The Theory and Practice of FX Risk Management

Session 1:

Currency Derivatives: An Introduction

Presenter:

Stuart Thomas

School of Economics and Finance, RMIT

2

Why Trade Foreign Exchange?

• International Trade in Goods and Services• Capital Movements

– Borrowing offshore– Investing offshore

• Hedging– Hedging value of foreign currency receivables

and payables

• Arbitrage & Speculation

3

Characteristics of Foreign Currencies and Markets

• The Nature of Exchange Rates– Definition: the rate at which the currency of

one country can be translated into the currency of another country.

– The price in one currency of purchasing another currency.

4

Currency Risk Management

• What is exchange rate Risk?

“The risk of a loss from an unexpected change in exchange rates”

• transaction exposures

• translation exposures

5

Forward FX Contracts

• An agreement between two parties to exchange one currency for another at an agreed future maturity or settlement date at a price/rate agreed to today– OTC and privately negotiated

– tailored specifically to customer needs

– value (payoff) of the forward contract is only known at maturity

– early or late delivery requires renegotiation

– Exit usually by cancellation

6

FX Forwards - Short hedge

• On June 29, US coy. with UK subsidiary knows it will need to transfer £10,000,000 from its London bank to its UK bank on September 28– concerned that USD will appreciate against

GBP– Spot GBP/USD = 1.362– 3mth Fwd GBP/USD = 1.375

7

FX forwards - short hedgeSpot Market Forward Market

June 29 Spot GBP/USD = 1.3620

Spot Value of funds = $13,620,000

Fwd GBP/USD = 1.3570

Sell GBP forwardForward value of funds = $13,570,000

September 28 Spot GBP/USD = 1.2375

Spot Value of Funds = $12,375,000

Deliver £10,000,000 against USD at 1.375

Receive USD $13,750,000

Outcome: $13,750,000 - $12,375,000 = $1,375,000 “gain” over unhedged GBP sale

8

FX Swap

• FX Swap is an agreement for a round trip exchange of currencies, where an outright forward is a one way exchange of currencies at the forward date, eg:

– Exchange of AUD for USD at spot, and

– Re-exchange of USD for USD at forward date

• Accounts for approx. 50% of FX trading volume

• No net FX position is created.

• Used for position management, and to adjust for cash flow mismatches.

9

Currency Futures

• An alternative to forward contracts• Exchange traded like all other futures contracts• Standardised wrt:

– size

– maturity

– currency (terms currency always USD)

• Long position in commodity currency also a short position in terms currency

10

Currency Futures Contracts

Contract Exchange Contract Size

AUD/USD CME AUD 100,000

AUD/USD IMM AUD 100,000

AUD/USD SFE AUD 100,000

DEM/USD SGX DEM 125,000

JPY/USD SGX JPY 12,500,000

GBP/USD SGX GBP 62,500

11

CCY futures hedge

• Australian company expects to receive $1mio USD in three months:– prevailing exchange rate AUD/USD 0.7000

– AUD futures on CME trading at 0.6928

– Brokerage $50 (USD) per contract

• Expected AUD receipt $1,428,571 (@0.70)• Concerned that AUD will appreciate• Buy 14 AUD futures on CME @ 0.6928

12

CCY futures hedgeCash Market Futures Market

NowExpected receipt of USD 1mioDo nothing.

Buy 14 March AUD/USD contracts atmarket price of 0.6928Cost Equals USD 969,920Brokerage $700 USD ($1000 AUD)

3 months’ timeReceive USD1,000,000, convert atAUD/USD 0.80 yielding AUD1,250,000

Sell 14 March futures @0.8005 =USD1,127,000 less purchase cost ofUSD969,920 = profit of USD157,080.(Convert at 0.8000, to equal AUD196,350)Brokerage $700 USD ($875 AUD)

Proceeds:AUD $1,250,000

Futures Profit:AUD $196,350

Outcome: AUD 1,250,000 + 196,350 - 1,875 = AUD 1,444,475

13

CCY Futures issues:

• Standardisation– $1,428,571 AUD exposure, 100k contract

• Basis Risk

• Commodity Basis Risk– eg NZD?

• Margin Calls

14

Currency Options

• Currency Option

“the right but not the obligation to buy or sell one currency against another currency at a specified price during a specified period”

15

Currency Options

• Lack of flexibility in FX futures, and the possibility of margin calls lead many FX hedgers to options

• Puts• Calls• Premium

– in USD per unit of Commodity CCY

• OTC & ETO markets

16

Currency Options

• OTC– usually sold by banks, tailored to requirements– illiquid secondary market

• ETO– Standardised like futures– liquid, competitively priced– Cash CCY options & Options on CCY futures

17

Currency Options

• Option premiums are usually expressed as either:

- a fixed number of exchange points or

- a percentage of the strike price

18

Currency Options

• Assume AUD put / USD call

• Face Value of US$20,000,000

• Spot = 0.7093

• Strike = 0.7124

• Percentage of Strike = 1.30890374

• US$ per A$ = 0.00932463

• A$ per US$ = 0.01845346

19

Currency OptionsUS$ per A$

strike of strike = A$ per US$ * spot * 100

Percentage of Strike method:

Premium = US$20,000,000*1.30890374

100

at strike of 0.7124

Premium = A$28,074,115.67 x 0.00932463 =

Premium = US$20,000,000 * 0.01845346 =

(at 0.7093)

* %

$261, .

$ $

$20, , $28, , .

$261, .

$ $

$369, .

: $261, . $369, .

100

780 75

000 000 074 115 67

780 75

069 15

780 75 069 15

US

US perA

US A

US

A perUS

A

Note US A

20

Options on CCY Futures• Call Option on Futures (at exercise):

– Buyer takes long position in nearest futures contract on Commodity CCY

– Seller takes short position in nearest futures contract on Commodity CCY

• Put Option on Futures (at exercise):– Buyer takes short position in nearest futures contract on

Commodity CCY– Seller Takes long position in nearest futures contract on

Commodity CCY

21

FX Option Products

• Cap• Floor• Collar

– combination of a cap and a floor– single maturity more common in FX– aka “range forward”

• Tunnel– in FX - a rolling series of collars

22

Currency Swaps• Definition

– an agreement between two parties in which one party will make a series of payments in one currency and other will make a series of payments in another currency.

– aka:• Cross-currency swaps• Cross-currency interest rate swaps

23

Currency Swap Example

An agreement to pay 11% on a sterling principal of £10,000,000 & receive 8% on a US$ principal of $15,000,000 every year for 5 years. GBP/USD fixed at 1.5000 for life of swap.

24

Exchange of Principal

• In an interest rate swap, the principal is not exchanged

• In a currency swap the principal is exchanged at the beginning &the end of the swap

25

CCY Swap Cash Flows

Years

Dollars Pounds$

------millions------

0 –15.00 +10.001 +1.20 –1.102 +1.20 –1.10

3 +1.20 –1.104 +1.20 –1.10

5 +16.20 -11.10

£

26

Swaps & Forwards

• A swap can be regarded as a package of forward contracts

• The “fixed for fixed” currency swap in our example consists of a spot/cash transaction & 5 forward contracts

27

Typical Uses of a Currency Swap

• Conversion from a liability in one currency to a liability in another currency

• Conversion from an investment in one currency to an investment in another currency

28

Swaps & Forwards

• The value of the swap is the sum of the values of the forward contracts underlying the swap

• Swaps are normally “at the money” initially– This means that it “costs

NOTHING” to enter into a swap

The Theory and Practice of FX Risk Management

Session 2:

Pricing Theory

30

Option Pricing

• Option Premium determined by:– Current Spot Price– Exercise (Strike) Price– Term to maturity– Short-term (risk free) interest rate– Put or Call– Volatility of underlying security– American or European Option

31

B-S Model – General Form

• Estimates “fair value” of an Option

– C = S.N(d1) - Xe-rt.N(d2)– P = -S.(1-N(d1)) + Xe-rt.(1- N(d2))

• And:ln(S/X) + (r + 2/2) * t

d1 = * t

– d2 = d1 - * t

32

B-S Inputs

• Where:– C=Call premium, P=Put premium– S = Spot Price, X = Exercise Price– r = Risk Free rate– t=time to maturity– N() = Cumulative Normal Distribution values

for d1 & d2

= volatility

33

B-S Assumptions

• Returns on Underlying asset are lognormally distributed

• Risk-Free Rate is constant through life of option

• Volatility is constant through life of option• European Option• Value at expiry is intrinsic value only• Value of option cannot be negative

34

Black-Scholes Model

• Note re Short Term Rates:

Short term interest rates are not quoted in markets as continuously compounding rates but rather as discretely compounded rates

• This introduces a Pricing Bias

35

Black Scholes ModelConversion to Continuously Compounding:

Number of Compound Periods p.a.

Continuosly Compounding Rate

Discretely Compounding Rate

cc

er

m

m

r

r

r mr

m

r dcm

cc

dc

dc

cc

1

1* ln

36

Rate ConversionExample

r m

r

r

r m

r

r

dc

cc

cc

dc

cc

cc

:

. , /

* ln.

. .

. , / .

* ln.

. .

0 08 12 3 4

4 10 08

4

0 079211 7 92%

0 08 365 90 4 055556

365

901

0 0836590

0 079221 7 92%

months

days

37

Black Scholes Model• Volatility Measurement:• Volatility must be Annualised• Variance is proportional to the time over which the

price change takes place Period Adjustment Annualised Vol

• 1 month *12 * 12 1 week *52 * 52 1 day *260 * 260

38

Black Scholes Model

• Volatility Measurement:

• Standard deviation of returns

1

1 1

2

( )( )

nr ri

i

n

39

Black Scholes Formula

• Historical vs Implied Volatility

t = 0 t = Xt = -n

Historical Implied

40

Sensitivity to Inputs• Value of a Call Option

- increases as share price increases- decreases as strike price increases - increases with time to maturity- increases as variance increases- increases as interest rates increase

• Value of a call is not dependent on personal preferences or expected asset returns

41

Sensitivity to Inputs• Value of a Put Option

- decreases as share price increases- increases as strike price increases - increases with time to maturity- increases as variance increases - decreases as interest rates increase

• Value of a put is not dependent on personal preferences or expected share returns

42

Biases in the Black-Scholes Model

• Bias in Moneyness– mispricing of deep in and out of the money

options relative to at the money options

• Time to Maturity Bias– mispricing of near to maturity options

• Volatility Bias– mispricing of high and low volatility options

43

Biases in the Black Scholes Model• B-S works best for:

– at the money– medium to long term maturity assets, with– mid quintile volatility

• B-S Underprices:– in the money calls– options on low variance assets– near to maturity options

• B-S Overprices:– out of the money calls– options on high variance assets

44

Biases in the Black Scholes Model• variance of returns is usually not constant

(non-stationary)• uses European option assumption to price

American options• Biases from the model inputs:

- volatility measurement- effective days to

maturity - appropriate risk free rate

45

Effective Days to Maturity

• 365 vs 360 vs 260 vs 250 day year– Implications for specification of time to

expiration and for risk free rate

46

Currency Options

• Factors effecting CCY option prices:- time to maturity

- volatility- spot

price -relative interest rates

47

Valuing Cash Currency Options

• A foreign currency is an asset that provides a continuous “dividend yield” equal to rf

• We can use the formula for an option on a stock paying a continuous dividend yield :

Set S = current exchange rate

Set q = rƒoreign

48

The Foreign Interest Rate in CCY Option Valuation

• We denote the foreign interest rate by rf

• When a U.S. company buys one unit of the foreign currency it has an investment of S0 dollars

• The return from investing at the foreign rate is rf S0 dollars

• This shows that the foreign currency provides a “dividend yield” at rate rf

49

Currency Options

Garman & Kohlhagen (1983) and Grabbe (1983)

C Se N d Ke N d

P Ke N d Se N d

d

SK

r r T

T

d d T

r T r T

r T r T

domestic foreign

foreign domestic

domestic foreign

( ) ( )

( ) ( )

ln . *

1 2

2 1

1

2

2 1

0 50

50

Alternative (Black, 1976)

F S e r r Tf

0 0 ( )Using

c e F N d XN d

p e XN d F N d

dF X T

T

d d T

rT

rT

[ ( ) ( )]

[ ( ) ( )]

ln( / ) /

0 1 2

2 0 1

10

2

2 1

2

We can use Black’s approach to value options on CCY Futures, where F = current Futures price

51 10

Pricing FX Forwards

• Interest Rate Parity– The relationship between spot and forward

prices of a currency. Same as cost of carry model in other forward and futures markets.

– If parity holds, one cannot convert a currency to another currency, sell a forward, earn the foreign risk-free rate and convert back (without risk), earning a rate higher than the domestic rate.

52

Pricing FX Forwards• 2 parties with funds in different currencies

they plan to exchange in FX market in 3mths. Each could do one of the following:– Exchange at spot now and invest in a risk-free

security (such as a treasury note), yielding a future amount in the desired currency

– Invest in a three month risk-free security in their domestic money market and exchange the proceeds in three months’ time.

53

Pricing FX Forwards

• Either way, interest rate parity and the law of one price dictate that they will acquire the future amount of the other currency:– FSGD = S(1 + rSGDt)

– FUSD = S(1 + rUSDt)

• The equivalent PV amounts reflect the spot rate, eg:– SGD10,000,000 x 0.6230 = USD6,230,000

54

Pricing FX Forwards

• The forward outright rate in three month’s time will be the ratio of the future value amounts in each country:

f SGD/USD = FUSD/FSGD

We derive a formula by adjusting the spot rate

for the ratio of the of the terms CCY FV to the commodity currency FV

55

Example:say the current rUSD is 3.9%current rSGD is 5.0%, t is 90 days and the spot rate is 0.6230:

6214.0

36590*0500.01

36090*0390.01

6230.0/

USDAUDf

Note that this gives a forward rate at a discount to spot - this is due to the interest rate in the commodity currency being higher than the interest rate in the terms currency

56

2-way Forward Price

tr

trSf

borrowcomm

lendtermsofferoffer

/

/

1

1

tr

trSf

lendcomm

borrowtermsbidbid

/

/

1

1

57

Example:

Bid Offer

Spot AUD/NZD 1.2800 1.2850

90 day money:

AUD 14.50 14.25

NZD 17.50 17.25

58

Example

f S

r t

r toffer offer

terms lend

comm borrow

1

1

/

/

=1.2950

36590*1425.01

36590*1750.01

2850.1

f S

r t

r tbid bid

terms borrow

comm lend

1

1

/

/

36590*1425.01

36590*1750.01

2800.1 =1.2884

2-way forward rate will be 1.2884-1.2950, showing a spread of 66 points compared to the spot spread of 50 points.

fin.

The Theory and Practice of FX Risk Management

Session 3:

Hedging with Currency Options

60

Building a Forward Curve

• Most straight-forward approach is to calculate covered-interest parity forward price over a range of maturities, using spot FX rate and current yield curve

• Forward curve will need to be re-estimated as yield curve changes

• Odd maturities can be interpolated between known CIP forward prices.

61

Option Delta• Measures the sensitivity of the option

premium to changes in the asset price

• CALL OPTIONS- always positive- direct relationship between call

and asset price- ranges between 0 to 1- at the money = 0.50- proxied by N(d1) in BS model

62

Delta Characteristics

• PUT Options- always negative

- indirect relationship between put and asset price

- ranges between 0 to -1 - ATM = -0.50 - Value = (N(d1)-1) in BS model

63

DeltaThe rate of change of option premium for a unit change in asset price:

Delta e N dr Tdomestic

( )1

64

Dynamic Delta Hedging Example

• A trader buys an ATM EUR Call/USD Put over €10mio, with 1mth to expiry, and is 0.5

• We can interpret this as implying a 50% chance that the buyer will exercise the option, so the option writer needs to buy in €5mio to cover

• A week later, the spot € moves the and is now 0.6 the writer needs to buy in another €1mio

• The next day, spot moves again and is 0.55, the option writer sells €500k, and so on

• This is Delta hedging, and when hedged in this way, the position is said to be delta hedged, or delta neutral (the “expected” payoff is zero), and insulated from small changes on the value of the position.

• Delta hedging is costly and difficult to do

65

Gamma

tTS

dNe tTr

)( 1

66

GAMMA Characteristics

• GAMMA is equal for put and calls for same time and strike

• GAMMA most sensitive for at te money options

• GAMMA can be positive or negative- positive GAMMA (gain

value) - negative GAMMA (lose value)

67

Gamma Hedging Principles• The more frequently an option’s hedge needs to be adjusted, the

higher will be the .

• Options with small are easy to hedge, b/c will not change much with spot rate

• Options with high , such as our short-dated ATM EUR can be difficult and costly to hedge: a very small swing in the spot, say 0.05%, might swing the option ITM, in which case the writer needs to have €10mio on hand for the holder not if but when he exercises, conversely, the spot rate moves back 0.07%, the writer now needs zero cover

68

Vega

)( 1)( dNetTSVega tTr

Change in Option Premium for a 1% change in volatility