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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