Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 98.2800 73.03 25.25 20.00% 0.35307232 -0.421878 2,042,893
20Y 72.00 USD CALL JPY PUT 10,000,000 720,000,000 98.2800 66.15 32.13 20.00% 0.35267203 -0.5421839 1,752,760
20Y 72.00 USD PUT JPY CALL 10,000,000 720,000,000 98.2800 66.15 32.13 20.00% 0.35267203 -0.5421839 2,240,535
25Y 78.50 USD CALL JPY PUT 10,000,000 785,000,000 98.2800 59.91 38.37 20.00% 0.23013095 -0.7703073 1,434,482
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 exotics 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
• An FX barrier option is identical to a vanilla FX option apart from the fact that its
terminal value is contingent on 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 hits 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
• Combinations of FX barrier options can replicate vanilla FX options
➢ , , + , , = ,
➢ , , + , , = ,
9. FX Options - Exotic
• 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
9. Term Project 2
1. Using QuantLib Python/Excel and the volatility market data provided on slide
24, along with a USDJPY FX spot rate of 105, 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 29 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 the same volatilities that are shown on slide 29.
3. Using QuantLib Python/Excel calculate the premium of a 3y Down and Out USD
Call JPY Put on USD 25m, with barrier 95 and strike 115. Assume a Black
Scholes Merton process (defined in QuantLib) and use a constant volatility of
15% with the same yield curve assumptions as in question 1 above.
9. Term Project 2 (Continued)
4. What FX Spot hedge would you put on as an initial hedge? Produce a vega, rega
and sega risk report showing the sensitivity of this FX option to the FX volatility
inputs shown on slide 24.
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 2 April 2021
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
• 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
• 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
7. FX Option Volatilities
Expiry Tenor-10D Put Vol -25D Put Vol ZDS/ATM Vol 25D Call Vol 10D Call Vol
O/N 17.530 15.565 15.000 14.7150 16.0700
1W 22.840 20.605 19.500 18.8550 19.7600
1M 23.235 20.445 18.500 17.2950 17.3650
2M 23.440 20.270 18.000 16.5700 16.4600
3M 23.765 20.105 17.500 15.8550 15.6350
6M 23.545 19.275 16.250 14.3250 13.9550
1Y 24.910 19.085 15.400 13.0350 12.8900
2Y 23.810 17.725 13.800 11.2750 10.9900
3Y 23.010 16.745 12.700 10.0950 9.7900
4Y 23.155 16.765 12.600 9.9150 9.5450
5Y 23.255 16.750 12.500 9.7500 9.3450
7Y 23.625 17.055 12.700 9.7050 9.3750
10Y 25.250 18.775 14.500 11.1250 10.8500
12Y 26.275 19.950 14.500 12.1500 11.6250
15Y 27.200 21.200 15.750 13.4000 12.9000
20Y 29.500 23.575 18.050 15.7250 15.2000
25Y 30.325 24.475 18.950 16.6250 16.0750
30Y 32.075 26.200 20.650 18.3000 17.8250
7. FX Option Volatilities
• 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
• 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
• 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 Options Volatilities
• We now have two separate grids that we can combine and interpolate volatilities
from
Expiry Tenor-10D Put Vol -25D Put Vol ZDS/ATM Vol 25D Call Vol 10D Call Vol Expiry Tenor 10D Put Strike 25D Put Strike ZDS/ATM Strike 25D Call Strike 10D Call Strike
O/N 17.53 15.57 15.00 14.72 16.07 O/N 96.29 97.33 98.25 99.15 100.12
1W 22.84 20.61 19.50 18.86 19.76 1W 94.36 96.37 98.21 99.99 101.77
1M 23.24 20.45 18.50 17.30 17.37 1M 90.58 94.53 98.01 101.30 104.31
2M 23.44 20.27 18.00 16.57 16.46 2M 86.72 92.60 97.67 102.57 106.95
3M 23.77 20.11 17.50 15.86 15.64 3M 84.26 91.42 97.41 103.16 108.22
6M 23.55 19.28 16.25 14.33 13.96 6M 78.98 88.74 96.62 104.23 110.73
1Y 24.91 19.09 15.40 13.04 12.89 1Y 70.92 84.76 95.15 105.21 114.08
2Y 23.81 17.73 13.80 11.28 10.99 2Y 62.86 79.88 92.57 105.14 115.83
3Y 23.01 16.75 12.70 10.10 9.79 3Y 57.59 76.43 90.39 104.27 115.88
4Y 23.16 16.77 12.60 9.92 9.55 4Y 52.65 72.80 87.96 103.87 116.97
5Y 23.26 16.75 12.50 9.75 9.35 5Y 48.61 69.63 85.61 103.21 117.55
7Y 23.63 17.06 12.70 9.71 9.38 7Y 41.94 63.77 80.86 101.83 119.28
10Y 25.25 18.78 14.50 11.13 10.85 10Y 33.43 55.12 72.58 102.30 128.64
12Y 26.28 19.95 14.50 12.15 11.63 12Y 29.04 50.04 68.30 102.93 134.86
15Y 27.20 21.20 15.75 13.40 12.90 15Y 24.22 43.83 60.62 103.33 146.85
20Y 29.50 23.58 18.05 15.73 15.20 20Y 17.87 35.11 47.74 104.49 176.08
25Y 30.33 24.48 18.95 16.63 16.08 25Y 14.15 29.30 38.23 101.00 195.14
30Y 32.08 26.20 20.65 18.30 17.83 30Y 11.08 24.17 28.61 97.09 236.85
7. FX Option Example
General Inputs
Quote Date 3-Apr-20
Fx Spot Shift 0.00
Fx Vol Shift 0.00%
Market Inputs
Dom Yield Curve IDJPY Yield Curve#0004
For Yield Curve IDUSD Yield Curve#0004
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 98.2800 97.78 0.50 20.00% 0.55217252 0.4523096 675,895
3M 93.00 USD PUT JPY CALL 10,000,000 930,000,000 98.2800 97.78 0.50 20.00% 0.55217252 0.4523096 190,395
3M 90.00 USD PUT JPY CALL 10,000,000 900,000,000 98.2800 97.78 0.50 20.00% 0.88052085 0.78065793 108,184
4M 92.00 USD CALL JPY PUT 10,000,000 920,000,000 98.2800 97.60 0.68 20.00% 0.56551976 0.44894772 787,155
4M 92.00 USD PUT JPY CALL 10,000,000 920,000,000 98.2800 97.60 0.68 20.00% 0.56551976 0.44894772 218,885
4M 90.00 USD PUT JPY CALL 10,000,000 900,000,000 98.2800 97.60 0.68 20.00% 0.75406331 0.63749127 159,282
4Y 44.41 USD CALL JPY PUT 10,000,000 444,100,000 98.2800 90.80 7.48 20.00% 1.98686169 1.58658781 4,571,299
4Y 44.41 USD PUT JPY CALL 10,000,000 444,100,000 98.2800 90.80 7.48 20.00% 1.98686169 1.58658781 36,122
5Y 62.00 USD CALL JPY PUT 10,000,000 620,000,000 98.2800 89.02 9.26 20.00% 1.03237474 0.58503864 2,990,989
5Y 62.00 USD PUT JPY CALL 10,000,000 620,000,000 98.2800 89.02 9.26 20.00% 1.03237474 0.58503864 375,211
7Y 75.00 USD CALL JPY PUT 10,000,000 750,000,000 98.2800 85.56 12.72 20.00% 0.51347101 -0.0159898 2,138,830
7Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 98.2800 85.56 12.72 20.00% 0.51347101 -0.0159898 1,137,152
10Y 111.99 USD CALL JPY PUT 10,000,000 1,119,886,873 98.2800 80.63 17.65 20.00% -0.202886 -0.8356014 1,035,577
10Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 98.2800 80.63 17.65 20.00% 0.43074758 -0.2019678 1,530,994
12Y 75.00 USD CALL JPY PUT 10,000,000 750,000,000 98.2800 77.50 20.78 20.00% 0.39384429 -0.2993714 1,980,497
12Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 98.2800 77.50 20.78 20.00% 0.39384429 -0.2993714 1,755,163
15Y 75.00 USD CALL JPY PUT 10,000,000 750,000,000 98.2800 73.03 25.25 20.00% 0.35307232 -0.421878 1,870,052
15Y 75.00 USD PUT JPY CALL 10,000,000 750,000,000 98.2800 73.03 25.25 20.00% 0.35307232 -0.421878 2,042,893
20Y 72.00 USD CALL JPY PUT 10,000,000 720,000,000 98.2800 66.15 32.13 20.00% 0.35267203 -0.5421839 1,752,760
20Y 72.00 USD PUT JPY CALL 10,000,000 720,000,000 98.2800 66.15 32.13 20.00% 0.35267203 -0.5421839 2,240,535
25Y 78.50 USD CALL JPY PUT 10,000,000 785,000,000 98.2800 59.91 38.37 20.00% 0.23013095 -0.7703073 1,434,482
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 exotics 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
• An FX barrier option is identical to a vanilla FX option apart from the fact that its
terminal value is contingent on 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 hits 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
• Combinations of FX barrier options can replicate vanilla FX options
➢ , , + , , = ,
➢ , , + , , = ,
9. FX Options - Exotic
• 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
9. Term Project 2
1. Using QuantLib Python/Excel and the volatility market data provided on slide
24, along with a USDJPY FX spot rate of 105, 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 29 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 the same volatilities that are shown on slide 29.
3. Using QuantLib Python/Excel calculate the premium of a 3y Down and Out USD
Call JPY Put on USD 25m, with barrier 95 and strike 115. Assume a Black
Scholes Merton process (defined in QuantLib) and use a constant volatility of
15% with the same yield curve assumptions as in question 1 above.
9. Term Project 2 (Continued)
4. What FX Spot hedge would you put on as an initial hedge? Produce a vega, rega
and sega risk report showing the sensitivity of this FX option to the FX volatility
inputs shown on slide 24.
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 2 April 2021
