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
Transcript
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
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
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
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
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
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