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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 hedge
Spot Market Forw ard Market
June 29
Spot GBP/USD = 1.3620
Spot Value of funds = $13,620,000
Fwd GBP/USD = 1.3570
Sell GBP forward
Forward 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 hedge
Cash Market Futures Market
Now
Expected receipt of USD 1mio
Do nothing.
Buy 14 March AUD/USD contracts at
market price of 0.6928
Cost Equals USD 969,920
Brokerage $700 USD ($1000 AUD)
3 months’ time
Receive USD1,000,000, convert at
AUD/USD 0.80
yielding AUD1,250,000
Sell 14 March futures @0.8005 =
USD1,127,000 less purchase cost of
USD969,920 = 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 Options
US$ 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 +10.00
1 +1.20 –1.10
2 +1.20 –1.10
3 +1.20 –1.10
4 +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 + σ2
/2) * t
d1= σ * √t
– d2 = d1 - σ * √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
σ = 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 Model
Conversion to Continuously Compounding:
Number of Compound Periods p.a.
Continuosly Compounding Rate
Discretely Compounding Rate
cc
e
r
m
m
r
r
r m
r
m
r dc
m
cc
dc
dc
cc
= +






=
=
=
= +






1
1* ln
36
Rate Conversion
Example
r m
r
r
r m
r
r
dc
cc
cc
dc
cc
cc
:
. , /
* ln
.
. .
. , / .
* ln
.
. .
= = =
= +






= =
= = =
= +


















= =
0 08 12 3 4
4 1
0 08
4
0 079211 7 92%
0 08 365 90 4 055556
365
90
1
0 08
365
90
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 σ * 12
1 week *52 σ * 52
1 day *260 σ * 260
38
Black Scholes Model
• Volatility Measurement:
• Standard deviation of returns
σ =
−
−∑






=
1
1 1
2
( )
( )
n
r 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 rfS0 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
S
K
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 σ
σ
σ
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
d
F X T
T
d d T
rT
rT
= −
= − − −
=
+
= −
−
−
[ ( ) ( )]
[ ( ) ( )]
ln( / ) /
0 1 2
2 0 1
1
0
2
2 1
2σ
σ
σ
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
365
90*0500.01
360
90*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
tr
Sf
borrowcomm
lendterms
offeroffer
/
/
1
1






+
+
=
tr
tr
Sf
lendcomm
borrowterms
bidbid
/
/
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 t
offer offer
terms lend
comm borrow
=
+
+








1
1
/
/
=1.2950
( )
( )







+
+
365
90*1425.01
365
90*1750.01
2850.1
( )
( )
f S
r t
r t
bid bid
terms borrow
comm lend
=
+
+








1
1
/
/
( )
( )







+
+
365
90*1425.01
365
90*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
Delta
The 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
−
=
−−
σ
γ
)( 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 γ.
• Options with small γ are easy to hedge, b/c ∆ will not change
much with spot rate
• Options with high γ, 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

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Currency Derivatives: A Practical Introduction

  • 1. The Theory and Practice of FX Risk Management Session 1: Currency Derivatives: An Introduction Presenter: Stuart Thomas School of Economics and Finance, RMIT
  • 2. 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. 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. 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. 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. 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. 7 FX forwards - short hedge Spot Market Forw ard Market June 29 Spot GBP/USD = 1.3620 Spot Value of funds = $13,620,000 Fwd GBP/USD = 1.3570 Sell GBP forward Forward 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. 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. 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. 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. 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. 12 CCY futures hedge Cash Market Futures Market Now Expected receipt of USD 1mio Do nothing. Buy 14 March AUD/USD contracts at market price of 0.6928 Cost Equals USD 969,920 Brokerage $700 USD ($1000 AUD) 3 months’ time Receive USD1,000,000, convert at AUD/USD 0.80 yielding AUD1,250,000 Sell 14 March futures @0.8005 = USD1,127,000 less purchase cost of USD969,920 = 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. 13 CCY Futures issues: • Standardisation – $1,428,571 AUD exposure, 100k contract • Basis Risk • Commodity Basis Risk – eg NZD? • Margin Calls
  • 14. 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. 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. 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. 17 Currency Options • Option premiums are usually expressed as either: - a fixed number of exchange points or - a percentage of the strike price
  • 18. 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. 19 Currency Options US$ 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, . $369, . 100 780 75 000 000 074 115 67 780 75 069 15 780 75 069 15 = = = = US US perA US A US A perUS A Note US A
  • 20. 20 Options on CCY Futures • Call Option on Futures (at exercise): – Buyer takes long position in nearest futures contract on Commodity CCY – Seller takes short position in nearest futures contract on Commodity CCY • Put Option on Futures (at exercise): – Buyer takes short position in nearest futures contract on Commodity CCY – Seller Takes long position in nearest futures contract on Commodity CCY
  • 21. 21 FX Option Products • Cap • Floor • Collar – combination of a cap and a floor – single maturity more common in FX – aka “range forward” • Tunnel – in FX - a rolling series of collars
  • 22. 22 Currency Swaps • Definition – an agreement between two parties in which one party will make a series of payments in one currency and other will make a series of payments in another currency. – aka: • Cross-currency swaps • Cross-currency interest rate swaps
  • 23. 23 Currency Swap Example An agreement to pay 11% on a sterling principal of £10,000,000 & receive 8% on a US$ principal of $15,000,000 every year for 5 years. GBP/USD fixed at 1.5000 for life of swap.
  • 24. 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. 25 CCY Swap Cash Flows Years Dollars Pounds $ ------millions------ 0 –15.00 +10.00 1 +1.20 –1.10 2 +1.20 –1.10 3 +1.20 –1.10 4 +1.20 –1.10 5 +16.20 -11.10 £
  • 26. 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. 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. 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
  • 29. The Theory and Practice of FX Risk Management Session 2: Pricing Theory
  • 30. 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. 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 + σ2 /2) * t d1= σ * √t – d2 = d1 - σ * √t
  • 32. 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 σ = volatility
  • 33. 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. 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. 35 Black Scholes Model Conversion to Continuously Compounding: Number of Compound Periods p.a. Continuosly Compounding Rate Discretely Compounding Rate cc e r m m r r r m r m r dc m cc dc dc cc = +       = = = = +       1 1* ln
  • 36. 36 Rate Conversion Example r m r r r m r r dc cc cc dc cc cc : . , / * ln . . . . , / . * ln . . . = = = = +       = = = = = = +                   = = 0 08 12 3 4 4 1 0 08 4 0 079211 7 92% 0 08 365 90 4 055556 365 90 1 0 08 365 90 0 079221 7 92% months days
  • 37. 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 σ * 12 1 week *52 σ * 52 1 day *260 σ * 260
  • 38. 38 Black Scholes Model • Volatility Measurement: • Standard deviation of returns σ = − −∑       = 1 1 1 2 ( ) ( ) n r ri i n
  • 39. 39 Black Scholes Formula • Historical vs Implied Volatility t = 0 t = Xt = -n Historical Implied
  • 40. 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. 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. 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. 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. 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. 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. 46 Currency Options • Factors effecting CCY option prices: - time to maturity - volatility - spot price - relative interest rates
  • 47. 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. 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 rfS0 dollars • This shows that the foreign currency provides a “dividend yield” at rate rf
  • 49. 49 Currency Options ( ) Garman & Kohlhagen (1983) and Grabbe (1983) C Se N d Ke N d P Ke N d Se N d d S K 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 σ σ σ
  • 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 d F X T T d d T rT rT = − = − − − = + = − − − [ ( ) ( )] [ ( ) ( )] ln( / ) / 0 1 2 2 0 1 1 0 2 2 1 2σ σ σ We can use Black’s approach to value options on CCY Futures, where F = current Futures price
  • 51. 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. 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. 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. 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. 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: 6214.0 365 90*0500.01 360 90*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
  • 57. 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. 58 Example ( ) ( ) f S r t r t offer offer terms lend comm borrow = + +         1 1 / / =1.2950 ( ) ( )        + + 365 90*1425.01 365 90*1750.01 2850.1 ( ) ( ) f S r t r t bid bid terms borrow comm lend = + +         1 1 / / ( ) ( )        + + 365 90*1425.01 365 90*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.
  • 59. The Theory and Practice of FX Risk Management Session 3: Hedging with Currency Options
  • 60. 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. 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. 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. 63 Delta The rate of change of option premium for a unit change in asset price: Delta e N dr Tdomestic = = − ∆ ( )1
  • 64. 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
  • 66. 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. 67 Gamma Hedging Principles • The more frequently an option’s hedge needs to be adjusted, the higher will be the γ. • Options with small γ are easy to hedge, b/c ∆ will not change much with spot rate • Options with high γ, 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. 68 Vega )( 1 )( dNetTSVega tTr −− −= Change in Option Premium for a 1% change in volatility

Editor's Notes

  1. 15
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  5. 10 > Variance of the rates of return are not constant - tendency for rates of return to Increase as share price falls - decrease price leads to decreased capitalisation leads to increase in leverage (debt/equity) & increase in shareholder risk > Using European to price American Options - strongest for in the money options - puts as they move in the money - calls for dividend paying shares > Inputs - constant vs stochastic volatility - volatility smile - term structure of volatility - trading vs non-trading periods - need for a risk premium due to lack of continuous trading